LAPACK
3.4.2
LAPACK: Linear Algebra PACKage

Functions/Subroutines  
subroutine  dbdt01 (M, N, KD, A, LDA, Q, LDQ, D, E, PT, LDPT, WORK, RESID) 
DBDT01  
subroutine  dbdt02 (M, N, B, LDB, C, LDC, U, LDU, WORK, RESID) 
DBDT02  
subroutine  dbdt03 (UPLO, N, KD, D, E, U, LDU, S, VT, LDVT, WORK, RESID) 
DBDT03  
subroutine  dchkbb (NSIZES, MVAL, NVAL, NWDTHS, KK, NTYPES, DOTYPE, NRHS, ISEED, THRESH, NOUNIT, A, LDA, AB, LDAB, BD, BE, Q, LDQ, P, LDP, C, LDC, CC, WORK, LWORK, RESULT, INFO) 
DCHKBB  
subroutine  dchkbd (NSIZES, MVAL, NVAL, NTYPES, DOTYPE, NRHS, ISEED, THRESH, A, LDA, BD, BE, S1, S2, X, LDX, Y, Z, Q, LDQ, PT, LDPT, U, VT, WORK, LWORK, IWORK, NOUT, INFO) 
DCHKBD  
subroutine  dchkbk (NIN, NOUT) 
DCHKBK  
subroutine  dchkbl (NIN, NOUT) 
DCHKBL  
subroutine  dchkec (THRESH, TSTERR, NIN, NOUT) 
DCHKEC  
program  dchkee 
DCHKEE  
subroutine  dchkgg (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, TSTDIF, THRSHN, NOUNIT, A, LDA, B, H, T, S1, S2, P1, P2, U, LDU, V, Q, Z, ALPHR1, ALPHI1, BETA1, ALPHR3, ALPHI3, BETA3, EVECTL, EVECTR, WORK, LWORK, LLWORK, RESULT, INFO) 
DCHKGG  
subroutine  dchkgk (NIN, NOUT) 
DCHKGK  
subroutine  dchkgl (NIN, NOUT) 
DCHKGL  
subroutine  dchkhs (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, H, T1, T2, U, LDU, Z, UZ, WR1, WI1, WR3, WI3, EVECTL, EVECTR, EVECTY, EVECTX, UU, TAU, WORK, NWORK, IWORK, SELECT, RESULT, INFO) 
DCHKHS  
subroutine  dchksb (NSIZES, NN, NWDTHS, KK, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, SD, SE, U, LDU, WORK, LWORK, RESULT, INFO) 
DCHKSB  
subroutine  dchkst (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, AP, SD, SE, D1, D2, D3, D4, D5, WA1, WA2, WA3, WR, U, LDU, V, VP, TAU, Z, WORK, LWORK, IWORK, LIWORK, RESULT, INFO) 
DCHKST  
subroutine  dckcsd (NM, MVAL, PVAL, QVAL, NMATS, ISEED, THRESH, MMAX, X, XF, U1, U2, V1T, V2T, THETA, IWORK, WORK, RWORK, NIN, NOUT, INFO) 
DCKCSD  
subroutine  dckglm (NN, MVAL, PVAL, NVAL, NMATS, ISEED, THRESH, NMAX, A, AF, B, BF, X, WORK, RWORK, NIN, NOUT, INFO) 
DCKGLM  
subroutine  dckgqr (NM, MVAL, NP, PVAL, NN, NVAL, NMATS, ISEED, THRESH, NMAX, A, AF, AQ, AR, TAUA, B, BF, BZ, BT, BWK, TAUB, WORK, RWORK, NIN, NOUT, INFO) 
DCKGQR  
subroutine  dckgsv (NM, MVAL, PVAL, NVAL, NMATS, ISEED, THRESH, NMAX, A, AF, B, BF, U, V, Q, ALPHA, BETA, R, IWORK, WORK, RWORK, NIN, NOUT, INFO) 
DCKGSV  
subroutine  dcklse (NN, MVAL, PVAL, NVAL, NMATS, ISEED, THRESH, NMAX, A, AF, B, BF, X, WORK, RWORK, NIN, NOUT, INFO) 
DCKLSE  
subroutine  dcsdts (M, P, Q, X, XF, LDX, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, THETA, IWORK, WORK, LWORK, RWORK, RESULT) 
DCSDTS  
subroutine  ddrges (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, B, S, T, Q, LDQ, Z, ALPHAR, ALPHAI, BETA, WORK, LWORK, RESULT, BWORK, INFO) 
DDRGES  
subroutine  ddrgev (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, B, S, T, Q, LDQ, Z, QE, LDQE, ALPHAR, ALPHAI, BETA, ALPHR1, ALPHI1, BETA1, WORK, LWORK, RESULT, INFO) 
DDRGEV  
subroutine  ddrgsx (NSIZE, NCMAX, THRESH, NIN, NOUT, A, LDA, B, AI, BI, Z, Q, ALPHAR, ALPHAI, BETA, C, LDC, S, WORK, LWORK, IWORK, LIWORK, BWORK, INFO) 
DDRGSX  
subroutine  ddrgvx (NSIZE, THRESH, NIN, NOUT, A, LDA, B, AI, BI, ALPHAR, ALPHAI, BETA, VL, VR, ILO, IHI, LSCALE, RSCALE, S, DTRU, DIF, DIFTRU, WORK, LWORK, IWORK, LIWORK, RESULT, BWORK, INFO) 
DDRGVX  
subroutine  ddrvbd (NSIZES, MM, NN, NTYPES, DOTYPE, ISEED, THRESH, A, LDA, U, LDU, VT, LDVT, ASAV, USAV, VTSAV, S, SSAV, E, WORK, LWORK, IWORK, NOUT, INFO) 
DDRVBD  
subroutine  ddrves (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, H, HT, WR, WI, WRT, WIT, VS, LDVS, RESULT, WORK, NWORK, IWORK, BWORK, INFO) 
DDRVES  
subroutine  ddrvev (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, H, WR, WI, WR1, WI1, VL, LDVL, VR, LDVR, LRE, LDLRE, RESULT, WORK, NWORK, IWORK, INFO) 
DDRVEV  
subroutine  ddrvgg (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, THRSHN, NOUNIT, A, LDA, B, S, T, S2, T2, Q, LDQ, Z, ALPHR1, ALPHI1, BETA1, ALPHR2, ALPHI2, BETA2, VL, VR, WORK, LWORK, RESULT, INFO) 
DDRVGG  
subroutine  ddrvsg (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, B, LDB, D, Z, LDZ, AB, BB, AP, BP, WORK, NWORK, IWORK, LIWORK, RESULT, INFO) 
DDRVSG  
subroutine  ddrvst (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, D1, D2, D3, D4, EVEIGS, WA1, WA2, WA3, U, LDU, V, TAU, Z, WORK, LWORK, IWORK, LIWORK, RESULT, INFO) 
DDRVST  
subroutine  ddrvsx (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NIUNIT, NOUNIT, A, LDA, H, HT, WR, WI, WRT, WIT, WRTMP, WITMP, VS, LDVS, VS1, RESULT, WORK, LWORK, IWORK, BWORK, INFO) 
DDRVSX  
subroutine  ddrvvx (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NIUNIT, NOUNIT, A, LDA, H, WR, WI, WR1, WI1, VL, LDVL, VR, LDVR, LRE, LDLRE, RCONDV, RCNDV1, RCDVIN, RCONDE, RCNDE1, RCDEIN, SCALE, SCALE1, RESULT, WORK, NWORK, IWORK, INFO) 
DDRVVX  
subroutine  derrbd (PATH, NUNIT) 
DERRBD  
subroutine  derrec (PATH, NUNIT) 
DERREC  
subroutine  derred (PATH, NUNIT) 
DERRED  
subroutine  derrgg (PATH, NUNIT) 
DERRGG  
subroutine  derrhs (PATH, NUNIT) 
DERRHS  
subroutine  derrst (PATH, NUNIT) 
DERRST  
subroutine  dget02 (TRANS, M, N, NRHS, A, LDA, X, LDX, B, LDB, RWORK, RESID) 
DGET02  
subroutine  dget10 (M, N, A, LDA, B, LDB, WORK, RESULT) 
DGET10  
subroutine  dget22 (TRANSA, TRANSE, TRANSW, N, A, LDA, E, LDE, WR, WI, WORK, RESULT) 
DGET22  
subroutine  dget23 (COMP, BALANC, JTYPE, THRESH, ISEED, NOUNIT, N, A, LDA, H, WR, WI, WR1, WI1, VL, LDVL, VR, LDVR, LRE, LDLRE, RCONDV, RCNDV1, RCDVIN, RCONDE, RCNDE1, RCDEIN, SCALE, SCALE1, RESULT, WORK, LWORK, IWORK, INFO) 
DGET23  
subroutine  dget24 (COMP, JTYPE, THRESH, ISEED, NOUNIT, N, A, LDA, H, HT, WR, WI, WRT, WIT, WRTMP, WITMP, VS, LDVS, VS1, RCDEIN, RCDVIN, NSLCT, ISLCT, RESULT, WORK, LWORK, IWORK, BWORK, INFO) 
DGET24  
subroutine  dget31 (RMAX, LMAX, NINFO, KNT) 
DGET31  
subroutine  dget32 (RMAX, LMAX, NINFO, KNT) 
DGET32  
subroutine  dget33 (RMAX, LMAX, NINFO, KNT) 
DGET33  
subroutine  dget34 (RMAX, LMAX, NINFO, KNT) 
DGET34  
subroutine  dget35 (RMAX, LMAX, NINFO, KNT) 
DGET35  
subroutine  dget36 (RMAX, LMAX, NINFO, KNT, NIN) 
DGET36  
subroutine  dget37 (RMAX, LMAX, NINFO, KNT, NIN) 
DGET37  
subroutine  dget38 (RMAX, LMAX, NINFO, KNT, NIN) 
DGET38  
subroutine  dget39 (RMAX, LMAX, NINFO, KNT) 
DGET39  
subroutine  dget51 (ITYPE, N, A, LDA, B, LDB, U, LDU, V, LDV, WORK, RESULT) 
DGET51  
subroutine  dget52 (LEFT, N, A, LDA, B, LDB, E, LDE, ALPHAR, ALPHAI, BETA, WORK, RESULT) 
DGET52  
subroutine  dget53 (A, LDA, B, LDB, SCALE, WR, WI, RESULT, INFO) 
DGET53  
subroutine  dget54 (N, A, LDA, B, LDB, S, LDS, T, LDT, U, LDU, V, LDV, WORK, RESULT) 
DGET54  
subroutine  dglmts (N, M, P, A, AF, LDA, B, BF, LDB, D, DF, X, U, WORK, LWORK, RWORK, RESULT) 
DGLMTS  
subroutine  dgqrts (N, M, P, A, AF, Q, R, LDA, TAUA, B, BF, Z, T, BWK, LDB, TAUB, WORK, LWORK, RWORK, RESULT) 
DGQRTS  
subroutine  dgrqts (M, P, N, A, AF, Q, R, LDA, TAUA, B, BF, Z, T, BWK, LDB, TAUB, WORK, LWORK, RWORK, RESULT) 
DGRQTS  
subroutine  dgsvts (M, P, N, A, AF, LDA, B, BF, LDB, U, LDU, V, LDV, Q, LDQ, ALPHA, BETA, R, LDR, IWORK, WORK, LWORK, RWORK, RESULT) 
DGSVTS  
subroutine  dhst01 (N, ILO, IHI, A, LDA, H, LDH, Q, LDQ, WORK, LWORK, RESULT) 
DHST01  
subroutine  dlafts (TYPE, M, N, IMAT, NTESTS, RESULT, ISEED, THRESH, IOUNIT, IE) 
DLAFTS  
subroutine  dlahd2 (IOUNIT, PATH) 
DLAHD2  
subroutine  dlarfy (UPLO, N, V, INCV, TAU, C, LDC, WORK) 
DLARFY  
subroutine  dlarhs (PATH, XTYPE, UPLO, TRANS, M, N, KL, KU, NRHS, A, LDA, X, LDX, B, LDB, ISEED, INFO) 
DLARHS  
subroutine  dlasum (TYPE, IOUNIT, IE, NRUN) 
DLASUM  
subroutine  dlatb9 (PATH, IMAT, M, P, N, TYPE, KLA, KUA, KLB, KUB, ANORM, BNORM, MODEA, MODEB, CNDNMA, CNDNMB, DISTA, DISTB) 
DLATB9  
subroutine  dlatm4 (ITYPE, N, NZ1, NZ2, ISIGN, AMAGN, RCOND, TRIANG, IDIST, ISEED, A, LDA) 
DLATM4  
LOGICAL function  dlctes (ZR, ZI, D) 
DLCTES  
LOGICAL function  dlctsx (AR, AI, BETA) 
DLCTSX  
subroutine  dlsets (M, P, N, A, AF, LDA, B, BF, LDB, C, CF, D, DF, X, WORK, LWORK, RWORK, RESULT) 
DLSETS  
subroutine  dort01 (ROWCOL, M, N, U, LDU, WORK, LWORK, RESID) 
DORT01  
subroutine  dort03 (RC, MU, MV, N, K, U, LDU, V, LDV, WORK, LWORK, RESULT, INFO) 
DORT03  
subroutine  dsbt21 (UPLO, N, KA, KS, A, LDA, D, E, U, LDU, WORK, RESULT) 
DSBT21  
subroutine  dsgt01 (ITYPE, UPLO, N, M, A, LDA, B, LDB, Z, LDZ, D, WORK, RESULT) 
DSGT01  
LOGICAL function  dslect (ZR, ZI) 
DSLECT  
subroutine  dspt21 (ITYPE, UPLO, N, KBAND, AP, D, E, U, LDU, VP, TAU, WORK, RESULT) 
DSPT21  
subroutine  dstech (N, A, B, EIG, TOL, WORK, INFO) 
DSTECH  
subroutine  dstect (N, A, B, SHIFT, NUM) 
DSTECT  
subroutine  dstt21 (N, KBAND, AD, AE, SD, SE, U, LDU, WORK, RESULT) 
DSTT21  
subroutine  dstt22 (N, M, KBAND, AD, AE, SD, SE, U, LDU, WORK, LDWORK, RESULT) 
DSTT22  
subroutine  dsvdch (N, S, E, SVD, TOL, INFO) 
DSVDCH  
subroutine  dsvdct (N, S, E, SHIFT, NUM) 
DSVDCT  
DOUBLE PRECISION function  dsxt1 (IJOB, D1, N1, D2, N2, ABSTOL, ULP, UNFL) 
DSXT1  
subroutine  dsyt21 (ITYPE, UPLO, N, KBAND, A, LDA, D, E, U, LDU, V, LDV, TAU, WORK, RESULT) 
DSYT21  
subroutine  dsyt22 (ITYPE, UPLO, N, M, KBAND, A, LDA, D, E, U, LDU, V, LDV, TAU, WORK, RESULT) 
DSYT22 
This is the group of double LAPACK TESTING EIG routines.
subroutine dbdt01  (  integer  M, 
integer  N,  
integer  KD,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( ldq, * )  Q,  
integer  LDQ,  
double precision, dimension( * )  D,  
double precision, dimension( * )  E,  
double precision, dimension( ldpt, * )  PT,  
integer  LDPT,  
double precision, dimension( * )  WORK,  
double precision  RESID  
) 
DBDT01
DBDT01 reconstructs a general matrix A from its bidiagonal form A = Q * B * P' where Q (m by min(m,n)) and P' (min(m,n) by n) are orthogonal matrices and B is bidiagonal. The test ratio to test the reduction is RESID = norm( A  Q * B * PT ) / ( n * norm(A) * EPS ) where PT = P' and EPS is the machine precision.
[in]  M  M is INTEGER The number of rows of the matrices A and Q. 
[in]  N  N is INTEGER The number of columns of the matrices A and P'. 
[in]  KD  KD is INTEGER If KD = 0, B is diagonal and the array E is not referenced. If KD = 1, the reduction was performed by xGEBRD; B is upper bidiagonal if M >= N, and lower bidiagonal if M < N. If KD = 1, the reduction was performed by xGBBRD; B is always upper bidiagonal. 
[in]  A  A is DOUBLE PRECISION array, dimension (LDA,N) The m by n matrix A. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). 
[in]  Q  Q is DOUBLE PRECISION array, dimension (LDQ,N) The m by min(m,n) orthogonal matrix Q in the reduction A = Q * B * P'. 
[in]  LDQ  LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,M). 
[in]  D  D is DOUBLE PRECISION array, dimension (min(M,N)) The diagonal elements of the bidiagonal matrix B. 
[in]  E  E is DOUBLE PRECISION array, dimension (min(M,N)1) The superdiagonal elements of the bidiagonal matrix B if m >= n, or the subdiagonal elements of B if m < n. 
[in]  PT  PT is DOUBLE PRECISION array, dimension (LDPT,N) The min(m,n) by n orthogonal matrix P' in the reduction A = Q * B * P'. 
[in]  LDPT  LDPT is INTEGER The leading dimension of the array PT. LDPT >= max(1,min(M,N)). 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (M+N) 
[out]  RESID  RESID is DOUBLE PRECISION The test ratio: norm(A  Q * B * P') / ( n * norm(A) * EPS ) 
Definition at line 140 of file dbdt01.f.
subroutine dbdt02  (  integer  M, 
integer  N,  
double precision, dimension( ldb, * )  B,  
integer  LDB,  
double precision, dimension( ldc, * )  C,  
integer  LDC,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( * )  WORK,  
double precision  RESID  
) 
DBDT02
DBDT02 tests the change of basis C = U' * B by computing the residual RESID = norm( B  U * C ) / ( max(m,n) * norm(B) * EPS ), where B and C are M by N matrices, U is an M by M orthogonal matrix, and EPS is the machine precision.
[in]  M  M is INTEGER The number of rows of the matrices B and C and the order of the matrix Q. 
[in]  N  N is INTEGER The number of columns of the matrices B and C. 
[in]  B  B is DOUBLE PRECISION array, dimension (LDB,N) The m by n matrix B. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,M). 
[in]  C  C is DOUBLE PRECISION array, dimension (LDC,N) The m by n matrix C, assumed to contain U' * B. 
[in]  LDC  LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M). 
[in]  U  U is DOUBLE PRECISION array, dimension (LDU,M) The m by m orthogonal matrix U. 
[in]  LDU  LDU is INTEGER The leading dimension of the array U. LDU >= max(1,M). 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (M) 
[out]  RESID  RESID is DOUBLE PRECISION RESID = norm( B  U * C ) / ( max(m,n) * norm(B) * EPS ), 
Definition at line 112 of file dbdt02.f.
subroutine dbdt03  (  character  UPLO, 
integer  N,  
integer  KD,  
double precision, dimension( * )  D,  
double precision, dimension( * )  E,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( * )  S,  
double precision, dimension( ldvt, * )  VT,  
integer  LDVT,  
double precision, dimension( * )  WORK,  
double precision  RESID  
) 
DBDT03
DBDT03 reconstructs a bidiagonal matrix B from its SVD: S = U' * B * V where U and V are orthogonal matrices and S is diagonal. The test ratio to test the singular value decomposition is RESID = norm( B  U * S * VT ) / ( n * norm(B) * EPS ) where VT = V' and EPS is the machine precision.
[in]  UPLO  UPLO is CHARACTER*1 Specifies whether the matrix B is upper or lower bidiagonal. = 'U': Upper bidiagonal = 'L': Lower bidiagonal 
[in]  N  N is INTEGER The order of the matrix B. 
[in]  KD  KD is INTEGER The bandwidth of the bidiagonal matrix B. If KD = 1, the matrix B is bidiagonal, and if KD = 0, B is diagonal and E is not referenced. If KD is greater than 1, it is assumed to be 1, and if KD is less than 0, it is assumed to be 0. 
[in]  D  D is DOUBLE PRECISION array, dimension (N) The n diagonal elements of the bidiagonal matrix B. 
[in]  E  E is DOUBLE PRECISION array, dimension (N1) The (n1) superdiagonal elements of the bidiagonal matrix B if UPLO = 'U', or the (n1) subdiagonal elements of B if UPLO = 'L'. 
[in]  U  U is DOUBLE PRECISION array, dimension (LDU,N) The n by n orthogonal matrix U in the reduction B = U'*A*P. 
[in]  LDU  LDU is INTEGER The leading dimension of the array U. LDU >= max(1,N) 
[in]  S  S is DOUBLE PRECISION array, dimension (N) The singular values from the SVD of B, sorted in decreasing order. 
[in]  VT  VT is DOUBLE PRECISION array, dimension (LDVT,N) The n by n orthogonal matrix V' in the reduction B = U * S * V'. 
[in]  LDVT  LDVT is INTEGER The leading dimension of the array VT. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (2*N) 
[out]  RESID  RESID is DOUBLE PRECISION The test ratio: norm(B  U * S * V') / ( n * norm(A) * EPS ) 
Definition at line 135 of file dbdt03.f.
subroutine dchkbb  (  integer  NSIZES, 
integer, dimension( * )  MVAL,  
integer, dimension( * )  NVAL,  
integer  NWDTHS,  
integer, dimension( * )  KK,  
integer  NTYPES,  
logical, dimension( * )  DOTYPE,  
integer  NRHS,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
integer  NOUNIT,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( ldab, * )  AB,  
integer  LDAB,  
double precision, dimension( * )  BD,  
double precision, dimension( * )  BE,  
double precision, dimension( ldq, * )  Q,  
integer  LDQ,  
double precision, dimension( ldp, * )  P,  
integer  LDP,  
double precision, dimension( ldc, * )  C,  
integer  LDC,  
double precision, dimension( ldc, * )  CC,  
double precision, dimension( * )  WORK,  
integer  LWORK,  
double precision, dimension( * )  RESULT,  
integer  INFO  
) 
DCHKBB
DCHKBB tests the reduction of a general real rectangular band matrix to bidiagonal form. DGBBRD factors a general band matrix A as Q B P* , where * means transpose, B is upper bidiagonal, and Q and P are orthogonal; DGBBRD can also overwrite a given matrix C with Q* C . For each pair of matrix dimensions (M,N) and each selected matrix type, an M by N matrix A and an M by NRHS matrix C are generated. The problem dimensions are as follows A: M x N Q: M x M P: N x N B: min(M,N) x min(M,N) C: M x NRHS For each generated matrix, 4 tests are performed: (1)  A  Q B PT  / ( A max(M,N) ulp ), PT = P' (2)  I  Q' Q  / ( M ulp ) (3)  I  PT PT'  / ( N ulp ) (4)  Y  Q' C  / ( Y max(M,NRHS) ulp ), where Y = Q' C. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: The possible matrix types are (1) The zero matrix. (2) The identity matrix. (3) A diagonal matrix with evenly spaced entries 1, ..., ULP and random signs. (ULP = (first number larger than 1)  1 ) (4) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random signs. (5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP and random signs. (6) Same as (3), but multiplied by SQRT( overflow threshold ) (7) Same as (3), but multiplied by SQRT( underflow threshold ) (8) A matrix of the form U D V, where U and V are orthogonal and D has evenly spaced entries 1, ..., ULP with random signs on the diagonal. (9) A matrix of the form U D V, where U and V are orthogonal and D has geometrically spaced entries 1, ..., ULP with random signs on the diagonal. (10) A matrix of the form U D V, where U and V are orthogonal and D has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal. (11) Same as (8), but multiplied by SQRT( overflow threshold ) (12) Same as (8), but multiplied by SQRT( underflow threshold ) (13) Rectangular matrix with random entries chosen from (1,1). (14) Same as (13), but multiplied by SQRT( overflow threshold ) (15) Same as (13), but multiplied by SQRT( underflow threshold )
[in]  NSIZES  NSIZES is INTEGER The number of values of M and N contained in the vectors MVAL and NVAL. The matrix sizes are used in pairs (M,N). If NSIZES is zero, DCHKBB does nothing. NSIZES must be at least zero. 
[in]  MVAL  MVAL is INTEGER array, dimension (NSIZES) The values of the matrix row dimension M. 
[in]  NVAL  NVAL is INTEGER array, dimension (NSIZES) The values of the matrix column dimension N. 
[in]  NWDTHS  NWDTHS is INTEGER The number of bandwidths to use. If it is zero, DCHKBB does nothing. It must be at least zero. 
[in]  KK  KK is INTEGER array, dimension (NWDTHS) An array containing the bandwidths to be used for the band matrices. The values must be at least zero. 
[in]  NTYPES  NTYPES is INTEGER The number of elements in DOTYPE. If it is zero, DCHKBB does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. . 
[in]  DOTYPE  DOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored. 
[in]  NRHS  NRHS is INTEGER The number of columns in the "righthand side" matrix C. If NRHS = 0, then the operations on the righthand side will not be tested. NRHS must be at least 0. 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to DCHKBB to continue the same random number sequence. 
[in]  THRESH  THRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero. 
[in]  NOUNIT  NOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) 
[in,out]  A  A is DOUBLE PRECISION array, dimension (LDA, max(NN)) Used to hold the matrix A. 
[in]  LDA  LDA is INTEGER The leading dimension of A. It must be at least 1 and at least max( NN ). 
[out]  AB  AB is DOUBLE PRECISION array, dimension (LDAB, max(NN)) Used to hold A in band storage format. 
[in]  LDAB  LDAB is INTEGER The leading dimension of AB. It must be at least 2 (not 1!) and at least max( KK )+1. 
[out]  BD  BD is DOUBLE PRECISION array, dimension (max(NN)) Used to hold the diagonal of the bidiagonal matrix computed by DGBBRD. 
[out]  BE  BE is DOUBLE PRECISION array, dimension (max(NN)) Used to hold the offdiagonal of the bidiagonal matrix computed by DGBBRD. 
[out]  Q  Q is DOUBLE PRECISION array, dimension (LDQ, max(NN)) Used to hold the orthogonal matrix Q computed by DGBBRD. 
[in]  LDQ  LDQ is INTEGER The leading dimension of Q. It must be at least 1 and at least max( NN ). 
[out]  P  P is DOUBLE PRECISION array, dimension (LDP, max(NN)) Used to hold the orthogonal matrix P computed by DGBBRD. 
[in]  LDP  LDP is INTEGER The leading dimension of P. It must be at least 1 and at least max( NN ). 
[out]  C  C is DOUBLE PRECISION array, dimension (LDC, max(NN)) Used to hold the matrix C updated by DGBBRD. 
[in]  LDC  LDC is INTEGER The leading dimension of U. It must be at least 1 and at least max( NN ). 
[out]  CC  CC is DOUBLE PRECISION array, dimension (LDC, max(NN)) Used to hold a copy of the matrix C. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER The number of entries in WORK. This must be at least max( LDA+1, max(NN)+1 )*max(NN). 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (4) The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. 
[out]  INFO  INFO is INTEGER If 0, then everything ran OK.  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. NTEST The number of tests performed, or which can be performed so far, for the current matrix. NTESTT The total number of tests performed so far. NMAX Largest value in NN. NMATS The number of matrices generated so far. NERRS The number of tests which have exceeded THRESH so far. COND, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. ULP, ULPINV Finest relative precision and its inverse. RTOVFL, RTUNFL Square roots of the previous 2 values. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) ) 
Definition at line 353 of file dchkbb.f.
subroutine dchkbd  (  integer  NSIZES, 
integer, dimension( * )  MVAL,  
integer, dimension( * )  NVAL,  
integer  NTYPES,  
logical, dimension( * )  DOTYPE,  
integer  NRHS,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( * )  BD,  
double precision, dimension( * )  BE,  
double precision, dimension( * )  S1,  
double precision, dimension( * )  S2,  
double precision, dimension( ldx, * )  X,  
integer  LDX,  
double precision, dimension( ldx, * )  Y,  
double precision, dimension( ldx, * )  Z,  
double precision, dimension( ldq, * )  Q,  
integer  LDQ,  
double precision, dimension( ldpt, * )  PT,  
integer  LDPT,  
double precision, dimension( ldpt, * )  U,  
double precision, dimension( ldpt, * )  VT,  
double precision, dimension( * )  WORK,  
integer  LWORK,  
integer, dimension( * )  IWORK,  
integer  NOUT,  
integer  INFO  
) 
DCHKBD
DCHKBD checks the singular value decomposition (SVD) routines. DGEBRD reduces a real general m by n matrix A to upper or lower bidiagonal form B by an orthogonal transformation: Q' * A * P = B (or A = Q * B * P'). The matrix B is upper bidiagonal if m >= n and lower bidiagonal if m < n. DORGBR generates the orthogonal matrices Q and P' from DGEBRD. Note that Q and P are not necessarily square. DBDSQR computes the singular value decomposition of the bidiagonal matrix B as B = U S V'. It is called three times to compute 1) B = U S1 V', where S1 is the diagonal matrix of singular values and the columns of the matrices U and V are the left and right singular vectors, respectively, of B. 2) Same as 1), but the singular values are stored in S2 and the singular vectors are not computed. 3) A = (UQ) S (P'V'), the SVD of the original matrix A. In addition, DBDSQR has an option to apply the left orthogonal matrix U to a matrix X, useful in least squares applications. DBDSDC computes the singular value decomposition of the bidiagonal matrix B as B = U S V' using divideandconquer. It is called twice to compute 1) B = U S1 V', where S1 is the diagonal matrix of singular values and the columns of the matrices U and V are the left and right singular vectors, respectively, of B. 2) Same as 1), but the singular values are stored in S2 and the singular vectors are not computed. For each pair of matrix dimensions (M,N) and each selected matrix type, an M by N matrix A and an M by NRHS matrix X are generated. The problem dimensions are as follows A: M x N Q: M x min(M,N) (but M x M if NRHS > 0) P: min(M,N) x N B: min(M,N) x min(M,N) U, V: min(M,N) x min(M,N) S1, S2 diagonal, order min(M,N) X: M x NRHS For each generated matrix, 14 tests are performed: Test DGEBRD and DORGBR (1)  A  Q B PT  / ( A max(M,N) ulp ), PT = P' (2)  I  Q' Q  / ( M ulp ) (3)  I  PT PT'  / ( N ulp ) Test DBDSQR on bidiagonal matrix B (4)  B  U S1 VT  / ( B min(M,N) ulp ), VT = V' (5)  Y  U Z  / ( Y max(min(M,N),k) ulp ), where Y = Q' X and Z = U' Y. (6)  I  U' U  / ( min(M,N) ulp ) (7)  I  VT VT'  / ( min(M,N) ulp ) (8) S1 contains min(M,N) nonnegative values in decreasing order. (Return 0 if true, 1/ULP if false.) (9)  S1  S2  / ( S1 ulp ), where S2 is computed without computing U and V. (10) 0 if the true singular values of B are within THRESH of those in S1. 2*THRESH if they are not. (Tested using DSVDCH) Test DBDSQR on matrix A (11)  A  (QU) S (VT PT)  / ( A max(M,N) ulp ) (12)  X  (QU) Z  / ( X max(M,k) ulp ) (13)  I  (QU)'(QU)  / ( M ulp ) (14)  I  (VT PT) (PT'VT')  / ( N ulp ) Test DBDSDC on bidiagonal matrix B (15)  B  U S1 VT  / ( B min(M,N) ulp ), VT = V' (16)  I  U' U  / ( min(M,N) ulp ) (17)  I  VT VT'  / ( min(M,N) ulp ) (18) S1 contains min(M,N) nonnegative values in decreasing order. (Return 0 if true, 1/ULP if false.) (19)  S1  S2  / ( S1 ulp ), where S2 is computed without computing U and V. The possible matrix types are (1) The zero matrix. (2) The identity matrix. (3) A diagonal matrix with evenly spaced entries 1, ..., ULP and random signs. (ULP = (first number larger than 1)  1 ) (4) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random signs. (5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP and random signs. (6) Same as (3), but multiplied by SQRT( overflow threshold ) (7) Same as (3), but multiplied by SQRT( underflow threshold ) (8) A matrix of the form U D V, where U and V are orthogonal and D has evenly spaced entries 1, ..., ULP with random signs on the diagonal. (9) A matrix of the form U D V, where U and V are orthogonal and D has geometrically spaced entries 1, ..., ULP with random signs on the diagonal. (10) A matrix of the form U D V, where U and V are orthogonal and D has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal. (11) Same as (8), but multiplied by SQRT( overflow threshold ) (12) Same as (8), but multiplied by SQRT( underflow threshold ) (13) Rectangular matrix with random entries chosen from (1,1). (14) Same as (13), but multiplied by SQRT( overflow threshold ) (15) Same as (13), but multiplied by SQRT( underflow threshold ) Special case: (16) A bidiagonal matrix with random entries chosen from a logarithmic distribution on [ulp^2,ulp^(2)] (I.e., each entry is e^x, where x is chosen uniformly on [ 2 log(ulp), 2 log(ulp) ] .) For *this* type: (a) DGEBRD is not called to reduce it to bidiagonal form. (b) the bidiagonal is min(M,N) x min(M,N); if M<N, the matrix will be lower bidiagonal, otherwise upper. (c) only tests 58 and 14 are performed. A subset of the full set of matrix types may be selected through the logical array DOTYPE.
[in]  NSIZES  NSIZES is INTEGER The number of values of M and N contained in the vectors MVAL and NVAL. The matrix sizes are used in pairs (M,N). 
[in]  MVAL  MVAL is INTEGER array, dimension (NM) The values of the matrix row dimension M. 
[in]  NVAL  NVAL is INTEGER array, dimension (NM) The values of the matrix column dimension N. 
[in]  NTYPES  NTYPES is INTEGER The number of elements in DOTYPE. If it is zero, DCHKBD does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrices are in A and B. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. . 
[in]  DOTYPE  DOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size (m,n), a matrix of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored. 
[in]  NRHS  NRHS is INTEGER The number of columns in the "righthand side" matrices X, Y, and Z, used in testing DBDSQR. If NRHS = 0, then the operations on the righthand side will not be tested. NRHS must be at least 0. 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The values of ISEED are changed on exit, and can be used in the next call to DCHKBD to continue the same random number sequence. 
[in]  THRESH  THRESH is DOUBLE PRECISION The threshold value for the test ratios. A result is included in the output file if RESULT >= THRESH. To have every test ratio printed, use THRESH = 0. Note that the expected value of the test ratios is O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. 
[out]  A  A is DOUBLE PRECISION array, dimension (LDA,NMAX) where NMAX is the maximum value of N in NVAL. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,MMAX), where MMAX is the maximum value of M in MVAL. 
[out]  BD  BD is DOUBLE PRECISION array, dimension (max(min(MVAL(j),NVAL(j)))) 
[out]  BE  BE is DOUBLE PRECISION array, dimension (max(min(MVAL(j),NVAL(j)))) 
[out]  S1  S1 is DOUBLE PRECISION array, dimension (max(min(MVAL(j),NVAL(j)))) 
[out]  S2  S2 is DOUBLE PRECISION array, dimension (max(min(MVAL(j),NVAL(j)))) 
[out]  X  X is DOUBLE PRECISION array, dimension (LDX,NRHS) 
[in]  LDX  LDX is INTEGER The leading dimension of the arrays X, Y, and Z. LDX >= max(1,MMAX) 
[out]  Y  Y is DOUBLE PRECISION array, dimension (LDX,NRHS) 
[out]  Z  Z is DOUBLE PRECISION array, dimension (LDX,NRHS) 
[out]  Q  Q is DOUBLE PRECISION array, dimension (LDQ,MMAX) 
[in]  LDQ  LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,MMAX). 
[out]  PT  PT is DOUBLE PRECISION array, dimension (LDPT,NMAX) 
[in]  LDPT  LDPT is INTEGER The leading dimension of the arrays PT, U, and V. LDPT >= max(1, max(min(MVAL(j),NVAL(j)))). 
[out]  U  U is DOUBLE PRECISION array, dimension (LDPT,max(min(MVAL(j),NVAL(j)))) 
[out]  VT  VT is DOUBLE PRECISION array, dimension (LDPT,max(min(MVAL(j),NVAL(j)))) 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER The number of entries in WORK. This must be at least 3(M+N) and M(M + max(M,N,k) + 1) + N*min(M,N) for all pairs (M,N)=(MM(j),NN(j)) 
[out]  IWORK  IWORK is INTEGER array, dimension at least 8*min(M,N) 
[in]  NOUT  NOUT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) 
[out]  INFO  INFO is INTEGER If 0, then everything ran OK. 1: NSIZES < 0 2: Some MM(j) < 0 3: Some NN(j) < 0 4: NTYPES < 0 6: NRHS < 0 8: THRESH < 0 11: LDA < 1 or LDA < MMAX, where MMAX is max( MM(j) ). 17: LDB < 1 or LDB < MMAX. 21: LDQ < 1 or LDQ < MMAX. 23: LDPT< 1 or LDPT< MNMAX. 27: LWORK too small. If DLATMR, SLATMS, DGEBRD, DORGBR, or DBDSQR, returns an error code, the absolute value of it is returned.  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. NTEST The number of tests performed, or which can be performed so far, for the current matrix. MMAX Largest value in NN. NMAX Largest value in NN. MNMIN min(MM(j), NN(j)) (the dimension of the bidiagonal matrix.) MNMAX The maximum value of MNMIN for j=1,...,NSIZES. NFAIL The number of tests which have exceeded THRESH COND, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. RTOVFL, RTUNFL Square roots of the previous 2 values. ULP, ULPINV Finest relative precision and its inverse. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) ) 
Definition at line 434 of file dchkbd.f.
subroutine dchkbk  (  integer  NIN, 
integer  NOUT  
) 
DCHKBK
DCHKBK tests DGEBAK, a routine for backward transformation of the computed right or left eigenvectors if the orginal matrix was preprocessed by balance subroutine DGEBAL.
[in]  NIN  NIN is INTEGER The logical unit number for input. NIN > 0. 
[in]  NOUT  NOUT is INTEGER The logical unit number for output. NOUT > 0. 
Definition at line 56 of file dchkbk.f.
subroutine dchkbl  (  integer  NIN, 
integer  NOUT  
) 
DCHKBL
DCHKBL tests DGEBAL, a routine for balancing a general real matrix and isolating some of its eigenvalues.
[in]  NIN  NIN is INTEGER The logical unit number for input. NIN > 0. 
[in]  NOUT  NOUT is INTEGER The logical unit number for output. NOUT > 0. 
Definition at line 55 of file dchkbl.f.
subroutine dchkec  (  double precision  THRESH, 
logical  TSTERR,  
integer  NIN,  
integer  NOUT  
) 
DCHKEC
DCHKEC tests eigen condition estimation routines DLALN2, DLASY2, DLANV2, DLAQTR, DLAEXC, DTRSYL, DTREXC, DTRSNA, DTRSEN In all cases, the routine runs through a fixed set of numerical examples, subjects them to various tests, and compares the test results to a threshold THRESH. In addition, DTREXC, DTRSNA and DTRSEN are tested by reading in precomputed examples from a file (on input unit NIN). Output is written to output unit NOUT.
[in]  THRESH  THRESH is DOUBLE PRECISION Threshold for residual tests. A computed test ratio passes the threshold if it is less than THRESH. 
[in]  TSTERR  TSTERR is LOGICAL Flag that indicates whether error exits are to be tested. 
[in]  NIN  NIN is INTEGER The logical unit number for input. 
[in]  NOUT  NOUT is INTEGER The logical unit number for output. 
Definition at line 77 of file dchkec.f.
program dchkee  (  ) 
DCHKEE
DCHKEE tests the DOUBLE PRECISION LAPACK subroutines for the matrix eigenvalue problem. The test paths in this version are NEP (Nonsymmetric Eigenvalue Problem): Test DGEHRD, DORGHR, DHSEQR, DTREVC, DHSEIN, and DORMHR SEP (Symmetric Eigenvalue Problem): Test DSYTRD, DORGTR, DSTEQR, DSTERF, DSTEIN, DSTEDC, and drivers DSYEV(X), DSBEV(X), DSPEV(X), DSTEV(X), DSYEVD, DSBEVD, DSPEVD, DSTEVD SVD (Singular Value Decomposition): Test DGEBRD, DORGBR, DBDSQR, DBDSDC and the drivers DGESVD, DGESDD DEV (Nonsymmetric Eigenvalue/eigenvector Driver): Test DGEEV DES (Nonsymmetric Schur form Driver): Test DGEES DVX (Nonsymmetric Eigenvalue/eigenvector Expert Driver): Test DGEEVX DSX (Nonsymmetric Schur form Expert Driver): Test DGEESX DGG (Generalized Nonsymmetric Eigenvalue Problem): Test DGGHRD, DGGBAL, DGGBAK, DHGEQZ, and DTGEVC and the driver routines DGEGS and DGEGV DGS (Generalized Nonsymmetric Schur form Driver): Test DGGES DGV (Generalized Nonsymmetric Eigenvalue/eigenvector Driver): Test DGGEV DGX (Generalized Nonsymmetric Schur form Expert Driver): Test DGGESX DXV (Generalized Nonsymmetric Eigenvalue/eigenvector Expert Driver): Test DGGEVX DSG (Symmetric Generalized Eigenvalue Problem): Test DSYGST, DSYGV, DSYGVD, DSYGVX, DSPGST, DSPGV, DSPGVD, DSPGVX, DSBGST, DSBGV, DSBGVD, and DSBGVX DSB (Symmetric Band Eigenvalue Problem): Test DSBTRD DBB (Band Singular Value Decomposition): Test DGBBRD DEC (Eigencondition estimation): Test DLALN2, DLASY2, DLAEQU, DLAEXC, DTRSYL, DTREXC, DTRSNA, DTRSEN, and DLAQTR DBL (Balancing a general matrix) Test DGEBAL DBK (Back transformation on a balanced matrix) Test DGEBAK DGL (Balancing a matrix pair) Test DGGBAL DGK (Back transformation on a matrix pair) Test DGGBAK GLM (Generalized Linear Regression Model): Tests DGGGLM GQR (Generalized QR and RQ factorizations): Tests DGGQRF and DGGRQF GSV (Generalized Singular Value Decomposition): Tests DGGSVD, DGGSVP, DTGSJA, DLAGS2, DLAPLL, and DLAPMT CSD (CS decomposition): Tests DORCSD LSE (Constrained Linear Least Squares): Tests DGGLSE Each test path has a different set of inputs, but the data sets for the driver routines xEV, xES, xVX, and xSX can be concatenated in a single input file. The first line of input should contain one of the 3character path names in columns 13. The number of remaining lines depends on what is found on the first line. The number of matrix types used in testing is often controllable from the input file. The number of matrix types for each path, and the test routine that describes them, is as follows: Path name(s) Types Test routine DHS or NEP 21 DCHKHS DST or SEP 21 DCHKST (routines) 18 DDRVST (drivers) DBD or SVD 16 DCHKBD (routines) 5 DDRVBD (drivers) DEV 21 DDRVEV DES 21 DDRVES DVX 21 DDRVVX DSX 21 DDRVSX DGG 26 DCHKGG (routines) 26 DDRVGG (drivers) DGS 26 DDRGES DGX 5 DDRGSX DGV 26 DDRGEV DXV 2 DDRGVX DSG 21 DDRVSG DSB 15 DCHKSB DBB 15 DCHKBB DEC  DCHKEC DBL  DCHKBL DBK  DCHKBK DGL  DCHKGL DGK  DCHKGK GLM 8 DCKGLM GQR 8 DCKGQR GSV 8 DCKGSV CSD 3 DCKCSD LSE 8 DCKLSE  NEP input file: line 2: NN, INTEGER Number of values of N. line 3: NVAL, INTEGER array, dimension (NN) The values for the matrix dimension N. line 4: NPARMS, INTEGER Number of values of the parameters NB, NBMIN, NX, NS, and MAXB. line 5: NBVAL, INTEGER array, dimension (NPARMS) The values for the blocksize NB. line 6: NBMIN, INTEGER array, dimension (NPARMS) The values for the minimum blocksize NBMIN. line 7: NXVAL, INTEGER array, dimension (NPARMS) The values for the crossover point NX. line 8: INMIN, INTEGER array, dimension (NPARMS) LAHQR vs TTQRE crossover point, >= 11 line 9: INWIN, INTEGER array, dimension (NPARMS) recommended deflation window size line 10: INIBL, INTEGER array, dimension (NPARMS) nibble crossover point line 11: ISHFTS, INTEGER array, dimension (NPARMS) number of simultaneous shifts) line 12: IACC22, INTEGER array, dimension (NPARMS) select structured matrix multiply: 0, 1 or 2) line 13: THRESH Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. To have all of the test ratios printed, use THRESH = 0.0 . line 14: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 14 was 2: line 15: INTEGER array, dimension (4) Four integer values for the random number seed. lines 15EOF: The remaining lines occur in sets of 1 or 2 and allow the user to specify the matrix types. Each line contains a 3character path name in columns 13, and the number of matrix types must be the first nonblank item in columns 480. If the number of matrix types is at least 1 but is less than the maximum number of possible types, a second line will be read to get the numbers of the matrix types to be used. For example, NEP 21 requests all of the matrix types for the nonsymmetric eigenvalue problem, while NEP 4 9 10 11 12 requests only matrices of type 9, 10, 11, and 12. The valid 3character path names are 'NEP' or 'SHS' for the nonsymmetric eigenvalue routines.  SEP or DSG input file: line 2: NN, INTEGER Number of values of N. line 3: NVAL, INTEGER array, dimension (NN) The values for the matrix dimension N. line 4: NPARMS, INTEGER Number of values of the parameters NB, NBMIN, and NX. line 5: NBVAL, INTEGER array, dimension (NPARMS) The values for the blocksize NB. line 6: NBMIN, INTEGER array, dimension (NPARMS) The values for the minimum blocksize NBMIN. line 7: NXVAL, INTEGER array, dimension (NPARMS) The values for the crossover point NX. line 8: THRESH Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 9: TSTCHK, LOGICAL Flag indicating whether or not to test the LAPACK routines. line 10: TSTDRV, LOGICAL Flag indicating whether or not to test the driver routines. line 11: TSTERR, LOGICAL Flag indicating whether or not to test the error exits for the LAPACK routines and driver routines. line 12: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 12 was 2: line 13: INTEGER array, dimension (4) Four integer values for the random number seed. lines 13EOF: Lines specifying matrix types, as for NEP. The 3character path names are 'SEP' or 'SST' for the symmetric eigenvalue routines and driver routines, and 'DSG' for the routines for the symmetric generalized eigenvalue problem.  SVD input file: line 2: NN, INTEGER Number of values of M and N. line 3: MVAL, INTEGER array, dimension (NN) The values for the matrix row dimension M. line 4: NVAL, INTEGER array, dimension (NN) The values for the matrix column dimension N. line 5: NPARMS, INTEGER Number of values of the parameter NB, NBMIN, NX, and NRHS. line 6: NBVAL, INTEGER array, dimension (NPARMS) The values for the blocksize NB. line 7: NBMIN, INTEGER array, dimension (NPARMS) The values for the minimum blocksize NBMIN. line 8: NXVAL, INTEGER array, dimension (NPARMS) The values for the crossover point NX. line 9: NSVAL, INTEGER array, dimension (NPARMS) The values for the number of right hand sides NRHS. line 10: THRESH Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 11: TSTCHK, LOGICAL Flag indicating whether or not to test the LAPACK routines. line 12: TSTDRV, LOGICAL Flag indicating whether or not to test the driver routines. line 13: TSTERR, LOGICAL Flag indicating whether or not to test the error exits for the LAPACK routines and driver routines. line 14: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 14 was 2: line 15: INTEGER array, dimension (4) Four integer values for the random number seed. lines 15EOF: Lines specifying matrix types, as for NEP. The 3character path names are 'SVD' or 'SBD' for both the SVD routines and the SVD driver routines.  DEV and DES data files: line 1: 'DEV' or 'DES' in columns 1 to 3. line 2: NSIZES, INTEGER Number of sizes of matrices to use. Should be at least 0 and at most 20. If NSIZES = 0, no testing is done (although the remaining 3 lines are still read). line 3: NN, INTEGER array, dimension(NSIZES) Dimensions of matrices to be tested. line 4: NB, NBMIN, NX, NS, NBCOL, INTEGERs These integer parameters determine how blocking is done (see ILAENV for details) NB : block size NBMIN : minimum block size NX : minimum dimension for blocking NS : number of shifts in xHSEQR NBCOL : minimum column dimension for blocking line 5: THRESH, REAL The test threshold against which computed residuals are compared. Should generally be in the range from 10. to 20. If it is 0., all test case data will be printed. line 6: TSTERR, LOGICAL Flag indicating whether or not to test the error exits. line 7: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 7 was 2: line 8: INTEGER array, dimension (4) Four integer values for the random number seed. lines 9 and following: Lines specifying matrix types, as for NEP. The 3character path name is 'DEV' to test SGEEV, or 'DES' to test SGEES.  The DVX data has two parts. The first part is identical to DEV, and the second part consists of test matrices with precomputed solutions. line 1: 'DVX' in columns 13. line 2: NSIZES, INTEGER If NSIZES = 0, no testing of randomly generated examples is done, but any precomputed examples are tested. line 3: NN, INTEGER array, dimension(NSIZES) line 4: NB, NBMIN, NX, NS, NBCOL, INTEGERs line 5: THRESH, REAL line 6: TSTERR, LOGICAL line 7: NEWSD, INTEGER If line 7 was 2: line 8: INTEGER array, dimension (4) lines 9 and following: The first line contains 'DVX' in columns 13 followed by the number of matrix types, possibly with a second line to specify certain matrix types. If the number of matrix types = 0, no testing of randomly generated examples is done, but any precomputed examples are tested. remaining lines : Each matrix is stored on 1+2*N lines, where N is its dimension. The first line contains the dimension (a single integer). The next N lines contain the matrix, one row per line. The last N lines correspond to each eigenvalue. Each of these last N lines contains 4 real values: the real part of the eigenvalue, the imaginary part of the eigenvalue, the reciprocal condition number of the eigenvalues, and the reciprocal condition number of the eigenvector. The end of data is indicated by dimension N=0. Even if no data is to be tested, there must be at least one line containing N=0.  The DSX data is like DVX. The first part is identical to DEV, and the second part consists of test matrices with precomputed solutions. line 1: 'DSX' in columns 13. line 2: NSIZES, INTEGER If NSIZES = 0, no testing of randomly generated examples is done, but any precomputed examples are tested. line 3: NN, INTEGER array, dimension(NSIZES) line 4: NB, NBMIN, NX, NS, NBCOL, INTEGERs line 5: THRESH, REAL line 6: TSTERR, LOGICAL line 7: NEWSD, INTEGER If line 7 was 2: line 8: INTEGER array, dimension (4) lines 9 and following: The first line contains 'DSX' in columns 13 followed by the number of matrix types, possibly with a second line to specify certain matrix types. If the number of matrix types = 0, no testing of randomly generated examples is done, but any precomputed examples are tested. remaining lines : Each matrix is stored on 3+N lines, where N is its dimension. The first line contains the dimension N and the dimension M of an invariant subspace. The second line contains M integers, identifying the eigenvalues in the invariant subspace (by their position in a list of eigenvalues ordered by increasing real part). The next N lines contain the matrix. The last line contains the reciprocal condition number for the average of the selected eigenvalues, and the reciprocal condition number for the corresponding right invariant subspace. The end of data is indicated by a line containing N=0 and M=0. Even if no data is to be tested, there must be at least one line containing N=0 and M=0.  DGG input file: line 2: NN, INTEGER Number of values of N. line 3: NVAL, INTEGER array, dimension (NN) The values for the matrix dimension N. line 4: NPARMS, INTEGER Number of values of the parameters NB, NBMIN, NS, MAXB, and NBCOL. line 5: NBVAL, INTEGER array, dimension (NPARMS) The values for the blocksize NB. line 6: NBMIN, INTEGER array, dimension (NPARMS) The values for NBMIN, the minimum row dimension for blocks. line 7: NSVAL, INTEGER array, dimension (NPARMS) The values for the number of shifts. line 8: MXBVAL, INTEGER array, dimension (NPARMS) The values for MAXB, used in determining minimum blocksize. line 9: NBCOL, INTEGER array, dimension (NPARMS) The values for NBCOL, the minimum column dimension for blocks. line 10: THRESH Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 11: TSTCHK, LOGICAL Flag indicating whether or not to test the LAPACK routines. line 12: TSTDRV, LOGICAL Flag indicating whether or not to test the driver routines. line 13: TSTERR, LOGICAL Flag indicating whether or not to test the error exits for the LAPACK routines and driver routines. line 14: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 14 was 2: line 15: INTEGER array, dimension (4) Four integer values for the random number seed. lines 15EOF: Lines specifying matrix types, as for NEP. The 3character path name is 'DGG' for the generalized eigenvalue problem routines and driver routines.  DGS and DGV input files: line 1: 'DGS' or 'DGV' in columns 1 to 3. line 2: NN, INTEGER Number of values of N. line 3: NVAL, INTEGER array, dimension(NN) Dimensions of matrices to be tested. line 4: NB, NBMIN, NX, NS, NBCOL, INTEGERs These integer parameters determine how blocking is done (see ILAENV for details) NB : block size NBMIN : minimum block size NX : minimum dimension for blocking NS : number of shifts in xHGEQR NBCOL : minimum column dimension for blocking line 5: THRESH, REAL The test threshold against which computed residuals are compared. Should generally be in the range from 10. to 20. If it is 0., all test case data will be printed. line 6: TSTERR, LOGICAL Flag indicating whether or not to test the error exits. line 7: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 17 was 2: line 7: INTEGER array, dimension (4) Four integer values for the random number seed. lines 7EOF: Lines specifying matrix types, as for NEP. The 3character path name is 'DGS' for the generalized eigenvalue problem routines and driver routines.  DXV input files: line 1: 'DXV' in columns 1 to 3. line 2: N, INTEGER Value of N. line 3: NB, NBMIN, NX, NS, NBCOL, INTEGERs These integer parameters determine how blocking is done (see ILAENV for details) NB : block size NBMIN : minimum block size NX : minimum dimension for blocking NS : number of shifts in xHGEQR NBCOL : minimum column dimension for blocking line 4: THRESH, REAL The test threshold against which computed residuals are compared. Should generally be in the range from 10. to 20. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 5: TSTERR, LOGICAL Flag indicating whether or not to test the error exits for the LAPACK routines and driver routines. line 6: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 6 was 2: line 7: INTEGER array, dimension (4) Four integer values for the random number seed. If line 2 was 0: line 7EOF: Precomputed examples are tested. remaining lines : Each example is stored on 3+2*N lines, where N is its dimension. The first line contains the dimension (a single integer). The next N lines contain the matrix A, one row per line. The next N lines contain the matrix B. The next line contains the reciprocals of the eigenvalue condition numbers. The last line contains the reciprocals of the eigenvector condition numbers. The end of data is indicated by dimension N=0. Even if no data is to be tested, there must be at least one line containing N=0.  DGX input files: line 1: 'DGX' in columns 1 to 3. line 2: N, INTEGER Value of N. line 3: NB, NBMIN, NX, NS, NBCOL, INTEGERs These integer parameters determine how blocking is done (see ILAENV for details) NB : block size NBMIN : minimum block size NX : minimum dimension for blocking NS : number of shifts in xHGEQR NBCOL : minimum column dimension for blocking line 4: THRESH, REAL The test threshold against which computed residuals are compared. Should generally be in the range from 10. to 20. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 5: TSTERR, LOGICAL Flag indicating whether or not to test the error exits for the LAPACK routines and driver routines. line 6: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 6 was 2: line 7: INTEGER array, dimension (4) Four integer values for the random number seed. If line 2 was 0: line 7EOF: Precomputed examples are tested. remaining lines : Each example is stored on 3+2*N lines, where N is its dimension. The first line contains the dimension (a single integer). The next line contains an integer k such that only the last k eigenvalues will be selected and appear in the leading diagonal blocks of $A$ and $B$. The next N lines contain the matrix A, one row per line. The next N lines contain the matrix B. The last line contains the reciprocal of the eigenvalue cluster condition number and the reciprocal of the deflating subspace (associated with the selected eigencluster) condition number. The end of data is indicated by dimension N=0. Even if no data is to be tested, there must be at least one line containing N=0.  DSB input file: line 2: NN, INTEGER Number of values of N. line 3: NVAL, INTEGER array, dimension (NN) The values for the matrix dimension N. line 4: NK, INTEGER Number of values of K. line 5: KVAL, INTEGER array, dimension (NK) The values for the matrix dimension K. line 6: THRESH Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 7: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 7 was 2: line 8: INTEGER array, dimension (4) Four integer values for the random number seed. lines 8EOF: Lines specifying matrix types, as for NEP. The 3character path name is 'DSB'.  DBB input file: line 2: NN, INTEGER Number of values of M and N. line 3: MVAL, INTEGER array, dimension (NN) The values for the matrix row dimension M. line 4: NVAL, INTEGER array, dimension (NN) The values for the matrix column dimension N. line 4: NK, INTEGER Number of values of K. line 5: KVAL, INTEGER array, dimension (NK) The values for the matrix bandwidth K. line 6: NPARMS, INTEGER Number of values of the parameter NRHS line 7: NSVAL, INTEGER array, dimension (NPARMS) The values for the number of right hand sides NRHS. line 8: THRESH Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 9: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 9 was 2: line 10: INTEGER array, dimension (4) Four integer values for the random number seed. lines 10EOF: Lines specifying matrix types, as for SVD. The 3character path name is 'DBB'.  DEC input file: line 2: THRESH, REAL Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. lines 3EOF: Input for testing the eigencondition routines consists of a set of specially constructed test cases and their solutions. The data format is not intended to be modified by the user.  DBL and DBK input files: line 1: 'DBL' in columns 13 to test SGEBAL, or 'DBK' in columns 13 to test SGEBAK. The remaining lines consist of specially constructed test cases.  DGL and DGK input files: line 1: 'DGL' in columns 13 to test DGGBAL, or 'DGK' in columns 13 to test DGGBAK. The remaining lines consist of specially constructed test cases.  GLM data file: line 1: 'GLM' in columns 1 to 3. line 2: NN, INTEGER Number of values of M, P, and N. line 3: MVAL, INTEGER array, dimension(NN) Values of M (row dimension). line 4: PVAL, INTEGER array, dimension(NN) Values of P (row dimension). line 5: NVAL, INTEGER array, dimension(NN) Values of N (column dimension), note M <= N <= M+P. line 6: THRESH, REAL Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 7: TSTERR, LOGICAL Flag indicating whether or not to test the error exits for the LAPACK routines and driver routines. line 8: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 8 was 2: line 9: INTEGER array, dimension (4) Four integer values for the random number seed. lines 9EOF: Lines specifying matrix types, as for NEP. The 3character path name is 'GLM' for the generalized linear regression model routines.  GQR data file: line 1: 'GQR' in columns 1 to 3. line 2: NN, INTEGER Number of values of M, P, and N. line 3: MVAL, INTEGER array, dimension(NN) Values of M. line 4: PVAL, INTEGER array, dimension(NN) Values of P. line 5: NVAL, INTEGER array, dimension(NN) Values of N. line 6: THRESH, REAL Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 7: TSTERR, LOGICAL Flag indicating whether or not to test the error exits for the LAPACK routines and driver routines. line 8: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 8 was 2: line 9: INTEGER array, dimension (4) Four integer values for the random number seed. lines 9EOF: Lines specifying matrix types, as for NEP. The 3character path name is 'GQR' for the generalized QR and RQ routines.  GSV data file: line 1: 'GSV' in columns 1 to 3. line 2: NN, INTEGER Number of values of M, P, and N. line 3: MVAL, INTEGER array, dimension(NN) Values of M (row dimension). line 4: PVAL, INTEGER array, dimension(NN) Values of P (row dimension). line 5: NVAL, INTEGER array, dimension(NN) Values of N (column dimension). line 6: THRESH, REAL Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 7: TSTERR, LOGICAL Flag indicating whether or not to test the error exits for the LAPACK routines and driver routines. line 8: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 8 was 2: line 9: INTEGER array, dimension (4) Four integer values for the random number seed. lines 9EOF: Lines specifying matrix types, as for NEP. The 3character path name is 'GSV' for the generalized SVD routines.  CSD data file: line 1: 'CSD' in columns 1 to 3. line 2: NM, INTEGER Number of values of M, P, and N. line 3: MVAL, INTEGER array, dimension(NM) Values of M (row and column dimension of orthogonal matrix). line 4: PVAL, INTEGER array, dimension(NM) Values of P (row dimension of topleft block). line 5: NVAL, INTEGER array, dimension(NM) Values of N (column dimension of topleft block). line 6: THRESH, REAL Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 7: TSTERR, LOGICAL Flag indicating whether or not to test the error exits for the LAPACK routines and driver routines. line 8: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 8 was 2: line 9: INTEGER array, dimension (4) Four integer values for the random number seed. lines 9EOF: Lines specifying matrix types, as for NEP. The 3character path name is 'CSD' for the CSD routine.  LSE data file: line 1: 'LSE' in columns 1 to 3. line 2: NN, INTEGER Number of values of M, P, and N. line 3: MVAL, INTEGER array, dimension(NN) Values of M. line 4: PVAL, INTEGER array, dimension(NN) Values of P. line 5: NVAL, INTEGER array, dimension(NN) Values of N, note P <= N <= P+M. line 6: THRESH, REAL Threshold value for the test ratios. Information will be printed about each test for which the test ratio is greater than or equal to the threshold. line 7: TSTERR, LOGICAL Flag indicating whether or not to test the error exits for the LAPACK routines and driver routines. line 8: NEWSD, INTEGER A code indicating how to set the random number seed. = 0: Set the seed to a default value before each run = 1: Initialize the seed to a default value only before the first run = 2: Like 1, but use the seed values on the next line If line 8 was 2: line 9: INTEGER array, dimension (4) Four integer values for the random number seed. lines 9EOF: Lines specifying matrix types, as for NEP. The 3character path name is 'GSV' for the generalized SVD routines.  NMAX is currently set to 132 and must be at least 12 for some of the precomputed examples, and LWORK = NMAX*(5*NMAX+5)+1 in the parameter statements below. For SVD, we assume NRHS may be as big as N. The parameter NEED is set to 14 to allow for 14 NbyN matrices for DGG.
Definition at line 1040 of file dchkee.f.
subroutine dchkgg  (  integer  NSIZES, 
integer, dimension( * )  NN,  
integer  NTYPES,  
logical, dimension( * )  DOTYPE,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
logical  TSTDIF,  
double precision  THRSHN,  
integer  NOUNIT,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( lda, * )  B,  
double precision, dimension( lda, * )  H,  
double precision, dimension( lda, * )  T,  
double precision, dimension( lda, * )  S1,  
double precision, dimension( lda, * )  S2,  
double precision, dimension( lda, * )  P1,  
double precision, dimension( lda, * )  P2,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( ldu, * )  V,  
double precision, dimension( ldu, * )  Q,  
double precision, dimension( ldu, * )  Z,  
double precision, dimension( * )  ALPHR1,  
double precision, dimension( * )  ALPHI1,  
double precision, dimension( * )  BETA1,  
double precision, dimension( * )  ALPHR3,  
double precision, dimension( * )  ALPHI3,  
double precision, dimension( * )  BETA3,  
double precision, dimension( ldu, * )  EVECTL,  
double precision, dimension( ldu, * )  EVECTR,  
double precision, dimension( * )  WORK,  
integer  LWORK,  
logical, dimension( * )  LLWORK,  
double precision, dimension( 15 )  RESULT,  
integer  INFO  
) 
DCHKGG
DCHKGG checks the nonsymmetric generalized eigenvalue problem routines. T T T DGGHRD factors A and B as U H V and U T V , where means transpose, H is hessenberg, T is triangular and U and V are orthogonal. T T DHGEQZ factors H and T as Q S Z and Q P Z , where P is upper triangular, S is in generalized Schur form (block upper triangular, with 1x1 and 2x2 blocks on the diagonal, the 2x2 blocks corresponding to complex conjugate pairs of generalized eigenvalues), and Q and Z are orthogonal. It also computes the generalized eigenvalues (alpha(1),beta(1)),...,(alpha(n),beta(n)), where alpha(j)=S(j,j) and beta(j)=P(j,j)  thus, w(j) = alpha(j)/beta(j) is a root of the generalized eigenvalue problem det( A  w(j) B ) = 0 and m(j) = beta(j)/alpha(j) is a root of the essentially equivalent problem det( m(j) A  B ) = 0 DTGEVC computes the matrix L of left eigenvectors and the matrix R of right eigenvectors for the matrix pair ( S, P ). In the description below, l and r are left and right eigenvectors corresponding to the generalized eigenvalues (alpha,beta). When DCHKGG is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified. For each size ("n") and each type of matrix, one matrix will be generated and used to test the nonsymmetric eigenroutines. For each matrix, 15 tests will be performed. The first twelve "test ratios" should be small  O(1). They will be compared with the threshhold THRESH: T (1)  A  U H V  / ( A n ulp ) T (2)  B  U T V  / ( B n ulp ) T (3)  I  UU  / ( n ulp ) T (4)  I  VV  / ( n ulp ) T (5)  H  Q S Z  / ( H n ulp ) T (6)  T  Q P Z  / ( T n ulp ) T (7)  I  QQ  / ( n ulp ) T (8)  I  ZZ  / ( n ulp ) (9) max over all left eigenvalue/vector pairs (beta/alpha,l) of  l**H * (beta S  alpha P)  / ( ulp max( beta S, alpha P ) ) (10) max over all left eigenvalue/vector pairs (beta/alpha,l') of T  l'**H * (beta H  alpha T)  / ( ulp max( beta H, alpha T ) ) where the eigenvectors l' are the result of passing Q to DTGEVC and back transforming (HOWMNY='B'). (11) max over all right eigenvalue/vector pairs (beta/alpha,r) of  (beta S  alpha T) r  / ( ulp max( beta S, alpha T ) ) (12) max over all right eigenvalue/vector pairs (beta/alpha,r') of  (beta H  alpha T) r'  / ( ulp max( beta H, alpha T ) ) where the eigenvectors r' are the result of passing Z to DTGEVC and back transforming (HOWMNY='B'). The last three test ratios will usually be small, but there is no mathematical requirement that they be so. They are therefore compared with THRESH only if TSTDIF is .TRUE. (13)  S(Q,Z computed)  S(Q,Z not computed)  / ( S ulp ) (14)  P(Q,Z computed)  P(Q,Z not computed)  / ( P ulp ) (15) max( alpha(Q,Z computed)  alpha(Q,Z not computed)/S , beta(Q,Z computed)  beta(Q,Z not computed)/P ) / ulp In addition, the normalization of L and R are checked, and compared with the threshhold THRSHN. Test Matrices   The sizes of the test matrices are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) ( 0, 0 ) (a pair of zero matrices) (2) ( I, 0 ) (an identity and a zero matrix) (3) ( 0, I ) (an identity and a zero matrix) (4) ( I, I ) (a pair of identity matrices) t t (5) ( J , J ) (a pair of transposed Jordan blocks) t ( I 0 ) (6) ( X, Y ) where X = ( J 0 ) and Y = ( t ) ( 0 I ) ( 0 J ) and I is a k x k identity and J a (k+1)x(k+1) Jordan block; k=(N1)/2 (7) ( D, I ) where D is diag( 0, 1,..., N1 ) (a diagonal matrix with those diagonal entries.) (8) ( I, D ) (9) ( big*D, small*I ) where "big" is near overflow and small=1/big (10) ( small*D, big*I ) (11) ( big*I, small*D ) (12) ( small*I, big*D ) (13) ( big*D, big*I ) (14) ( small*D, small*I ) (15) ( D1, D2 ) where D1 is diag( 0, 0, 1, ..., N3, 0 ) and D2 is diag( 0, N3, N4,..., 1, 0, 0 ) t t (16) U ( J , J ) V where U and V are random orthogonal matrices. (17) U ( T1, T2 ) V where T1 and T2 are upper triangular matrices with random O(1) entries above the diagonal and diagonal entries diag(T1) = ( 0, 0, 1, ..., N3, 0 ) and diag(T2) = ( 0, N3, N4,..., 1, 0, 0 ) (18) U ( T1, T2 ) V diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 ) diag(T2) = ( 0, 1, 0, 1,..., 1, 0 ) s = machine precision. (19) U ( T1, T2 ) V diag(T1)=( 0,0,1,1, 1d, ..., 1(N5)*d=s, 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 ) N5 (20) U ( T1, T2 ) V diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) (21) U ( T1, T2 ) V diag(T1)=( 0, 0, 1, r1, r2, ..., r(N4), 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) where r1,..., r(N4) are random. (22) U ( big*T1, small*T2 ) V diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (23) U ( small*T1, big*T2 ) V diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (24) U ( small*T1, small*T2 ) V diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (25) U ( big*T1, big*T2 ) V diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (26) U ( T1, T2 ) V where T1 and T2 are random uppertriangular matrices.
[in]  NSIZES  NSIZES is INTEGER The number of sizes of matrices to use. If it is zero, DCHKGG does nothing. It must be at least zero. 
[in]  NN  NN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero. 
[in]  NTYPES  NTYPES is INTEGER The number of elements in DOTYPE. If it is zero, DCHKGG does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. . 
[in]  DOTYPE  DOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored. 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to DCHKGG to continue the same random number sequence. 
[in]  THRESH  THRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero. 
[in]  TSTDIF  TSTDIF is LOGICAL Specifies whether test ratios 1315 will be computed and compared with THRESH. = .FALSE.: Only test ratios 112 will be computed and tested. Ratios 1315 will be set to zero. = .TRUE.: All the test ratios 115 will be computed and tested. 
[in]  THRSHN  THRSHN is DOUBLE PRECISION Threshhold for reporting eigenvector normalization error. If the normalization of any eigenvector differs from 1 by more than THRSHN*ulp, then a special error message will be printed. (This is handled separately from the other tests, since only a compiler or programming error should cause an error message, at least if THRSHN is at least 510.) 
[in]  NOUNIT  NOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) 
[in,out]  A  A is DOUBLE PRECISION array, dimension (LDA, max(NN)) Used to hold the original A matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. 
[in]  LDA  LDA is INTEGER The leading dimension of A, B, H, T, S1, P1, S2, and P2. It must be at least 1 and at least max( NN ). 
[in,out]  B  B is DOUBLE PRECISION array, dimension (LDA, max(NN)) Used to hold the original B matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. 
[out]  H  H is DOUBLE PRECISION array, dimension (LDA, max(NN)) The upper Hessenberg matrix computed from A by DGGHRD. 
[out]  T  T is DOUBLE PRECISION array, dimension (LDA, max(NN)) The upper triangular matrix computed from B by DGGHRD. 
[out]  S1  S1 is DOUBLE PRECISION array, dimension (LDA, max(NN)) The Schur (block upper triangular) matrix computed from H by DHGEQZ when Q and Z are also computed. 
[out]  S2  S2 is DOUBLE PRECISION array, dimension (LDA, max(NN)) The Schur (block upper triangular) matrix computed from H by DHGEQZ when Q and Z are not computed. 
[out]  P1  P1 is DOUBLE PRECISION array, dimension (LDA, max(NN)) The upper triangular matrix computed from T by DHGEQZ when Q and Z are also computed. 
[out]  P2  P2 is DOUBLE PRECISION array, dimension (LDA, max(NN)) The upper triangular matrix computed from T by DHGEQZ when Q and Z are not computed. 
[out]  U  U is DOUBLE PRECISION array, dimension (LDU, max(NN)) The (left) orthogonal matrix computed by DGGHRD. 
[in]  LDU  LDU is INTEGER The leading dimension of U, V, Q, Z, EVECTL, and EVEZTR. It must be at least 1 and at least max( NN ). 
[out]  V  V is DOUBLE PRECISION array, dimension (LDU, max(NN)) The (right) orthogonal matrix computed by DGGHRD. 
[out]  Q  Q is DOUBLE PRECISION array, dimension (LDU, max(NN)) The (left) orthogonal matrix computed by DHGEQZ. 
[out]  Z  Z is DOUBLE PRECISION array, dimension (LDU, max(NN)) The (left) orthogonal matrix computed by DHGEQZ. 
[out]  ALPHR1  ALPHR1 is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  ALPHI1  ALPHI1 is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  BETA1  BETA1 is DOUBLE PRECISION array, dimension (max(NN)) The generalized eigenvalues of (A,B) computed by DHGEQZ when Q, Z, and the full Schur matrices are computed. On exit, ( ALPHR1(k)+ALPHI1(k)*i ) / BETA1(k) is the kth generalized eigenvalue of the matrices in A and B. 
[out]  ALPHR3  ALPHR3 is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  ALPHI3  ALPHI3 is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  BETA3  BETA3 is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  EVECTL  EVECTL is DOUBLE PRECISION array, dimension (LDU, max(NN)) The (block lower triangular) left eigenvector matrix for the matrices in S1 and P1. (See DTGEVC for the format.) 
[out]  EVECTR  EVECTR is DOUBLE PRECISION array, dimension (LDU, max(NN)) The (block upper triangular) right eigenvector matrix for the matrices in S1 and P1. (See DTGEVC for the format.) 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER The number of entries in WORK. This must be at least max( 2 * N**2, 6*N, 1 ), for all N=NN(j). 
[out]  LLWORK  LLWORK is LOGICAL array, dimension (max(NN)) 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (15) The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: A routine returned an error code. INFO is the absolute value of the INFO value returned. 
Definition at line 508 of file dchkgg.f.
subroutine dchkgk  (  integer  NIN, 
integer  NOUT  
) 
DCHKGK
DCHKGK tests DGGBAK, a routine for backward balancing of a matrix pair (A, B).
[in]  NIN  NIN is INTEGER The logical unit number for input. NIN > 0. 
[in]  NOUT  NOUT is INTEGER The logical unit number for output. NOUT > 0. 
Definition at line 55 of file dchkgk.f.
subroutine dchkgl  (  integer  NIN, 
integer  NOUT  
) 
DCHKGL
DCHKGL tests DGGBAL, a routine for balancing a matrix pair (A, B).
[in]  NIN  NIN is INTEGER The logical unit number for input. NIN > 0. 
[in]  NOUT  NOUT is INTEGER The logical unit number for output. NOUT > 0. 
Definition at line 54 of file dchkgl.f.
subroutine dchkhs  (  integer  NSIZES, 
integer, dimension( * )  NN,  
integer  NTYPES,  
logical, dimension( * )  DOTYPE,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
integer  NOUNIT,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( lda, * )  H,  
double precision, dimension( lda, * )  T1,  
double precision, dimension( lda, * )  T2,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( ldu, * )  Z,  
double precision, dimension( ldu, * )  UZ,  
double precision, dimension( * )  WR1,  
double precision, dimension( * )  WI1,  
double precision, dimension( * )  WR3,  
double precision, dimension( * )  WI3,  
double precision, dimension( ldu, * )  EVECTL,  
double precision, dimension( ldu, * )  EVECTR,  
double precision, dimension( ldu, * )  EVECTY,  
double precision, dimension( ldu, * )  EVECTX,  
double precision, dimension( ldu, * )  UU,  
double precision, dimension( * )  TAU,  
double precision, dimension( * )  WORK,  
integer  NWORK,  
integer, dimension( * )  IWORK,  
logical, dimension( * )  SELECT,  
double precision, dimension( 14 )  RESULT,  
integer  INFO  
) 
DCHKHS
DCHKHS checks the nonsymmetric eigenvalue problem routines. DGEHRD factors A as U H U' , where ' means transpose, H is hessenberg, and U is an orthogonal matrix. DORGHR generates the orthogonal matrix U. DORMHR multiplies a matrix by the orthogonal matrix U. DHSEQR factors H as Z T Z' , where Z is orthogonal and T is "quasitriangular", and the eigenvalue vector W. DTREVC computes the left and right eigenvector matrices L and R for T. DHSEIN computes the left and right eigenvector matrices Y and X for H, using inverse iteration. When DCHKHS is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified. For each size ("n") and each type of matrix, one matrix will be generated and used to test the nonsymmetric eigenroutines. For each matrix, 14 tests will be performed: (1)  A  U H U**T  / ( A n ulp ) (2)  I  UU**T  / ( n ulp ) (3)  H  Z T Z**T  / ( H n ulp ) (4)  I  ZZ**T  / ( n ulp ) (5)  A  UZ H (UZ)**T  / ( A n ulp ) (6)  I  UZ (UZ)**T  / ( n ulp ) (7)  T(Z computed)  T(Z not computed)  / ( T ulp ) (8)  W(Z computed)  W(Z not computed)  / ( W ulp ) (9)  TR  RW  / ( T R ulp ) (10)  L**H T  W**H L  / ( T L ulp ) (11)  HX  XW  / ( H X ulp ) (12)  Y**H H  W**H Y  / ( H Y ulp ) (13)  AX  XW  / ( A X ulp ) (14)  Y**H A  W**H Y  / ( A Y ulp ) The "sizes" are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) The zero matrix. (2) The identity matrix. (3) A (transposed) Jordan block, with 1's on the diagonal. (4) A diagonal matrix with evenly spaced entries 1, ..., ULP and random signs. (ULP = (first number larger than 1)  1 ) (5) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random signs. (6) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP and random signs. (7) Same as (4), but multiplied by SQRT( overflow threshold ) (8) Same as (4), but multiplied by SQRT( underflow threshold ) (9) A matrix of the form U' T U, where U is orthogonal and T has evenly spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (10) A matrix of the form U' T U, where U is orthogonal and T has geometrically spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (11) A matrix of the form U' T U, where U is orthogonal and T has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (12) A matrix of the form U' T U, where U is orthogonal and T has real or complex conjugate paired eigenvalues randomly chosen from ( ULP, 1 ) and random O(1) entries in the upper triangle. (13) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has evenly spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (14) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has geometrically spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (15) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (16) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has real or complex conjugate paired eigenvalues randomly chosen from ( ULP, 1 ) and random O(1) entries in the upper triangle. (17) Same as (16), but multiplied by SQRT( overflow threshold ) (18) Same as (16), but multiplied by SQRT( underflow threshold ) (19) Nonsymmetric matrix with random entries chosen from (1,1). (20) Same as (19), but multiplied by SQRT( overflow threshold ) (21) Same as (19), but multiplied by SQRT( underflow threshold )
NSIZES  INTEGER The number of sizes of matrices to use. If it is zero, DCHKHS does nothing. It must be at least zero. Not modified. NN  INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero. Not modified. NTYPES  INTEGER The number of elements in DOTYPE. If it is zero, DCHKHS does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. . Not modified. DOTYPE  LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored. Not modified. ISEED  INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to DCHKHS to continue the same random number sequence. Modified. THRESH  DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero. Not modified. NOUNIT  INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) Not modified. A  DOUBLE PRECISION array, dimension (LDA,max(NN)) Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used. Modified. LDA  INTEGER The leading dimension of A, H, T1 and T2. It must be at least 1 and at least max( NN ). Not modified. H  DOUBLE PRECISION array, dimension (LDA,max(NN)) The upper hessenberg matrix computed by DGEHRD. On exit, H contains the Hessenberg form of the matrix in A. Modified. T1  DOUBLE PRECISION array, dimension (LDA,max(NN)) The Schur (="quasitriangular") matrix computed by DHSEQR if Z is computed. On exit, T1 contains the Schur form of the matrix in A. Modified. T2  DOUBLE PRECISION array, dimension (LDA,max(NN)) The Schur matrix computed by DHSEQR when Z is not computed. This should be identical to T1. Modified. LDU  INTEGER The leading dimension of U, Z, UZ and UU. It must be at least 1 and at least max( NN ). Not modified. U  DOUBLE PRECISION array, dimension (LDU,max(NN)) The orthogonal matrix computed by DGEHRD. Modified. Z  DOUBLE PRECISION array, dimension (LDU,max(NN)) The orthogonal matrix computed by DHSEQR. Modified. UZ  DOUBLE PRECISION array, dimension (LDU,max(NN)) The product of U times Z. Modified. WR1  DOUBLE PRECISION array, dimension (max(NN)) WI1  DOUBLE PRECISION array, dimension (max(NN)) The real and imaginary parts of the eigenvalues of A, as computed when Z is computed. On exit, WR1 + WI1*i are the eigenvalues of the matrix in A. Modified. WR3  DOUBLE PRECISION array, dimension (max(NN)) WI3  DOUBLE PRECISION array, dimension (max(NN)) Like WR1, WI1, these arrays contain the eigenvalues of A, but those computed when DHSEQR only computes the eigenvalues, i.e., not the Schur vectors and no more of the Schur form than is necessary for computing the eigenvalues. Modified. EVECTL  DOUBLE PRECISION array, dimension (LDU,max(NN)) The (upper triangular) left eigenvector matrix for the matrix in T1. For complex conjugate pairs, the real part is stored in one row and the imaginary part in the next. Modified. EVEZTR  DOUBLE PRECISION array, dimension (LDU,max(NN)) The (upper triangular) right eigenvector matrix for the matrix in T1. For complex conjugate pairs, the real part is stored in one column and the imaginary part in the next. Modified. EVECTY  DOUBLE PRECISION array, dimension (LDU,max(NN)) The left eigenvector matrix for the matrix in H. For complex conjugate pairs, the real part is stored in one row and the imaginary part in the next. Modified. EVECTX  DOUBLE PRECISION array, dimension (LDU,max(NN)) The right eigenvector matrix for the matrix in H. For complex conjugate pairs, the real part is stored in one column and the imaginary part in the next. Modified. UU  DOUBLE PRECISION array, dimension (LDU,max(NN)) Details of the orthogonal matrix computed by DGEHRD. Modified. TAU  DOUBLE PRECISION array, dimension(max(NN)) Further details of the orthogonal matrix computed by DGEHRD. Modified. WORK  DOUBLE PRECISION array, dimension (NWORK) Workspace. Modified. NWORK  INTEGER The number of entries in WORK. NWORK >= 4*NN(j)*NN(j) + 2. IWORK  INTEGER array, dimension (max(NN)) Workspace. Modified. SELECT  LOGICAL array, dimension (max(NN)) Workspace. Modified. RESULT  DOUBLE PRECISION array, dimension (14) The values computed by the fourteen tests described above. The values are currently limited to 1/ulp, to avoid overflow. Modified. INFO  INTEGER If 0, then everything ran OK. 1: NSIZES < 0 2: Some NN(j) < 0 3: NTYPES < 0 6: THRESH < 0 9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ). 14: LDU < 1 or LDU < NMAX. 28: NWORK too small. If DLATMR, SLATMS, or SLATME returns an error code, the absolute value of it is returned. If 1, then DHSEQR could not find all the shifts. If 2, then the EISPACK code (for small blocks) failed. If >2, then 30*N iterations were not enough to find an eigenvalue or to decompose the problem. Modified.  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. MTEST The number of tests defined: care must be taken that (1) the size of RESULT, (2) the number of tests actually performed, and (3) MTEST agree. NTEST The number of tests performed on this matrix so far. This should be less than MTEST, and equal to it by the last test. It will be less if any of the routines being tested indicates that it could not compute the matrices that would be tested. NMAX Largest value in NN. NMATS The number of matrices generated so far. NERRS The number of tests which have exceeded THRESH so far (computed by DLAFTS). COND, CONDS, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. ULP, ULPINV Finest relative precision and its inverse. RTOVFL, RTUNFL, RTULP, RTULPI Square roots of the previous 4 values. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) ) KCONDS(j) Selects whether CONDS is to be 1 or 1/sqrt(ulp). (0 means irrelevant.)
Definition at line 402 of file dchkhs.f.
subroutine dchksb  (  integer  NSIZES, 
integer, dimension( * )  NN,  
integer  NWDTHS,  
integer, dimension( * )  KK,  
integer  NTYPES,  
logical, dimension( * )  DOTYPE,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
integer  NOUNIT,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( * )  SD,  
double precision, dimension( * )  SE,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( * )  WORK,  
integer  LWORK,  
double precision, dimension( * )  RESULT,  
integer  INFO  
) 
DCHKSB
DCHKSB tests the reduction of a symmetric band matrix to tridiagonal form, used with the symmetric eigenvalue problem. DSBTRD factors a symmetric band matrix A as U S U' , where ' means transpose, S is symmetric tridiagonal, and U is orthogonal. DSBTRD can use either just the lower or just the upper triangle of A; DCHKSB checks both cases. When DCHKSB is called, a number of matrix "sizes" ("n's"), a number of bandwidths ("k's"), and a number of matrix "types" are specified. For each size ("n"), each bandwidth ("k") less than or equal to "n", and each type of matrix, one matrix will be generated and used to test the symmetric banded reduction routine. For each matrix, a number of tests will be performed: (1)  A  V S V'  / ( A n ulp ) computed by DSBTRD with UPLO='U' (2)  I  UU'  / ( n ulp ) (3)  A  V S V'  / ( A n ulp ) computed by DSBTRD with UPLO='L' (4)  I  UU'  / ( n ulp ) The "sizes" are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) The zero matrix. (2) The identity matrix. (3) A diagonal matrix with evenly spaced entries 1, ..., ULP and random signs. (ULP = (first number larger than 1)  1 ) (4) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random signs. (5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP and random signs. (6) Same as (4), but multiplied by SQRT( overflow threshold ) (7) Same as (4), but multiplied by SQRT( underflow threshold ) (8) A matrix of the form U' D U, where U is orthogonal and D has evenly spaced entries 1, ..., ULP with random signs on the diagonal. (9) A matrix of the form U' D U, where U is orthogonal and D has geometrically spaced entries 1, ..., ULP with random signs on the diagonal. (10) A matrix of the form U' D U, where U is orthogonal and D has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal. (11) Same as (8), but multiplied by SQRT( overflow threshold ) (12) Same as (8), but multiplied by SQRT( underflow threshold ) (13) Symmetric matrix with random entries chosen from (1,1). (14) Same as (13), but multiplied by SQRT( overflow threshold ) (15) Same as (13), but multiplied by SQRT( underflow threshold )
[in]  NSIZES  NSIZES is INTEGER The number of sizes of matrices to use. If it is zero, DCHKSB does nothing. It must be at least zero. 
[in]  NN  NN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero. 
[in]  NWDTHS  NWDTHS is INTEGER The number of bandwidths to use. If it is zero, DCHKSB does nothing. It must be at least zero. 
[in]  KK  KK is INTEGER array, dimension (NWDTHS) An array containing the bandwidths to be used for the band matrices. The values must be at least zero. 
[in]  NTYPES  NTYPES is INTEGER The number of elements in DOTYPE. If it is zero, DCHKSB does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. . 
[in]  DOTYPE  DOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored. 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to DCHKSB to continue the same random number sequence. 
[in]  THRESH  THRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero. 
[in]  NOUNIT  NOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) 
[in,out]  A  A is DOUBLE PRECISION array, dimension (LDA, max(NN)) Used to hold the matrix whose eigenvalues are to be computed. 
[in]  LDA  LDA is INTEGER The leading dimension of A. It must be at least 2 (not 1!) and at least max( KK )+1. 
[out]  SD  SD is DOUBLE PRECISION array, dimension (max(NN)) Used to hold the diagonal of the tridiagonal matrix computed by DSBTRD. 
[out]  SE  SE is DOUBLE PRECISION array, dimension (max(NN)) Used to hold the offdiagonal of the tridiagonal matrix computed by DSBTRD. 
[out]  U  U is DOUBLE PRECISION array, dimension (LDU, max(NN)) Used to hold the orthogonal matrix computed by DSBTRD. 
[in]  LDU  LDU is INTEGER The leading dimension of U. It must be at least 1 and at least max( NN ). 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER The number of entries in WORK. This must be at least max( LDA+1, max(NN)+1 )*max(NN). 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (4) The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. 
[out]  INFO  INFO is INTEGER If 0, then everything ran OK.  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. NTEST The number of tests performed, or which can be performed so far, for the current matrix. NTESTT The total number of tests performed so far. NMAX Largest value in NN. NMATS The number of matrices generated so far. NERRS The number of tests which have exceeded THRESH so far. COND, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. ULP, ULPINV Finest relative precision and its inverse. RTOVFL, RTUNFL Square roots of the previous 2 values. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) ) 
Definition at line 292 of file dchksb.f.
subroutine dchkst  (  integer  NSIZES, 
integer, dimension( * )  NN,  
integer  NTYPES,  
logical, dimension( * )  DOTYPE,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
integer  NOUNIT,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( * )  AP,  
double precision, dimension( * )  SD,  
double precision, dimension( * )  SE,  
double precision, dimension( * )  D1,  
double precision, dimension( * )  D2,  
double precision, dimension( * )  D3,  
double precision, dimension( * )  D4,  
double precision, dimension( * )  D5,  
double precision, dimension( * )  WA1,  
double precision, dimension( * )  WA2,  
double precision, dimension( * )  WA3,  
double precision, dimension( * )  WR,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( ldu, * )  V,  
double precision, dimension( * )  VP,  
double precision, dimension( * )  TAU,  
double precision, dimension( ldu, * )  Z,  
double precision, dimension( * )  WORK,  
integer  LWORK,  
integer, dimension( * )  IWORK,  
integer  LIWORK,  
double precision, dimension( * )  RESULT,  
integer  INFO  
) 
DCHKST
DCHKST checks the symmetric eigenvalue problem routines. DSYTRD factors A as U S U' , where ' means transpose, S is symmetric tridiagonal, and U is orthogonal. DSYTRD can use either just the lower or just the upper triangle of A; DCHKST checks both cases. U is represented as a product of Householder transformations, whose vectors are stored in the first n1 columns of V, and whose scale factors are in TAU. DSPTRD does the same as DSYTRD, except that A and V are stored in "packed" format. DORGTR constructs the matrix U from the contents of V and TAU. DOPGTR constructs the matrix U from the contents of VP and TAU. DSTEQR factors S as Z D1 Z' , where Z is the orthogonal matrix of eigenvectors and D1 is a diagonal matrix with the eigenvalues on the diagonal. D2 is the matrix of eigenvalues computed when Z is not computed. DSTERF computes D3, the matrix of eigenvalues, by the PWK method, which does not yield eigenvectors. DPTEQR factors S as Z4 D4 Z4' , for a symmetric positive definite tridiagonal matrix. D5 is the matrix of eigenvalues computed when Z is not computed. DSTEBZ computes selected eigenvalues. WA1, WA2, and WA3 will denote eigenvalues computed to high absolute accuracy, with different range options. WR will denote eigenvalues computed to high relative accuracy. DSTEIN computes Y, the eigenvectors of S, given the eigenvalues. DSTEDC factors S as Z D1 Z' , where Z is the orthogonal matrix of eigenvectors and D1 is a diagonal matrix with the eigenvalues on the diagonal ('I' option). It may also update an input orthogonal matrix, usually the output from DSYTRD/DORGTR or DSPTRD/DOPGTR ('V' option). It may also just compute eigenvalues ('N' option). DSTEMR factors S as Z D1 Z' , where Z is the orthogonal matrix of eigenvectors and D1 is a diagonal matrix with the eigenvalues on the diagonal ('I' option). DSTEMR uses the Relatively Robust Representation whenever possible. When DCHKST is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified. For each size ("n") and each type of matrix, one matrix will be generated and used to test the symmetric eigenroutines. For each matrix, a number of tests will be performed: (1)  A  V S V'  / ( A n ulp ) DSYTRD( UPLO='U', ... ) (2)  I  UV'  / ( n ulp ) DORGTR( UPLO='U', ... ) (3)  A  V S V'  / ( A n ulp ) DSYTRD( UPLO='L', ... ) (4)  I  UV'  / ( n ulp ) DORGTR( UPLO='L', ... ) (58) Same as 14, but for DSPTRD and DOPGTR. (9)  S  Z D Z'  / ( S n ulp ) DSTEQR('V',...) (10)  I  ZZ'  / ( n ulp ) DSTEQR('V',...) (11)  D1  D2  / ( D1 ulp ) DSTEQR('N',...) (12)  D1  D3  / ( D1 ulp ) DSTERF (13) 0 if the true eigenvalues (computed by sturm count) of S are within THRESH of those in D1. 2*THRESH if they are not. (Tested using DSTECH) For S positive definite, (14)  S  Z4 D4 Z4'  / ( S n ulp ) DPTEQR('V',...) (15)  I  Z4 Z4'  / ( n ulp ) DPTEQR('V',...) (16)  D4  D5  / ( 100 D4 ulp ) DPTEQR('N',...) When S is also diagonally dominant by the factor gamma < 1, (17) max  D4(i)  WR(i)  / ( D4(i) omega ) , i omega = 2 (2n1) ULP (1 + 8 gamma**2) / (1  gamma)**4 DSTEBZ( 'A', 'E', ...) (18)  WA1  D3  / ( D3 ulp ) DSTEBZ( 'A', 'E', ...) (19) ( max { min  WA2(i)WA3(j)  } + i j max { min  WA3(i)WA2(j)  } ) / ( D3 ulp ) i j DSTEBZ( 'I', 'E', ...) (20)  S  Y WA1 Y'  / ( S n ulp ) DSTEBZ, SSTEIN (21)  I  Y Y'  / ( n ulp ) DSTEBZ, SSTEIN (22)  S  Z D Z'  / ( S n ulp ) DSTEDC('I') (23)  I  ZZ'  / ( n ulp ) DSTEDC('I') (24)  S  Z D Z'  / ( S n ulp ) DSTEDC('V') (25)  I  ZZ'  / ( n ulp ) DSTEDC('V') (26)  D1  D2  / ( D1 ulp ) DSTEDC('V') and DSTEDC('N') Test 27 is disabled at the moment because DSTEMR does not guarantee high relatvie accuracy. (27) max  D6(i)  WR(i)  / ( D6(i) omega ) , i omega = 2 (2n1) ULP (1 + 8 gamma**2) / (1  gamma)**4 DSTEMR('V', 'A') (28) max  D6(i)  WR(i)  / ( D6(i) omega ) , i omega = 2 (2n1) ULP (1 + 8 gamma**2) / (1  gamma)**4 DSTEMR('V', 'I') Tests 29 through 34 are disable at present because DSTEMR does not handle partial specturm requests. (29)  S  Z D Z'  / ( S n ulp ) DSTEMR('V', 'I') (30)  I  ZZ'  / ( n ulp ) DSTEMR('V', 'I') (31) ( max { min  WA2(i)WA3(j)  } + i j max { min  WA3(i)WA2(j)  } ) / ( D3 ulp ) i j DSTEMR('N', 'I') vs. SSTEMR('V', 'I') (32)  S  Z D Z'  / ( S n ulp ) DSTEMR('V', 'V') (33)  I  ZZ'  / ( n ulp ) DSTEMR('V', 'V') (34) ( max { min  WA2(i)WA3(j)  } + i j max { min  WA3(i)WA2(j)  } ) / ( D3 ulp ) i j DSTEMR('N', 'V') vs. SSTEMR('V', 'V') (35)  S  Z D Z'  / ( S n ulp ) DSTEMR('V', 'A') (36)  I  ZZ'  / ( n ulp ) DSTEMR('V', 'A') (37) ( max { min  WA2(i)WA3(j)  } + i j max { min  WA3(i)WA2(j)  } ) / ( D3 ulp ) i j DSTEMR('N', 'A') vs. SSTEMR('V', 'A') The "sizes" are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) The zero matrix. (2) The identity matrix. (3) A diagonal matrix with evenly spaced entries 1, ..., ULP and random signs. (ULP = (first number larger than 1)  1 ) (4) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random signs. (5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP and random signs. (6) Same as (4), but multiplied by SQRT( overflow threshold ) (7) Same as (4), but multiplied by SQRT( underflow threshold ) (8) A matrix of the form U' D U, where U is orthogonal and D has evenly spaced entries 1, ..., ULP with random signs on the diagonal. (9) A matrix of the form U' D U, where U is orthogonal and D has geometrically spaced entries 1, ..., ULP with random signs on the diagonal. (10) A matrix of the form U' D U, where U is orthogonal and D has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal. (11) Same as (8), but multiplied by SQRT( overflow threshold ) (12) Same as (8), but multiplied by SQRT( underflow threshold ) (13) Symmetric matrix with random entries chosen from (1,1). (14) Same as (13), but multiplied by SQRT( overflow threshold ) (15) Same as (13), but multiplied by SQRT( underflow threshold ) (16) Same as (8), but diagonal elements are all positive. (17) Same as (9), but diagonal elements are all positive. (18) Same as (10), but diagonal elements are all positive. (19) Same as (16), but multiplied by SQRT( overflow threshold ) (20) Same as (16), but multiplied by SQRT( underflow threshold ) (21) A diagonally dominant tridiagonal matrix with geometrically spaced diagonal entries 1, ..., ULP.
[in]  NSIZES  NSIZES is INTEGER The number of sizes of matrices to use. If it is zero, DCHKST does nothing. It must be at least zero. 
[in]  NN  NN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero. 
[in]  NTYPES  NTYPES is INTEGER The number of elements in DOTYPE. If it is zero, DCHKST does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. . 
[in]  DOTYPE  DOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored. 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to DCHKST to continue the same random number sequence. 
[in]  THRESH  THRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero. 
[in]  NOUNIT  NOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) 
[in,out]  A  A is DOUBLE PRECISION array of dimension ( LDA , max(NN) ) Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used. 
[in]  LDA  LDA is INTEGER The leading dimension of A. It must be at least 1 and at least max( NN ). 
[out]  AP  AP is DOUBLE PRECISION array of dimension( max(NN)*max(NN+1)/2 ) The matrix A stored in packed format. 
[out]  SD  SD is DOUBLE PRECISION array of dimension( max(NN) ) The diagonal of the tridiagonal matrix computed by DSYTRD. On exit, SD and SE contain the tridiagonal form of the matrix in A. 
[out]  SE  SE is DOUBLE PRECISION array of dimension( max(NN) ) The offdiagonal of the tridiagonal matrix computed by DSYTRD. On exit, SD and SE contain the tridiagonal form of the matrix in A. 
[out]  D1  D1 is DOUBLE PRECISION array of dimension( max(NN) ) The eigenvalues of A, as computed by DSTEQR simlutaneously with Z. On exit, the eigenvalues in D1 correspond with the matrix in A. 
[out]  D2  D2 is DOUBLE PRECISION array of dimension( max(NN) ) The eigenvalues of A, as computed by DSTEQR if Z is not computed. On exit, the eigenvalues in D2 correspond with the matrix in A. 
[out]  D3  D3 is DOUBLE PRECISION array of dimension( max(NN) ) The eigenvalues of A, as computed by DSTERF. On exit, the eigenvalues in D3 correspond with the matrix in A. 
[out]  D4  D4 is DOUBLE PRECISION array of dimension( max(NN) ) The eigenvalues of A, as computed by DPTEQR(V). DPTEQR factors S as Z4 D4 Z4* On exit, the eigenvalues in D4 correspond with the matrix in A. 
[out]  D5  D5 is DOUBLE PRECISION array of dimension( max(NN) ) The eigenvalues of A, as computed by DPTEQR(N) when Z is not computed. On exit, the eigenvalues in D4 correspond with the matrix in A. 
[out]  WA1  WA1 is DOUBLE PRECISION array of dimension( max(NN) ) All eigenvalues of A, computed to high absolute accuracy, with different range options. as computed by DSTEBZ. 
[out]  WA2  WA2 is DOUBLE PRECISION array of dimension( max(NN) ) Selected eigenvalues of A, computed to high absolute accuracy, with different range options. as computed by DSTEBZ. Choose random values for IL and IU, and ask for the ILth through IUth eigenvalues. 
[out]  WA3  WA3 is DOUBLE PRECISION array of dimension( max(NN) ) Selected eigenvalues of A, computed to high absolute accuracy, with different range options. as computed by DSTEBZ. Determine the values VL and VU of the ILth and IUth eigenvalues and ask for all eigenvalues in this range. 
[out]  WR  WR is DOUBLE PRECISION array of dimension( max(NN) ) All eigenvalues of A, computed to high absolute accuracy, with different options. as computed by DSTEBZ. 
[out]  U  U is DOUBLE PRECISION array of dimension( LDU, max(NN) ). The orthogonal matrix computed by DSYTRD + DORGTR. 
[in]  LDU  LDU is INTEGER The leading dimension of U, Z, and V. It must be at least 1 and at least max( NN ). 
[out]  V  V is DOUBLE PRECISION array of dimension( LDU, max(NN) ). The Housholder vectors computed by DSYTRD in reducing A to tridiagonal form. The vectors computed with UPLO='U' are in the upper triangle, and the vectors computed with UPLO='L' are in the lower triangle. (As described in DSYTRD, the sub and superdiagonal are not set to 1, although the true Householder vector has a 1 in that position. The routines that use V, such as DORGTR, set those entries to 1 before using them, and then restore them later.) 
[out]  VP  VP is DOUBLE PRECISION array of dimension( max(NN)*max(NN+1)/2 ) The matrix V stored in packed format. 
[out]  TAU  TAU is DOUBLE PRECISION array of dimension( max(NN) ) The Householder factors computed by DSYTRD in reducing A to tridiagonal form. 
[out]  Z  Z is DOUBLE PRECISION array of dimension( LDU, max(NN) ). The orthogonal matrix of eigenvectors computed by DSTEQR, DPTEQR, and DSTEIN. 
[out]  WORK  WORK is DOUBLE PRECISION array of dimension( LWORK ) 
[in]  LWORK  LWORK is INTEGER The number of entries in WORK. This must be at least 1 + 4 * Nmax + 2 * Nmax * lg Nmax + 3 * Nmax**2 where Nmax = max( NN(j), 2 ) and lg = log base 2. 
[out]  IWORK  IWORK is INTEGER array, Workspace. 
[out]  LIWORK  LIWORK is INTEGER The number of entries in IWORK. This must be at least 6 + 6*Nmax + 5 * Nmax * lg Nmax where Nmax = max( NN(j), 2 ) and lg = log base 2. 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (26) The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. 
[out]  INFO  INFO is INTEGER If 0, then everything ran OK. 1: NSIZES < 0 2: Some NN(j) < 0 3: NTYPES < 0 5: THRESH < 0 9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ). 23: LDU < 1 or LDU < NMAX. 29: LWORK too small. If DLATMR, SLATMS, DSYTRD, DORGTR, DSTEQR, SSTERF, or DORMC2 returns an error code, the absolute value of it is returned.  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. NTEST The number of tests performed, or which can be performed so far, for the current matrix. NTESTT The total number of tests performed so far. NBLOCK Blocksize as returned by ENVIR. NMAX Largest value in NN. NMATS The number of matrices generated so far. NERRS The number of tests which have exceeded THRESH so far. COND, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. ULP, ULPINV Finest relative precision and its inverse. RTOVFL, RTUNFL Square roots of the previous 2 values. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) ) 
Definition at line 589 of file dchkst.f.
subroutine dckcsd  (  integer  NM, 
integer, dimension( * )  MVAL,  
integer, dimension( * )  PVAL,  
integer, dimension( * )  QVAL,  
integer  NMATS,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
integer  MMAX,  
double precision, dimension( * )  X,  
double precision, dimension( * )  XF,  
double precision, dimension( * )  U1,  
double precision, dimension( * )  U2,  
double precision, dimension( * )  V1T,  
double precision, dimension( * )  V2T,  
double precision, dimension( * )  THETA,  
integer, dimension( * )  IWORK,  
double precision, dimension( * )  WORK,  
double precision, dimension( * )  RWORK,  
integer  NIN,  
integer  NOUT,  
integer  INFO  
) 
DCKCSD
DCKCSD tests DORCSD: the CSD for an MbyM orthogonal matrix X partitioned as [ X11 X12; X21 X22 ]. X11 is PbyQ.
[in]  NM  NM is INTEGER The number of values of M contained in the vector MVAL. 
[in]  MVAL  MVAL is INTEGER array, dimension (NM) The values of the matrix row dimension M. 
[in]  PVAL  PVAL is INTEGER array, dimension (NM) The values of the matrix row dimension P. 
[in]  QVAL  QVAL is INTEGER array, dimension (NM) The values of the matrix column dimension Q. 
[in]  NMATS  NMATS is INTEGER The number of matrix types to be tested for each combination of matrix dimensions. If NMATS >= NTYPES (the maximum number of matrix types), then all the different types are generated for testing. If NMATS < NTYPES, another input line is read to get the numbers of the matrix types to be used. 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry, the seed of the random number generator. The array elements should be between 0 and 4095, otherwise they will be reduced mod 4096, and ISEED(4) must be odd. On exit, the next seed in the random number sequence after all the test matrices have been generated. 
[in]  THRESH  THRESH is DOUBLE PRECISION The threshold value for the test ratios. A result is included in the output file if RESULT >= THRESH. To have every test ratio printed, use THRESH = 0. 
[in]  MMAX  MMAX is INTEGER The maximum value permitted for M, used in dimensioning the work arrays. 
[out]  X  X is DOUBLE PRECISION array, dimension (MMAX*MMAX) 
[out]  XF  XF is DOUBLE PRECISION array, dimension (MMAX*MMAX) 
[out]  U1  U1 is DOUBLE PRECISION array, dimension (MMAX*MMAX) 
[out]  U2  U2 is DOUBLE PRECISION array, dimension (MMAX*MMAX) 
[out]  V1T  V1T is DOUBLE PRECISION array, dimension (MMAX*MMAX) 
[out]  V2T  V2T is DOUBLE PRECISION array, dimension (MMAX*MMAX) 
[out]  THETA  THETA is DOUBLE PRECISION array, dimension (MMAX) 
[out]  IWORK  IWORK is INTEGER array, dimension (MMAX) 
[out]  WORK  WORK is DOUBLE PRECISION array 
[out]  RWORK  RWORK is DOUBLE PRECISION array 
[in]  NIN  NIN is INTEGER The unit number for input. 
[in]  NOUT  NOUT is INTEGER The unit number for output. 
[out]  INFO  INFO is INTEGER = 0 : successful exit > 0 : If DLAROR returns an error code, the absolute value of it is returned. 
Definition at line 183 of file dckcsd.f.
subroutine dckglm  (  integer  NN, 
integer, dimension( * )  MVAL,  
integer, dimension( * )  PVAL,  
integer, dimension( * )  NVAL,  
integer  NMATS,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
integer  NMAX,  
double precision, dimension( * )  A,  
double precision, dimension( * )  AF,  
double precision, dimension( * )  B,  
double precision, dimension( * )  BF,  
double precision, dimension( * )  X,  
double precision, dimension( * )  WORK,  
double precision, dimension( * )  RWORK,  
integer  NIN,  
integer  NOUT,  
integer  INFO  
) 
DCKGLM
DCKGLM tests DGGGLM  subroutine for solving generalized linear model problem.
[in]  NN  NN is INTEGER The number of values of N, M and P contained in the vectors NVAL, MVAL and PVAL. 
[in]  MVAL  MVAL is INTEGER array, dimension (NN) The values of the matrix column dimension M. 
[in]  PVAL  PVAL is INTEGER array, dimension (NN) The values of the matrix column dimension P. 
[in]  NVAL  NVAL is INTEGER array, dimension (NN) The values of the matrix row dimension N. 
[in]  NMATS  NMATS is INTEGER The number of matrix types to be tested for each combination of matrix dimensions. If NMATS >= NTYPES (the maximum number of matrix types), then all the different types are generated for testing. If NMATS < NTYPES, another input line is read to get the numbers of the matrix types to be used. 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry, the seed of the random number generator. The array elements should be between 0 and 4095, otherwise they will be reduced mod 4096, and ISEED(4) must be odd. On exit, the next seed in the random number sequence after all the test matrices have been generated. 
[in]  THRESH  THRESH is DOUBLE PRECISION The threshold value for the test ratios. A result is included in the output file if RESID >= THRESH. To have every test ratio printed, use THRESH = 0. 
[in]  NMAX  NMAX is INTEGER The maximum value permitted for M or N, used in dimensioning the work arrays. 
[out]  A  A is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  AF  AF is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  B  B is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  BF  BF is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  X  X is DOUBLE PRECISION array, dimension (4*NMAX) 
[out]  RWORK  RWORK is DOUBLE PRECISION array, dimension (NMAX) 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[in]  NIN  NIN is INTEGER The unit number for input. 
[in]  NOUT  NOUT is INTEGER The unit number for output. 
[out]  INFO  INFO is INTEGER = 0 : successful exit > 0 : If DLATMS returns an error code, the absolute value of it is returned. 
Definition at line 166 of file dckglm.f.
subroutine dckgqr  (  integer  NM, 
integer, dimension( * )  MVAL,  
integer  NP,  
integer, dimension( * )  PVAL,  
integer  NN,  
integer, dimension( * )  NVAL,  
integer  NMATS,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
integer  NMAX,  
double precision, dimension( * )  A,  
double precision, dimension( * )  AF,  
double precision, dimension( * )  AQ,  
double precision, dimension( * )  AR,  
double precision, dimension( * )  TAUA,  
double precision, dimension( * )  B,  
double precision, dimension( * )  BF,  
double precision, dimension( * )  BZ,  
double precision, dimension( * )  BT,  
double precision, dimension( * )  BWK,  
double precision, dimension( * )  TAUB,  
double precision, dimension( * )  WORK,  
double precision, dimension( * )  RWORK,  
integer  NIN,  
integer  NOUT,  
integer  INFO  
) 
DCKGQR
DCKGQR tests DGGQRF: GQR factorization for NbyM matrix A and NbyP matrix B, DGGRQF: GRQ factorization for MbyN matrix A and PbyN matrix B.
[in]  NM  NM is INTEGER The number of values of M contained in the vector MVAL. 
[in]  MVAL  MVAL is INTEGER array, dimension (NM) The values of the matrix row(column) dimension M. 
[in]  NP  NP is INTEGER The number of values of P contained in the vector PVAL. 
[in]  PVAL  PVAL is INTEGER array, dimension (NP) The values of the matrix row(column) dimension P. 
[in]  NN  NN is INTEGER The number of values of N contained in the vector NVAL. 
[in]  NVAL  NVAL is INTEGER array, dimension (NN) The values of the matrix column(row) dimension N. 
[in]  NMATS  NMATS is INTEGER The number of matrix types to be tested for each combination of matrix dimensions. If NMATS >= NTYPES (the maximum number of matrix types), then all the different types are generated for testing. If NMATS < NTYPES, another input line is read to get the numbers of the matrix types to be used. 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry, the seed of the random number generator. The array elements should be between 0 and 4095, otherwise they will be reduced mod 4096, and ISEED(4) must be odd. On exit, the next seed in the random number sequence after all the test matrices have been generated. 
[in]  THRESH  THRESH is DOUBLE PRECISION The threshold value for the test ratios. A result is included in the output file if RESULT >= THRESH. To have every test ratio printed, use THRESH = 0. 
[in]  NMAX  NMAX is INTEGER The maximum value permitted for M or N, used in dimensioning the work arrays. 
[out]  A  A is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  AF  AF is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  AQ  AQ is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  AR  AR is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  TAUA  TAUA is DOUBLE PRECISION array, dimension (NMAX) 
[out]  B  B is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  BF  BF is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  BZ  BZ is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  BT  BT is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  BWK  BWK is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  TAUB  TAUB is DOUBLE PRECISION array, dimension (NMAX) 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  RWORK  RWORK is DOUBLE PRECISION array, dimension (NMAX) 
[in]  NIN  NIN is INTEGER The unit number for input. 
[in]  NOUT  NOUT is INTEGER The unit number for output. 
[out]  INFO  INFO is INTEGER = 0 : successful exit > 0 : If DLATMS returns an error code, the absolute value of it is returned. 
Definition at line 209 of file dckgqr.f.
subroutine dckgsv  (  integer  NM, 
integer, dimension( * )  MVAL,  
integer, dimension( * )  PVAL,  
integer, dimension( * )  NVAL,  
integer  NMATS,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
integer  NMAX,  
double precision, dimension( * )  A,  
double precision, dimension( * )  AF,  
double precision, dimension( * )  B,  
double precision, dimension( * )  BF,  
double precision, dimension( * )  U,  
double precision, dimension( * )  V,  
double precision, dimension( * )  Q,  
double precision, dimension( * )  ALPHA,  
double precision, dimension( * )  BETA,  
double precision, dimension( * )  R,  
integer, dimension( * )  IWORK,  
double precision, dimension( * )  WORK,  
double precision, dimension( * )  RWORK,  
integer  NIN,  
integer  NOUT,  
integer  INFO  
) 
DCKGSV
DCKGSV tests DGGSVD: the GSVD for MbyN matrix A and PbyN matrix B.
[in]  NM  NM is INTEGER The number of values of M contained in the vector MVAL. 
[in]  MVAL  MVAL is INTEGER array, dimension (NM) The values of the matrix row dimension M. 
[in]  PVAL  PVAL is INTEGER array, dimension (NP) The values of the matrix row dimension P. 
[in]  NVAL  NVAL is INTEGER array, dimension (NN) The values of the matrix column dimension N. 
[in]  NMATS  NMATS is INTEGER The number of matrix types to be tested for each combination of matrix dimensions. If NMATS >= NTYPES (the maximum number of matrix types), then all the different types are generated for testing. If NMATS < NTYPES, another input line is read to get the numbers of the matrix types to be used. 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry, the seed of the random number generator. The array elements should be between 0 and 4095, otherwise they will be reduced mod 4096, and ISEED(4) must be odd. On exit, the next seed in the random number sequence after all the test matrices have been generated. 
[in]  THRESH  THRESH is DOUBLE PRECISION The threshold value for the test ratios. A result is included in the output file if RESULT >= THRESH. To have every test ratio printed, use THRESH = 0. 
[in]  NMAX  NMAX is INTEGER The maximum value permitted for M or N, used in dimensioning the work arrays. 
[out]  A  A is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  AF  AF is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  B  B is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  BF  BF is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  U  U is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  V  V is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  Q  Q is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  ALPHA  ALPHA is DOUBLE PRECISION array, dimension (NMAX) 
[out]  BETA  BETA is DOUBLE PRECISION array, dimension (NMAX) 
[out]  R  R is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  IWORK  IWORK is INTEGER array, dimension (NMAX) 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  RWORK  RWORK is DOUBLE PRECISION array, dimension (NMAX) 
[in]  NIN  NIN is INTEGER The unit number for input. 
[in]  NOUT  NOUT is INTEGER The unit number for output. 
[out]  INFO  INFO is INTEGER = 0 : successful exit > 0 : If DLATMS returns an error code, the absolute value of it is returned. 
Definition at line 197 of file dckgsv.f.
subroutine dcklse  (  integer  NN, 
integer, dimension( * )  MVAL,  
integer, dimension( * )  PVAL,  
integer, dimension( * )  NVAL,  
integer  NMATS,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
integer  NMAX,  
double precision, dimension( * )  A,  
double precision, dimension( * )  AF,  
double precision, dimension( * )  B,  
double precision, dimension( * )  BF,  
double precision, dimension( * )  X,  
double precision, dimension( * )  WORK,  
double precision, dimension( * )  RWORK,  
integer  NIN,  
integer  NOUT,  
integer  INFO  
) 
DCKLSE
DCKLSE tests DGGLSE  a subroutine for solving linear equality constrained least square problem (LSE).
[in]  NN  NN is INTEGER The number of values of (M,P,N) contained in the vectors (MVAL, PVAL, NVAL). 
[in]  MVAL  MVAL is INTEGER array, dimension (NN) The values of the matrix row(column) dimension M. 
[in]  PVAL  PVAL is INTEGER array, dimension (NN) The values of the matrix row(column) dimension P. 
[in]  NVAL  NVAL is INTEGER array, dimension (NN) The values of the matrix column(row) dimension N. 
[in]  NMATS  NMATS is INTEGER The number of matrix types to be tested for each combination of matrix dimensions. If NMATS >= NTYPES (the maximum number of matrix types), then all the different types are generated for testing. If NMATS < NTYPES, another input line is read to get the numbers of the matrix types to be used. 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry, the seed of the random number generator. The array elements should be between 0 and 4095, otherwise they will be reduced mod 4096, and ISEED(4) must be odd. On exit, the next seed in the random number sequence after all the test matrices have been generated. 
[in]  THRESH  THRESH is DOUBLE PRECISION The threshold value for the test ratios. A result is included in the output file if RESULT >= THRESH. To have every test ratio printed, use THRESH = 0. 
[in]  NMAX  NMAX is INTEGER The maximum value permitted for M or N, used in dimensioning the work arrays. 
[out]  A  A is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  AF  AF is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  B  B is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  BF  BF is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  X  X is DOUBLE PRECISION array, dimension (5*NMAX) 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (NMAX*NMAX) 
[out]  RWORK  RWORK is DOUBLE PRECISION array, dimension (NMAX) 
[in]  NIN  NIN is INTEGER The unit number for input. 
[in]  NOUT  NOUT is INTEGER The unit number for output. 
[out]  INFO  INFO is INTEGER = 0 : successful exit > 0 : If DLATMS returns an error code, the absolute value of it is returned. 
Definition at line 166 of file dcklse.f.
subroutine dcsdts  (  integer  M, 
integer  P,  
integer  Q,  
double precision, dimension( ldx, * )  X,  
double precision, dimension( ldx, * )  XF,  
integer  LDX,  
double precision, dimension( ldu1, * )  U1,  
integer  LDU1,  
double precision, dimension( ldu2, * )  U2,  
integer  LDU2,  
double precision, dimension( ldv1t, * )  V1T,  
integer  LDV1T,  
double precision, dimension( ldv2t, * )  V2T,  
integer  LDV2T,  
double precision, dimension( * )  THETA,  
integer, dimension( * )  IWORK,  
double precision, dimension( lwork )  WORK,  
integer  LWORK,  
double precision, dimension( * )  RWORK,  
double precision, dimension( 9 )  RESULT  
) 
DCSDTS
DCSDTS tests DORCSD, which, given an MbyM partitioned orthogonal matrix X, Q MQ X = [ X11 X12 ] P , [ X21 X22 ] MP computes the CSD [ U1 ]**T * [ X11 X12 ] * [ V1 ] [ U2 ] [ X21 X22 ] [ V2 ] [ I 0 0  0 0 0 ] [ 0 C 0  0 S 0 ] [ 0 0 0  0 0 I ] = [] = [ D11 D12 ] . [ 0 0 0  I 0 0 ] [ D21 D22 ] [ 0 S 0  0 C 0 ] [ 0 0 I  0 0 0 ]
[in]  M  M is INTEGER The number of rows of the matrix X. M >= 0. 
[in]  P  P is INTEGER The number of rows of the matrix X11. P >= 0. 
[in]  Q  Q is INTEGER The number of columns of the matrix X11. Q >= 0. 
[in]  X  X is DOUBLE PRECISION array, dimension (LDX,M) The MbyM matrix X. 
[out]  XF  XF is DOUBLE PRECISION array, dimension (LDX,M) Details of the CSD of X, as returned by DORCSD; see DORCSD for further details. 
[in]  LDX  LDX is INTEGER The leading dimension of the arrays X and XF. LDX >= max( 1,M ). 
[out]  U1  U1 is DOUBLE PRECISION array, dimension(LDU1,P) The PbyP orthogonal matrix U1. 
[in]  LDU1  LDU1 is INTEGER The leading dimension of the array U1. LDU >= max(1,P). 
[out]  U2  U2 is DOUBLE PRECISION array, dimension(LDU2,MP) The (MP)by(MP) orthogonal matrix U2. 
[in]  LDU2  LDU2 is INTEGER The leading dimension of the array U2. LDU >= max(1,MP). 
[out]  V1T  V1T is DOUBLE PRECISION array, dimension(LDV1T,Q) The QbyQ orthogonal matrix V1T. 
[in]  LDV1T  LDV1T is INTEGER The leading dimension of the array V1T. LDV1T >= max(1,Q). 
[out]  V2T  V2T is DOUBLE PRECISION array, dimension(LDV2T,MQ) The (MQ)by(MQ) orthogonal matrix V2T. 
[in]  LDV2T  LDV2T is INTEGER The leading dimension of the array V2T. LDV2T >= max(1,MQ). 
[out]  THETA  THETA is DOUBLE PRECISION array, dimension MIN(P,MP,Q,MQ) The CS values of X; the essentially diagonal matrices C and S are constructed from THETA; see subroutine DORCSD for details. 
[out]  IWORK  IWORK is INTEGER array, dimension (M) 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER The dimension of the array WORK 
[out]  RWORK  RWORK is DOUBLE PRECISION array 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (9) The test ratios: RESULT(1) = norm( U1'*X11*V1  D11 ) / ( MAX(1,P,Q)*EPS2 ) RESULT(2) = norm( U1'*X12*V2  D12 ) / ( MAX(1,P,MQ)*EPS2 ) RESULT(3) = norm( U2'*X21*V1  D21 ) / ( MAX(1,MP,Q)*EPS2 ) RESULT(4) = norm( U2'*X22*V2  D22 ) / ( MAX(1,MP,MQ)*EPS2 ) RESULT(5) = norm( I  U1'*U1 ) / ( MAX(1,P)*ULP ) RESULT(6) = norm( I  U2'*U2 ) / ( MAX(1,MP)*ULP ) RESULT(7) = norm( I  V1T'*V1T ) / ( MAX(1,Q)*ULP ) RESULT(8) = norm( I  V2T'*V2T ) / ( MAX(1,MQ)*ULP ) RESULT(9) = 0 if THETA is in increasing order and all angles are in [0,pi/2]; = ULPINV otherwise. ( EPS2 = MAX( norm( I  X'*X ) / M, ULP ). ) 
Definition at line 203 of file dcsdts.f.
subroutine ddrges  (  integer  NSIZES, 
integer, dimension( * )  NN,  
integer  NTYPES,  
logical, dimension( * )  DOTYPE,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
integer  NOUNIT,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( lda, * )  B,  
double precision, dimension( lda, * )  S,  
double precision, dimension( lda, * )  T,  
double precision, dimension( ldq, * )  Q,  
integer  LDQ,  
double precision, dimension( ldq, * )  Z,  
double precision, dimension( * )  ALPHAR,  
double precision, dimension( * )  ALPHAI,  
double precision, dimension( * )  BETA,  
double precision, dimension( * )  WORK,  
integer  LWORK,  
double precision, dimension( 13 )  RESULT,  
logical, dimension( * )  BWORK,  
integer  INFO  
) 
DDRGES
DDRGES checks the nonsymmetric generalized eigenvalue (Schur form) problem driver DGGES. DGGES factors A and B as Q S Z' and Q T Z' , where ' means transpose, T is upper triangular, S is in generalized Schur form (block upper triangular, with 1x1 and 2x2 blocks on the diagonal, the 2x2 blocks corresponding to complex conjugate pairs of generalized eigenvalues), and Q and Z are orthogonal. It also computes the generalized eigenvalues (alpha(j),beta(j)), j=1,...,n, Thus, w(j) = alpha(j)/beta(j) is a root of the characteristic equation det( A  w(j) B ) = 0 Optionally it also reorder the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal block of the Schur forms. When DDRGES is called, a number of matrix "sizes" ("N's") and a number of matrix "TYPES" are specified. For each size ("N") and each TYPE of matrix, a pair of matrices (A, B) will be generated and used for testing. For each matrix pair, the following 13 tests will be performed and compared with the threshhold THRESH except the tests (5), (11) and (13). (1)  A  Q S Z'  / ( A n ulp ) (no sorting of eigenvalues) (2)  B  Q T Z'  / ( B n ulp ) (no sorting of eigenvalues) (3)  I  QQ'  / ( n ulp ) (no sorting of eigenvalues) (4)  I  ZZ'  / ( n ulp ) (no sorting of eigenvalues) (5) if A is in Schur form (i.e. quasitriangular form) (no sorting of eigenvalues) (6) if eigenvalues = diagonal blocks of the Schur form (S, T), i.e., test the maximum over j of D(j) where: if alpha(j) is real: alpha(j)  S(j,j) beta(j)  T(j,j) D(j) =  +  max(alpha(j),S(j,j)) max(beta(j),T(j,j)) if alpha(j) is complex:  det( s S  w T )  D(j) =  ulp max( s norm(S), w norm(T) )*norm( s S  w T ) and S and T are here the 2 x 2 diagonal blocks of S and T corresponding to the jth and j+1th eigenvalues. (no sorting of eigenvalues) (7)  (A,B)  Q (S,T) Z'  / (  (A,B)  n ulp ) (with sorting of eigenvalues). (8)  I  QQ'  / ( n ulp ) (with sorting of eigenvalues). (9)  I  ZZ'  / ( n ulp ) (with sorting of eigenvalues). (10) if A is in Schur form (i.e. quasitriangular form) (with sorting of eigenvalues). (11) if eigenvalues = diagonal blocks of the Schur form (S, T), i.e. test the maximum over j of D(j) where: if alpha(j) is real: alpha(j)  S(j,j) beta(j)  T(j,j) D(j) =  +  max(alpha(j),S(j,j)) max(beta(j),T(j,j)) if alpha(j) is complex:  det( s S  w T )  D(j) =  ulp max( s norm(S), w norm(T) )*norm( s S  w T ) and S and T are here the 2 x 2 diagonal blocks of S and T corresponding to the jth and j+1th eigenvalues. (with sorting of eigenvalues). (12) if sorting worked and SDIM is the number of eigenvalues which were SELECTed. Test Matrices ============= The sizes of the test matrices are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) ( 0, 0 ) (a pair of zero matrices) (2) ( I, 0 ) (an identity and a zero matrix) (3) ( 0, I ) (an identity and a zero matrix) (4) ( I, I ) (a pair of identity matrices) t t (5) ( J , J ) (a pair of transposed Jordan blocks) t ( I 0 ) (6) ( X, Y ) where X = ( J 0 ) and Y = ( t ) ( 0 I ) ( 0 J ) and I is a k x k identity and J a (k+1)x(k+1) Jordan block; k=(N1)/2 (7) ( D, I ) where D is diag( 0, 1,..., N1 ) (a diagonal matrix with those diagonal entries.) (8) ( I, D ) (9) ( big*D, small*I ) where "big" is near overflow and small=1/big (10) ( small*D, big*I ) (11) ( big*I, small*D ) (12) ( small*I, big*D ) (13) ( big*D, big*I ) (14) ( small*D, small*I ) (15) ( D1, D2 ) where D1 is diag( 0, 0, 1, ..., N3, 0 ) and D2 is diag( 0, N3, N4,..., 1, 0, 0 ) t t (16) Q ( J , J ) Z where Q and Z are random orthogonal matrices. (17) Q ( T1, T2 ) Z where T1 and T2 are upper triangular matrices with random O(1) entries above the diagonal and diagonal entries diag(T1) = ( 0, 0, 1, ..., N3, 0 ) and diag(T2) = ( 0, N3, N4,..., 1, 0, 0 ) (18) Q ( T1, T2 ) Z diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 ) diag(T2) = ( 0, 1, 0, 1,..., 1, 0 ) s = machine precision. (19) Q ( T1, T2 ) Z diag(T1)=( 0,0,1,1, 1d, ..., 1(N5)*d=s, 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 ) N5 (20) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) (21) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, r1, r2, ..., r(N4), 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) where r1,..., r(N4) are random. (22) Q ( big*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (23) Q ( small*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (24) Q ( small*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (25) Q ( big*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (26) Q ( T1, T2 ) Z where T1 and T2 are random uppertriangular matrices.
[in]  NSIZES  NSIZES is INTEGER The number of sizes of matrices to use. If it is zero, DDRGES does nothing. NSIZES >= 0. 
[in]  NN  NN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. NN >= 0. 
[in]  NTYPES  NTYPES is INTEGER The number of elements in DOTYPE. If it is zero, DDRGES does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A on input. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. . 
[in]  DOTYPE  DOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored. 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to DDRGES to continue the same random number sequence. 
[in]  THRESH  THRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. THRESH >= 0. 
[in]  NOUNIT  NOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) 
[in,out]  A  A is DOUBLE PRECISION array, dimension(LDA, max(NN)) Used to hold the original A matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. 
[in]  LDA  LDA is INTEGER The leading dimension of A, B, S, and T. It must be at least 1 and at least max( NN ). 
[in,out]  B  B is DOUBLE PRECISION array, dimension(LDA, max(NN)) Used to hold the original B matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. 
[out]  S  S is DOUBLE PRECISION array, dimension (LDA, max(NN)) The Schur form matrix computed from A by DGGES. On exit, S contains the Schur form matrix corresponding to the matrix in A. 
[out]  T  T is DOUBLE PRECISION array, dimension (LDA, max(NN)) The upper triangular matrix computed from B by DGGES. 
[out]  Q  Q is DOUBLE PRECISION array, dimension (LDQ, max(NN)) The (left) orthogonal matrix computed by DGGES. 
[in]  LDQ  LDQ is INTEGER The leading dimension of Q and Z. It must be at least 1 and at least max( NN ). 
[out]  Z  Z is DOUBLE PRECISION array, dimension( LDQ, max(NN) ) The (right) orthogonal matrix computed by DGGES. 
[out]  ALPHAR  ALPHAR is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  ALPHAI  ALPHAI is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  BETA  BETA is DOUBLE PRECISION array, dimension (max(NN)) The generalized eigenvalues of (A,B) computed by DGGES. ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the kth generalized eigenvalue of A and B. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER The dimension of the array WORK. LWORK >= MAX( 10*(N+1), 3*N*N ), where N is the largest matrix dimension. 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (15) The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. 
[out]  BWORK  BWORK is LOGICAL array, dimension (N) 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value. > 0: A routine returned an error code. INFO is the absolute value of the INFO value returned. 
Definition at line 401 of file ddrges.f.
subroutine ddrgev  (  integer  NSIZES, 
integer, dimension( * )  NN,  
integer  NTYPES,  
logical, dimension( * )  DOTYPE,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
integer  NOUNIT,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( lda, * )  B,  
double precision, dimension( lda, * )  S,  
double precision, dimension( lda, * )  T,  
double precision, dimension( ldq, * )  Q,  
integer  LDQ,  
double precision, dimension( ldq, * )  Z,  
double precision, dimension( ldqe, * )  QE,  
integer  LDQE,  
double precision, dimension( * )  ALPHAR,  
double precision, dimension( * )  ALPHAI,  
double precision, dimension( * )  BETA,  
double precision, dimension( * )  ALPHR1,  
double precision, dimension( * )  ALPHI1,  
double precision, dimension( * )  BETA1,  
double precision, dimension( * )  WORK,  
integer  LWORK,  
double precision, dimension( * )  RESULT,  
integer  INFO  
) 
DDRGEV
DDRGEV checks the nonsymmetric generalized eigenvalue problem driver routine DGGEV. DGGEV computes for a pair of nbyn nonsymmetric matrices (A,B) the generalized eigenvalues and, optionally, the left and right eigenvectors. A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A  w*B is singular. It is usually represented as the pair (alpha,beta), as there is reasonalbe interpretation for beta=0, and even for both being zero. A right generalized eigenvector corresponding to a generalized eigenvalue w for a pair of matrices (A,B) is a vector r such that (A  wB) * r = 0. A left generalized eigenvector is a vector l such that l**H * (A  wB) = 0, where l**H is the conjugatetranspose of l. When DDRGEV is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified. For each size ("n") and each type of matrix, a pair of matrices (A, B) will be generated and used for testing. For each matrix pair, the following tests will be performed and compared with the threshhold THRESH. Results from DGGEV: (1) max over all left eigenvalue/vector pairs (alpha/beta,l) of  VL**H * (beta A  alpha B) /( ulp max(beta A, alpha B) ) where VL**H is the conjugatetranspose of VL. (2)  VL(i)  1  / ulp and whether largest component real VL(i) denotes the ith column of VL. (3) max over all left eigenvalue/vector pairs (alpha/beta,r) of  (beta A  alpha B) * VR  / ( ulp max(beta A, alpha B) ) (4)  VR(i)  1  / ulp and whether largest component real VR(i) denotes the ith column of VR. (5) W(full) = W(partial) W(full) denotes the eigenvalues computed when both l and r are also computed, and W(partial) denotes the eigenvalues computed when only W, only W and r, or only W and l are computed. (6) VL(full) = VL(partial) VL(full) denotes the left eigenvectors computed when both l and r are computed, and VL(partial) denotes the result when only l is computed. (7) VR(full) = VR(partial) VR(full) denotes the right eigenvectors computed when both l and r are also computed, and VR(partial) denotes the result when only l is computed. Test Matrices   The sizes of the test matrices are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) ( 0, 0 ) (a pair of zero matrices) (2) ( I, 0 ) (an identity and a zero matrix) (3) ( 0, I ) (an identity and a zero matrix) (4) ( I, I ) (a pair of identity matrices) t t (5) ( J , J ) (a pair of transposed Jordan blocks) t ( I 0 ) (6) ( X, Y ) where X = ( J 0 ) and Y = ( t ) ( 0 I ) ( 0 J ) and I is a k x k identity and J a (k+1)x(k+1) Jordan block; k=(N1)/2 (7) ( D, I ) where D is diag( 0, 1,..., N1 ) (a diagonal matrix with those diagonal entries.) (8) ( I, D ) (9) ( big*D, small*I ) where "big" is near overflow and small=1/big (10) ( small*D, big*I ) (11) ( big*I, small*D ) (12) ( small*I, big*D ) (13) ( big*D, big*I ) (14) ( small*D, small*I ) (15) ( D1, D2 ) where D1 is diag( 0, 0, 1, ..., N3, 0 ) and D2 is diag( 0, N3, N4,..., 1, 0, 0 ) t t (16) Q ( J , J ) Z where Q and Z are random orthogonal matrices. (17) Q ( T1, T2 ) Z where T1 and T2 are upper triangular matrices with random O(1) entries above the diagonal and diagonal entries diag(T1) = ( 0, 0, 1, ..., N3, 0 ) and diag(T2) = ( 0, N3, N4,..., 1, 0, 0 ) (18) Q ( T1, T2 ) Z diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 ) diag(T2) = ( 0, 1, 0, 1,..., 1, 0 ) s = machine precision. (19) Q ( T1, T2 ) Z diag(T1)=( 0,0,1,1, 1d, ..., 1(N5)*d=s, 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 ) N5 (20) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) (21) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, r1, r2, ..., r(N4), 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) where r1,..., r(N4) are random. (22) Q ( big*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (23) Q ( small*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (24) Q ( small*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (25) Q ( big*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (26) Q ( T1, T2 ) Z where T1 and T2 are random uppertriangular matrices.
[in]  NSIZES  NSIZES is INTEGER The number of sizes of matrices to use. If it is zero, DDRGES does nothing. NSIZES >= 0. 
[in]  NN  NN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. NN >= 0. 
[in]  NTYPES  NTYPES is INTEGER The number of elements in DOTYPE. If it is zero, DDRGES does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. . 
[in]  DOTYPE  DOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored. 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to DDRGES to continue the same random number sequence. 
[in]  THRESH  THRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero. 
[in]  NOUNIT  NOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IERR not equal to 0.) 
[in,out]  A  A is DOUBLE PRECISION array, dimension(LDA, max(NN)) Used to hold the original A matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. 
[in]  LDA  LDA is INTEGER The leading dimension of A, B, S, and T. It must be at least 1 and at least max( NN ). 
[in,out]  B  B is DOUBLE PRECISION array, dimension(LDA, max(NN)) Used to hold the original B matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. 
[out]  S  S is DOUBLE PRECISION array, dimension (LDA, max(NN)) The Schur form matrix computed from A by DGGES. On exit, S contains the Schur form matrix corresponding to the matrix in A. 
[out]  T  T is DOUBLE PRECISION array, dimension (LDA, max(NN)) The upper triangular matrix computed from B by DGGES. 
[out]  Q  Q is DOUBLE PRECISION array, dimension (LDQ, max(NN)) The (left) eigenvectors matrix computed by DGGEV. 
[in]  LDQ  LDQ is INTEGER The leading dimension of Q and Z. It must be at least 1 and at least max( NN ). 
[out]  Z  Z is DOUBLE PRECISION array, dimension( LDQ, max(NN) ) The (right) orthogonal matrix computed by DGGES. 
[out]  QE  QE is DOUBLE PRECISION array, dimension( LDQ, max(NN) ) QE holds the computed right or left eigenvectors. 
[in]  LDQE  LDQE is INTEGER The leading dimension of QE. LDQE >= max(1,max(NN)). 
[out]  ALPHAR  ALPHAR is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  ALPHAI  ALPHAI is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  BETA  BETA is DOUBLE PRECISION array, dimension (max(NN)) The generalized eigenvalues of (A,B) computed by DGGEV. ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the kth generalized eigenvalue of A and B. 
[out]  ALPHR1  ALPHR1 is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  ALPHI1  ALPHI1 is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  BETA1  BETA1 is DOUBLE PRECISION array, dimension (max(NN)) Like ALPHAR, ALPHAI, BETA, these arrays contain the eigenvalues of A and B, but those computed when DGGEV only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER The number of entries in WORK. LWORK >= MAX( 8*N, N*(N+1) ). 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (2) The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value. > 0: A routine returned an error code. INFO is the absolute value of the INFO value returned. 
Definition at line 406 of file ddrgev.f.
subroutine ddrgsx  (  integer  NSIZE, 
integer  NCMAX,  
double precision  THRESH,  
integer  NIN,  
integer  NOUT,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( lda, * )  B,  
double precision, dimension( lda, * )  AI,  
double precision, dimension( lda, * )  BI,  
double precision, dimension( lda, * )  Z,  
double precision, dimension( lda, * )  Q,  
double precision, dimension( * )  ALPHAR,  
double precision, dimension( * )  ALPHAI,  
double precision, dimension( * )  BETA,  
double precision, dimension( ldc, * )  C,  
integer  LDC,  
double precision, dimension( * )  S,  
double precision, dimension( * )  WORK,  
integer  LWORK,  
integer, dimension( * )  IWORK,  
integer  LIWORK,  
logical, dimension( * )  BWORK,  
integer  INFO  
) 
DDRGSX
DDRGSX checks the nonsymmetric generalized eigenvalue (Schur form) problem expert driver DGGESX. DGGESX factors A and B as Q S Z' and Q T Z', where ' means transpose, T is upper triangular, S is in generalized Schur form (block upper triangular, with 1x1 and 2x2 blocks on the diagonal, the 2x2 blocks corresponding to complex conjugate pairs of generalized eigenvalues), and Q and Z are orthogonal. It also computes the generalized eigenvalues (alpha(1),beta(1)), ..., (alpha(n),beta(n)). Thus, w(j) = alpha(j)/beta(j) is a root of the characteristic equation det( A  w(j) B ) = 0 Optionally it also reorders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal block of the Schur forms; computes a reciprocal condition number for the average of the selected eigenvalues; and computes a reciprocal condition number for the right and left deflating subspaces corresponding to the selected eigenvalues. When DDRGSX is called with NSIZE > 0, five (5) types of builtin matrix pairs are used to test the routine DGGESX. When DDRGSX is called with NSIZE = 0, it reads in test matrix data to test DGGESX. For each matrix pair, the following tests will be performed and compared with the threshhold THRESH except for the tests (7) and (9): (1)  A  Q S Z'  / ( A n ulp ) (2)  B  Q T Z'  / ( B n ulp ) (3)  I  QQ'  / ( n ulp ) (4)  I  ZZ'  / ( n ulp ) (5) if A is in Schur form (i.e. quasitriangular form) (6) maximum over j of D(j) where: if alpha(j) is real: alpha(j)  S(j,j) beta(j)  T(j,j) D(j) =  +  max(alpha(j),S(j,j)) max(beta(j),T(j,j)) if alpha(j) is complex:  det( s S  w T )  D(j) =  ulp max( s norm(S), w norm(T) )*norm( s S  w T ) and S and T are here the 2 x 2 diagonal blocks of S and T corresponding to the jth and j+1th eigenvalues. (7) if sorting worked and SDIM is the number of eigenvalues which were selected. (8) the estimated value DIF does not differ from the true values of Difu and Difl more than a factor 10*THRESH. If the estimate DIF equals zero the corresponding true values of Difu and Difl should be less than EPS*norm(A, B). If the true value of Difu and Difl equal zero, the estimate DIF should be less than EPS*norm(A, B). (9) If INFO = N+3 is returned by DGGESX, the reordering "failed" and we check that DIF = PL = PR = 0 and that the true value of Difu and Difl is < EPS*norm(A, B). We count the events when INFO=N+3. For readin test matrices, the above tests are run except that the exact value for DIF (and PL) is input data. Additionally, there is one more test run for readin test matrices: (10) the estimated value PL does not differ from the true value of PLTRU more than a factor THRESH. If the estimate PL equals zero the corresponding true value of PLTRU should be less than EPS*norm(A, B). If the true value of PLTRU equal zero, the estimate PL should be less than EPS*norm(A, B). Note that for the builtin tests, a total of 10*NSIZE*(NSIZE1) matrix pairs are generated and tested. NSIZE should be kept small. SVD (routine DGESVD) is used for computing the true value of DIF_u and DIF_l when testing the builtin test problems. Builtin Test Matrices ====================== All builtin test matrices are the 2 by 2 block of triangular matrices A = [ A11 A12 ] and B = [ B11 B12 ] [ A22 ] [ B22 ] where for different type of A11 and A22 are given as the following. A12 and B12 are chosen so that the generalized Sylvester equation A11*R  L*A22 = A12 B11*R  L*B22 = B12 have prescribed solution R and L. Type 1: A11 = J_m(1,1) and A_22 = J_k(1a,1). B11 = I_m, B22 = I_k where J_k(a,b) is the kbyk Jordan block with ``a'' on diagonal and ``b'' on superdiagonal. Type 2: A11 = (a_ij) = ( 2(.5sin(i)) ) and B11 = (b_ij) = ( 2(.5sin(ij)) ) for i=1,...,m, j=i,...,m A22 = (a_ij) = ( 2(.5sin(i+j)) ) and B22 = (b_ij) = ( 2(.5sin(ij)) ) for i=m+1,...,k, j=i,...,k Type 3: A11, A22 and B11, B22 are chosen as for Type 2, but each second diagonal block in A_11 and each third diagonal block in A_22 are made as 2 by 2 blocks. Type 4: A11 = ( 20(.5  sin(ij)) ) and B22 = ( 2(.5  sin(i+j)) ) for i=1,...,m, j=1,...,m and A22 = ( 20(.5  sin(i+j)) ) and B22 = ( 2(.5  sin(ij)) ) for i=m+1,...,k, j=m+1,...,k Type 5: (A,B) and have potentially close or common eigenvalues and very large departure from block diagonality A_11 is chosen as the m x m leading submatrix of A_1:  1 b   b 1   1+d b   b 1+d  A_1 =  d 1   1 d   d 1   1 d   1  and A_22 is chosen as the k x k leading submatrix of A_2:  1 b   b 1   1d b   b 1d  A_2 =  d 1+b   1b d   d 1+b   1+b d   1d  and matrix B are chosen as identity matrices (see DLATM5).
[in]  NSIZE  NSIZE is INTEGER The maximum size of the matrices to use. NSIZE >= 0. If NSIZE = 0, no builtin tests matrices are used, but readin test matrices are used to test DGGESX. 
[in]  NCMAX  NCMAX is INTEGER Maximum allowable NMAX for generating Kroneker matrix in call to DLAKF2 
[in]  THRESH  THRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. THRESH >= 0. 
[in]  NIN  NIN is INTEGER The FORTRAN unit number for reading in the data file of problems to solve. 
[in]  NOUT  NOUT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) 
[out]  A  A is DOUBLE PRECISION array, dimension (LDA, NSIZE) Used to store the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used. 
[in]  LDA  LDA is INTEGER The leading dimension of A, B, AI, BI, Z and Q, LDA >= max( 1, NSIZE ). For the readin test, LDA >= max( 1, N ), N is the size of the test matrices. 
[out]  B  B is DOUBLE PRECISION array, dimension (LDA, NSIZE) Used to store the matrix whose eigenvalues are to be computed. On exit, B contains the last matrix actually used. 
[out]  AI  AI is DOUBLE PRECISION array, dimension (LDA, NSIZE) Copy of A, modified by DGGESX. 
[out]  BI  BI is DOUBLE PRECISION array, dimension (LDA, NSIZE) Copy of B, modified by DGGESX. 
[out]  Z  Z is DOUBLE PRECISION array, dimension (LDA, NSIZE) Z holds the left Schur vectors computed by DGGESX. 
[out]  Q  Q is DOUBLE PRECISION array, dimension (LDA, NSIZE) Q holds the right Schur vectors computed by DGGESX. 
[out]  ALPHAR  ALPHAR is DOUBLE PRECISION array, dimension (NSIZE) 
[out]  ALPHAI  ALPHAI is DOUBLE PRECISION array, dimension (NSIZE) 
[out]  BETA  BETA is DOUBLE PRECISION array, dimension (NSIZE) On exit, (ALPHAR + ALPHAI*i)/BETA are the eigenvalues. 
[out]  C  C is DOUBLE PRECISION array, dimension (LDC, LDC) Store the matrix generated by subroutine DLAKF2, this is the matrix formed by Kronecker products used for estimating DIF. 
[in]  LDC  LDC is INTEGER The leading dimension of C. LDC >= max(1, LDA*LDA/2 ). 
[out]  S  S is DOUBLE PRECISION array, dimension (LDC) Singular values of C 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER The dimension of the array WORK. LWORK >= MAX( 5*NSIZE*NSIZE/2  2, 10*(NSIZE+1) ) 
[out]  IWORK  IWORK is INTEGER array, dimension (LIWORK) 
[in]  LIWORK  LIWORK is INTEGER The dimension of the array IWORK. LIWORK >= NSIZE + 6. 
[out]  BWORK  BWORK is LOGICAL array, dimension (LDA) 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value. > 0: A routine returned an error code. 
Definition at line 358 of file ddrgsx.f.
subroutine ddrgvx  (  integer  NSIZE, 
double precision  THRESH,  
integer  NIN,  
integer  NOUT,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( lda, * )  B,  
double precision, dimension( lda, * )  AI,  
double precision, dimension( lda, * )  BI,  
double precision, dimension( * )  ALPHAR,  
double precision, dimension( * )  ALPHAI,  
double precision, dimension( * )  BETA,  
double precision, dimension( lda, * )  VL,  
double precision, dimension( lda, * )  VR,  
integer  ILO,  
integer  IHI,  
double precision, dimension( * )  LSCALE,  
double precision, dimension( * )  RSCALE,  
double precision, dimension( * )  S,  
double precision, dimension( * )  DTRU,  
double precision, dimension( * )  DIF,  
double precision, dimension( * )  DIFTRU,  
double precision, dimension( * )  WORK,  
integer  LWORK,  
integer, dimension( * )  IWORK,  
integer  LIWORK,  
double precision, dimension( 4 )  RESULT,  
logical, dimension( * )  BWORK,  
integer  INFO  
) 
DDRGVX
DDRGVX checks the nonsymmetric generalized eigenvalue problem expert driver DGGEVX. DGGEVX computes the generalized eigenvalues, (optionally) the left and/or right eigenvectors, (optionally) computes a balancing transformation to improve the conditioning, and (optionally) reciprocal condition numbers for the eigenvalues and eigenvectors. When DDRGVX is called with NSIZE > 0, two types of test matrix pairs are generated by the subroutine DLATM6 and test the driver DGGEVX. The test matrices have the known exact condition numbers for eigenvalues. For the condition numbers of the eigenvectors corresponding the first and last eigenvalues are also know ``exactly'' (see DLATM6). For each matrix pair, the following tests will be performed and compared with the threshhold THRESH. (1) max over all left eigenvalue/vector pairs (beta/alpha,l) of  l**H * (beta A  alpha B)  / ( ulp max( beta A, alpha B ) ) where l**H is the conjugate tranpose of l. (2) max over all right eigenvalue/vector pairs (beta/alpha,r) of  (beta A  alpha B) r  / ( ulp max( beta A, alpha B ) ) (3) The condition number S(i) of eigenvalues computed by DGGEVX differs less than a factor THRESH from the exact S(i) (see DLATM6). (4) DIF(i) computed by DTGSNA differs less than a factor 10*THRESH from the exact value (for the 1st and 5th vectors only). Test Matrices ============= Two kinds of test matrix pairs (A, B) = inverse(YH) * (Da, Db) * inverse(X) are used in the tests: 1: Da = 1+a 0 0 0 0 Db = 1 0 0 0 0 0 2+a 0 0 0 0 1 0 0 0 0 0 3+a 0 0 0 0 1 0 0 0 0 0 4+a 0 0 0 0 1 0 0 0 0 0 5+a , 0 0 0 0 1 , and 2: Da = 1 1 0 0 0 Db = 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1+a 1+b 0 0 0 1 0 0 0 0 1b 1+a , 0 0 0 0 1 . In both cases the same inverse(YH) and inverse(X) are used to compute (A, B), giving the exact eigenvectors to (A,B) as (YH, X): YH: = 1 0 y y y X = 1 0 x x x 0 1 y y y 0 1 x x x 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1, 0 0 0 0 1 , where a, b, x and y will have all values independently of each other from { sqrt(sqrt(ULP)), 0.1, 1, 10, 1/sqrt(sqrt(ULP)) }.
[in]  NSIZE  NSIZE is INTEGER The number of sizes of matrices to use. NSIZE must be at least zero. If it is zero, no randomly generated matrices are tested, but any test matrices read from NIN will be tested. 
[in]  THRESH  THRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero. 
[in]  NIN  NIN is INTEGER The FORTRAN unit number for reading in the data file of problems to solve. 
[in]  NOUT  NOUT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) 
[out]  A  A is DOUBLE PRECISION array, dimension (LDA, NSIZE) Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used. 
[in]  LDA  LDA is INTEGER The leading dimension of A, B, AI, BI, Ao, and Bo. It must be at least 1 and at least NSIZE. 
[out]  B  B is DOUBLE PRECISION array, dimension (LDA, NSIZE) Used to hold the matrix whose eigenvalues are to be computed. On exit, B contains the last matrix actually used. 
[out]  AI  AI is DOUBLE PRECISION array, dimension (LDA, NSIZE) Copy of A, modified by DGGEVX. 
[out]  BI  BI is DOUBLE PRECISION array, dimension (LDA, NSIZE) Copy of B, modified by DGGEVX. 
[out]  ALPHAR  ALPHAR is DOUBLE PRECISION array, dimension (NSIZE) 
[out]  ALPHAI  ALPHAI is DOUBLE PRECISION array, dimension (NSIZE) 
[out]  BETA  BETA is DOUBLE PRECISION array, dimension (NSIZE) On exit, (ALPHAR + ALPHAI*i)/BETA are the eigenvalues. 
[out]  VL  VL is DOUBLE PRECISION array, dimension (LDA, NSIZE) VL holds the left eigenvectors computed by DGGEVX. 
[out]  VR  VR is DOUBLE PRECISION array, dimension (LDA, NSIZE) VR holds the right eigenvectors computed by DGGEVX. 
[out]  ILO  ILO is INTEGER 
[out]  IHI  IHI is INTEGER 
[out]  LSCALE  LSCALE is DOUBLE PRECISION array, dimension (N) 
[out]  RSCALE  RSCALE is DOUBLE PRECISION array, dimension (N) 
[out]  S  S is DOUBLE PRECISION array, dimension (N) 
[out]  DTRU  DTRU is DOUBLE PRECISION array, dimension (N) 
[out]  DIF  DIF is DOUBLE PRECISION array, dimension (N) 
[out]  DIFTRU  DIFTRU is DOUBLE PRECISION array, dimension (N) 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER Leading dimension of WORK. LWORK >= 2*N*N+12*N+16. 
[out]  IWORK  IWORK is INTEGER array, dimension (LIWORK) 
[in]  LIWORK  LIWORK is INTEGER Leading dimension of IWORK. Must be at least N+6. 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (4) 
[out]  BWORK  BWORK is LOGICAL array, dimension (N) 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value. > 0: A routine returned an error code. 
Definition at line 298 of file ddrgvx.f.
subroutine ddrvbd  (  integer  NSIZES, 
integer, dimension( * )  MM,  
integer, dimension( * )  NN,  
integer  NTYPES,  
logical, dimension( * )  DOTYPE,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( ldvt, * )  VT,  
integer  LDVT,  
double precision, dimension( lda, * )  ASAV,  
double precision, dimension( ldu, * )  USAV,  
double precision, dimension( ldvt, * )  VTSAV,  
double precision, dimension( * )  S,  
double precision, dimension( * )  SSAV,  
double precision, dimension( * )  E,  
double precision, dimension( * )  WORK,  
integer  LWORK,  
integer, dimension( * )  IWORK,  
integer  NOUT,  
integer  INFO  
) 
DDRVBD
DDRVBD checks the singular value decomposition (SVD) drivers DGESVD, DGESDD, DGESVJ, and DGEJSV. Both DGESVD and DGESDD factor A = U diag(S) VT, where U and VT are orthogonal and diag(S) is diagonal with the entries of the array S on its diagonal. The entries of S are the singular values, nonnegative and stored in decreasing order. U and VT can be optionally not computed, overwritten on A, or computed partially. A is M by N. Let MNMIN = min( M, N ). S has dimension MNMIN. U can be M by M or M by MNMIN. VT can be N by N or MNMIN by N. When DDRVBD is called, a number of matrix "sizes" (M's and N's) and a number of matrix "types" are specified. For each size (M,N) and each type of matrix, and for the minimal workspace as well as workspace adequate to permit blocking, an M x N matrix "A" will be generated and used to test the SVD routines. For each matrix, A will be factored as A = U diag(S) VT and the following 12 tests computed: Test for DGESVD: (1)  A  U diag(S) VT  / ( A max(M,N) ulp ) (2)  I  U'U  / ( M ulp ) (3)  I  VT VT'  / ( N ulp ) (4) S contains MNMIN nonnegative values in decreasing order. (Return 0 if true, 1/ULP if false.) (5)  U  Upartial  / ( M ulp ) where Upartial is a partially computed U. (6)  VT  VTpartial  / ( N ulp ) where VTpartial is a partially computed VT. (7)  S  Spartial  / ( MNMIN ulp S ) where Spartial is the vector of singular values from the partial SVD Test for DGESDD: (8)  A  U diag(S) VT  / ( A max(M,N) ulp ) (9)  I  U'U  / ( M ulp ) (10)  I  VT VT'  / ( N ulp ) (11) S contains MNMIN nonnegative values in decreasing order. (Return 0 if true, 1/ULP if false.) (12)  U  Upartial  / ( M ulp ) where Upartial is a partially computed U. (13)  VT  VTpartial  / ( N ulp ) where VTpartial is a partially computed VT. (14)  S  Spartial  / ( MNMIN ulp S ) where Spartial is the vector of singular values from the partial SVD Test for SGESVJ: (15)  A  U diag(S) VT  / ( A max(M,N) ulp ) (16)  I  U'U  / ( M ulp ) (17)  I  VT VT'  / ( N ulp ) (18) S contains MNMIN nonnegative values in decreasing order. (Return 0 if true, 1/ULP if false.) Test for SGEJSV: (19)  A  U diag(S) VT  / ( A max(M,N) ulp ) (20)  I  U'U  / ( M ulp ) (21)  I  VT VT'  / ( N ulp ) (22) S contains MNMIN nonnegative values in decreasing order. (Return 0 if true, 1/ULP if false.) The "sizes" are specified by the arrays MM(1:NSIZES) and NN(1:NSIZES); the value of each element pair (MM(j),NN(j)) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) The zero matrix. (2) The identity matrix. (3) A matrix of the form U D V, where U and V are orthogonal and D has evenly spaced entries 1, ..., ULP with random signs on the diagonal. (4) Same as (3), but multiplied by the underflowthreshold / ULP. (5) Same as (3), but multiplied by the overflowthreshold * ULP.
[in]  NSIZES  NSIZES is INTEGER The number of matrix sizes (M,N) contained in the vectors MM and NN. 
[in]  MM  MM is INTEGER array, dimension (NSIZES) The values of the matrix row dimension M. 
[in]  NN  NN is INTEGER array, dimension (NSIZES) The values of the matrix column dimension N. 
[in]  NTYPES  NTYPES is INTEGER The number of elements in DOTYPE. If it is zero, DDRVBD does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrices are in A and B. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. . 
[in]  DOTYPE  DOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size (m,n), a matrix of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored. 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry, the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. On exit, ISEED is changed and can be used in the next call to DDRVBD to continue the same random number sequence. 
[in]  THRESH  THRESH is DOUBLE PRECISION The threshold value for the test ratios. A result is included in the output file if RESULT >= THRESH. The test ratios are scaled to be O(1), so THRESH should be a small multiple of 1, e.g., 10 or 100. To have every test ratio printed, use THRESH = 0. 
[out]  A  A is DOUBLE PRECISION array, dimension (LDA,NMAX) where NMAX is the maximum value of N in NN. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,MMAX), where MMAX is the maximum value of M in MM. 
[out]  U  U is DOUBLE PRECISION array, dimension (LDU,MMAX) 
[in]  LDU  LDU is INTEGER The leading dimension of the array U. LDU >= max(1,MMAX). 
[out]  VT  VT is DOUBLE PRECISION array, dimension (LDVT,NMAX) 
[in]  LDVT  LDVT is INTEGER The leading dimension of the array VT. LDVT >= max(1,NMAX). 
[out]  ASAV  ASAV is DOUBLE PRECISION array, dimension (LDA,NMAX) 
[out]  USAV  USAV is DOUBLE PRECISION array, dimension (LDU,MMAX) 
[out]  VTSAV  VTSAV is DOUBLE PRECISION array, dimension (LDVT,NMAX) 
[out]  S  S is DOUBLE PRECISION array, dimension (max(min(MM,NN))) 
[out]  SSAV  SSAV is DOUBLE PRECISION array, dimension (max(min(MM,NN))) 
[out]  E  E is DOUBLE PRECISION array, dimension (max(min(MM,NN))) 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER The number of entries in WORK. This must be at least max(3*MN+MX,5*MN4)+2*MN**2 for all pairs pairs (MN,MX)=( min(MM(j),NN(j), max(MM(j),NN(j)) ) 
[out]  IWORK  IWORK is INTEGER array, dimension at least 8*min(M,N) 
[in]  NOUT  NOUT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) 
[out]  INFO  INFO is INTEGER If 0, then everything ran OK. 1: NSIZES < 0 2: Some MM(j) < 0 3: Some NN(j) < 0 4: NTYPES < 0 7: THRESH < 0 10: LDA < 1 or LDA < MMAX, where MMAX is max( MM(j) ). 12: LDU < 1 or LDU < MMAX. 14: LDVT < 1 or LDVT < NMAX, where NMAX is max( NN(j) ). 21: LWORK too small. If DLATMS, or DGESVD returns an error code, the absolute value of it is returned. 
Definition at line 318 of file ddrvbd.f.
subroutine ddrves  (  integer  NSIZES, 
integer, dimension( * )  NN,  
integer  NTYPES,  
logical, dimension( * )  DOTYPE,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
integer  NOUNIT,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( lda, * )  H,  
double precision, dimension( lda, * )  HT,  
double precision, dimension( * )  WR,  
double precision, dimension( * )  WI,  
double precision, dimension( * )  WRT,  
double precision, dimension( * )  WIT,  
double precision, dimension( ldvs, * )  VS,  
integer  LDVS,  
double precision, dimension( 13 )  RESULT,  
double precision, dimension( * )  WORK,  
integer  NWORK,  
integer, dimension( * )  IWORK,  
logical, dimension( * )  BWORK,  
integer  INFO  
) 
DDRVES
DDRVES checks the nonsymmetric eigenvalue (Schur form) problem driver DGEES. When DDRVES is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified. For each size ("n") and each type of matrix, one matrix will be generated and used to test the nonsymmetric eigenroutines. For each matrix, 13 tests will be performed: (1) 0 if T is in Schur form, 1/ulp otherwise (no sorting of eigenvalues) (2)  A  VS T VS'  / ( n A ulp ) Here VS is the matrix of Schur eigenvectors, and T is in Schur form (no sorting of eigenvalues). (3)  I  VS VS'  / ( n ulp ) (no sorting of eigenvalues). (4) 0 if WR+sqrt(1)*WI are eigenvalues of T 1/ulp otherwise (no sorting of eigenvalues) (5) 0 if T(with VS) = T(without VS), 1/ulp otherwise (no sorting of eigenvalues) (6) 0 if eigenvalues(with VS) = eigenvalues(without VS), 1/ulp otherwise (no sorting of eigenvalues) (7) 0 if T is in Schur form, 1/ulp otherwise (with sorting of eigenvalues) (8)  A  VS T VS'  / ( n A ulp ) Here VS is the matrix of Schur eigenvectors, and T is in Schur form (with sorting of eigenvalues). (9)  I  VS VS'  / ( n ulp ) (with sorting of eigenvalues). (10) 0 if WR+sqrt(1)*WI are eigenvalues of T 1/ulp otherwise (with sorting of eigenvalues) (11) 0 if T(with VS) = T(without VS), 1/ulp otherwise (with sorting of eigenvalues) (12) 0 if eigenvalues(with VS) = eigenvalues(without VS), 1/ulp otherwise (with sorting of eigenvalues) (13) if sorting worked and SDIM is the number of eigenvalues which were SELECTed The "sizes" are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) The zero matrix. (2) The identity matrix. (3) A (transposed) Jordan block, with 1's on the diagonal. (4) A diagonal matrix with evenly spaced entries 1, ..., ULP and random signs. (ULP = (first number larger than 1)  1 ) (5) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random signs. (6) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP and random signs. (7) Same as (4), but multiplied by a constant near the overflow threshold (8) Same as (4), but multiplied by a constant near the underflow threshold (9) A matrix of the form U' T U, where U is orthogonal and T has evenly spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (10) A matrix of the form U' T U, where U is orthogonal and T has geometrically spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (11) A matrix of the form U' T U, where U is orthogonal and T has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (12) A matrix of the form U' T U, where U is orthogonal and T has real or complex conjugate paired eigenvalues randomly chosen from ( ULP, 1 ) and random O(1) entries in the upper triangle. (13) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has evenly spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (14) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has geometrically spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (15) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (16) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has real or complex conjugate paired eigenvalues randomly chosen from ( ULP, 1 ) and random O(1) entries in the upper triangle. (17) Same as (16), but multiplied by a constant near the overflow threshold (18) Same as (16), but multiplied by a constant near the underflow threshold (19) Nonsymmetric matrix with random entries chosen from (1,1). If N is at least 4, all entries in first two rows and last row, and first column and last two columns are zero. (20) Same as (19), but multiplied by a constant near the overflow threshold (21) Same as (19), but multiplied by a constant near the underflow threshold
[in]  NSIZES  NSIZES is INTEGER The number of sizes of matrices to use. If it is zero, DDRVES does nothing. It must be at least zero. 
[in]  NN  NN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero. 
[in]  NTYPES  NTYPES is INTEGER The number of elements in DOTYPE. If it is zero, DDRVES does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. . 
[in]  DOTYPE  DOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored. 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to DDRVES to continue the same random number sequence. 
[in]  THRESH  THRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero. 
[in]  NOUNIT  NOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns INFO not equal to 0.) 
[out]  A  A is DOUBLE PRECISION array, dimension (LDA, max(NN)) Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used. 
[in]  LDA  LDA is INTEGER The leading dimension of A, and H. LDA must be at least 1 and at least max(NN). 
[out]  H  H is DOUBLE PRECISION array, dimension (LDA, max(NN)) Another copy of the test matrix A, modified by DGEES. 
[out]  HT  HT is DOUBLE PRECISION array, dimension (LDA, max(NN)) Yet another copy of the test matrix A, modified by DGEES. 
[out]  WR  WR is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  WI  WI is DOUBLE PRECISION array, dimension (max(NN)) The real and imaginary parts of the eigenvalues of A. On exit, WR + WI*i are the eigenvalues of the matrix in A. 
[out]  WRT  WRT is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  WIT  WIT is DOUBLE PRECISION array, dimension (max(NN)) Like WR, WI, these arrays contain the eigenvalues of A, but those computed when DGEES only computes a partial eigendecomposition, i.e. not Schur vectors 
[out]  VS  VS is DOUBLE PRECISION array, dimension (LDVS, max(NN)) VS holds the computed Schur vectors. 
[in]  LDVS  LDVS is INTEGER Leading dimension of VS. Must be at least max(1,max(NN)). 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (13) The values computed by the 13 tests described above. The values are currently limited to 1/ulp, to avoid overflow. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (NWORK) 
[in]  NWORK  NWORK is INTEGER The number of entries in WORK. This must be at least 5*NN(j)+2*NN(j)**2 for all j. 
[out]  IWORK  IWORK is INTEGER array, dimension (max(NN)) 
[out]  BWORK  BWORK is LOGICAL array, dimension (max(NN)) 
[out]  INFO  INFO is INTEGER If 0, then everything ran OK. 1: NSIZES < 0 2: Some NN(j) < 0 3: NTYPES < 0 6: THRESH < 0 9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ). 17: LDVS < 1 or LDVS < NMAX, where NMAX is max( NN(j) ). 20: NWORK too small. If DLATMR, SLATMS, SLATME or DGEES returns an error code, the absolute value of it is returned.  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. NMAX Largest value in NN. NERRS The number of tests which have exceeded THRESH COND, CONDS, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. ULP, ULPINV Finest relative precision and its inverse. RTULP, RTULPI Square roots of the previous 4 values. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) ) KCONDS(j) Selectw whether CONDS is to be 1 or 1/sqrt(ulp). (0 means irrelevant.) 
Definition at line 387 of file ddrves.f.
subroutine ddrvev  (  integer  NSIZES, 
integer, dimension( * )  NN,  
integer  NTYPES,  
logical, dimension( * )  DOTYPE,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
integer  NOUNIT,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( lda, * )  H,  
double precision, dimension( * )  WR,  
double precision, dimension( * )  WI,  
double precision, dimension( * )  WR1,  
double precision, dimension( * )  WI1,  
double precision, dimension( ldvl, * )  VL,  
integer  LDVL,  
double precision, dimension( ldvr, * )  VR,  
integer  LDVR,  
double precision, dimension( ldlre, * )  LRE,  
integer  LDLRE,  
double precision, dimension( 7 )  RESULT,  
double precision, dimension( * )  WORK,  
integer  NWORK,  
integer, dimension( * )  IWORK,  
integer  INFO  
) 
DDRVEV
DDRVEV checks the nonsymmetric eigenvalue problem driver DGEEV. When DDRVEV is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified. For each size ("n") and each type of matrix, one matrix will be generated and used to test the nonsymmetric eigenroutines. For each matrix, 7 tests will be performed: (1)  A * VR  VR * W  / ( n A ulp ) Here VR is the matrix of unit right eigenvectors. W is a block diagonal matrix, with a 1x1 block for each real eigenvalue and a 2x2 block for each complex conjugate pair. If eigenvalues j and j+1 are a complex conjugate pair, so WR(j) = WR(j+1) = wr and WI(j) =  WI(j+1) = wi, then the 2 x 2 block corresponding to the pair will be: ( wr wi ) ( wi wr ) Such a block multiplying an n x 2 matrix ( ur ui ) on the right will be the same as multiplying ur + i*ui by wr + i*wi. (2)  A**H * VL  VL * W**H  / ( n A ulp ) Here VL is the matrix of unit left eigenvectors, A**H is the conjugate transpose of A, and W is as above. (3)  VR(i)  1  / ulp and whether largest component real VR(i) denotes the ith column of VR. (4)  VL(i)  1  / ulp and whether largest component real VL(i) denotes the ith column of VL. (5) W(full) = W(partial) W(full) denotes the eigenvalues computed when both VR and VL are also computed, and W(partial) denotes the eigenvalues computed when only W, only W and VR, or only W and VL are computed. (6) VR(full) = VR(partial) VR(full) denotes the right eigenvectors computed when both VR and VL are computed, and VR(partial) denotes the result when only VR is computed. (7) VL(full) = VL(partial) VL(full) denotes the left eigenvectors computed when both VR and VL are also computed, and VL(partial) denotes the result when only VL is computed. The "sizes" are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) The zero matrix. (2) The identity matrix. (3) A (transposed) Jordan block, with 1's on the diagonal. (4) A diagonal matrix with evenly spaced entries 1, ..., ULP and random signs. (ULP = (first number larger than 1)  1 ) (5) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random signs. (6) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP and random signs. (7) Same as (4), but multiplied by a constant near the overflow threshold (8) Same as (4), but multiplied by a constant near the underflow threshold (9) A matrix of the form U' T U, where U is orthogonal and T has evenly spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (10) A matrix of the form U' T U, where U is orthogonal and T has geometrically spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (11) A matrix of the form U' T U, where U is orthogonal and T has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (12) A matrix of the form U' T U, where U is orthogonal and T has real or complex conjugate paired eigenvalues randomly chosen from ( ULP, 1 ) and random O(1) entries in the upper triangle. (13) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has evenly spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (14) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has geometrically spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (15) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (16) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has real or complex conjugate paired eigenvalues randomly chosen from ( ULP, 1 ) and random O(1) entries in the upper triangle. (17) Same as (16), but multiplied by a constant near the overflow threshold (18) Same as (16), but multiplied by a constant near the underflow threshold (19) Nonsymmetric matrix with random entries chosen from (1,1). If N is at least 4, all entries in first two rows and last row, and first column and last two columns are zero. (20) Same as (19), but multiplied by a constant near the overflow threshold (21) Same as (19), but multiplied by a constant near the underflow threshold
[in]  NSIZES  NSIZES is INTEGER The number of sizes of matrices to use. If it is zero, DDRVEV does nothing. It must be at least zero. 
[in]  NN  NN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero. 
[in]  NTYPES  NTYPES is INTEGER The number of elements in DOTYPE. If it is zero, DDRVEV does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. . 
[in]  DOTYPE  DOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored. 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to DDRVEV to continue the same random number sequence. 
[in]  THRESH  THRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero. 
[in]  NOUNIT  NOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns INFO not equal to 0.) 
[out]  A  A is DOUBLE PRECISION array, dimension (LDA, max(NN)) Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used. 
[in]  LDA  LDA is INTEGER The leading dimension of A, and H. LDA must be at least 1 and at least max(NN). 
[out]  H  H is DOUBLE PRECISION array, dimension (LDA, max(NN)) Another copy of the test matrix A, modified by DGEEV. 
[out]  WR  WR is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  WI  WI is DOUBLE PRECISION array, dimension (max(NN)) The real and imaginary parts of the eigenvalues of A. On exit, WR + WI*i are the eigenvalues of the matrix in A. 
[out]  WR1  WR1 is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  WI1  WI1 is DOUBLE PRECISION array, dimension (max(NN)) Like WR, WI, these arrays contain the eigenvalues of A, but those computed when DGEEV only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors. 
[out]  VL  VL is DOUBLE PRECISION array, dimension (LDVL, max(NN)) VL holds the computed left eigenvectors. 
[in]  LDVL  LDVL is INTEGER Leading dimension of VL. Must be at least max(1,max(NN)). 
[out]  VR  VR is DOUBLE PRECISION array, dimension (LDVR, max(NN)) VR holds the computed right eigenvectors. 
[in]  LDVR  LDVR is INTEGER Leading dimension of VR. Must be at least max(1,max(NN)). 
[out]  LRE  LRE is DOUBLE PRECISION array, dimension (LDLRE,max(NN)) LRE holds the computed right or left eigenvectors. 
[in]  LDLRE  LDLRE is INTEGER Leading dimension of LRE. Must be at least max(1,max(NN)). 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (7) The values computed by the seven tests described above. The values are currently limited to 1/ulp, to avoid overflow. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (NWORK) 
[in]  NWORK  NWORK is INTEGER The number of entries in WORK. This must be at least 5*NN(j)+2*NN(j)**2 for all j. 
[out]  IWORK  IWORK is INTEGER array, dimension (max(NN)) 
[out]  INFO  INFO is INTEGER If 0, then everything ran OK. 1: NSIZES < 0 2: Some NN(j) < 0 3: NTYPES < 0 6: THRESH < 0 9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ). 16: LDVL < 1 or LDVL < NMAX, where NMAX is max( NN(j) ). 18: LDVR < 1 or LDVR < NMAX, where NMAX is max( NN(j) ). 20: LDLRE < 1 or LDLRE < NMAX, where NMAX is max( NN(j) ). 23: NWORK too small. If DLATMR, SLATMS, SLATME or DGEEV returns an error code, the absolute value of it is returned.  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. NMAX Largest value in NN. NERRS The number of tests which have exceeded THRESH COND, CONDS, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. ULP, ULPINV Finest relative precision and its inverse. RTULP, RTULPI Square roots of the previous 4 values. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) ) KCONDS(j) Selectw whether CONDS is to be 1 or 1/sqrt(ulp). (0 means irrelevant.) 
Definition at line 404 of file ddrvev.f.
subroutine ddrvgg  (  integer  NSIZES, 
integer, dimension( * )  NN,  
integer  NTYPES,  
logical, dimension( * )  DOTYPE,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
double precision  THRSHN,  
integer  NOUNIT,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( lda, * )  B,  
double precision, dimension( lda, * )  S,  
double precision, dimension( lda, * )  T,  
double precision, dimension( lda, * )  S2,  
double precision, dimension( lda, * )  T2,  
double precision, dimension( ldq, * )  Q,  
integer  LDQ,  
double precision, dimension( ldq, * )  Z,  
double precision, dimension( * )  ALPHR1,  
double precision, dimension( * )  ALPHI1,  
double precision, dimension( * )  BETA1,  
double precision, dimension( * )  ALPHR2,  
double precision, dimension( * )  ALPHI2,  
double precision, dimension( * )  BETA2,  
double precision, dimension( ldq, * )  VL,  
double precision, dimension( ldq, * )  VR,  
double precision, dimension( * )  WORK,  
integer  LWORK,  
double precision, dimension( * )  RESULT,  
integer  INFO  
) 
DDRVGG
DDRVGG checks the nonsymmetric generalized eigenvalue driver routines. T T T DGEGS factors A and B as Q S Z and Q T Z , where means transpose, T is upper triangular, S is in generalized Schur form (block upper triangular, with 1x1 and 2x2 blocks on the diagonal, the 2x2 blocks corresponding to complex conjugate pairs of generalized eigenvalues), and Q and Z are orthogonal. It also computes the generalized eigenvalues (alpha(1),beta(1)), ..., (alpha(n),beta(n)), where alpha(j)=S(j,j) and beta(j)=P(j,j)  thus, w(j) = alpha(j)/beta(j) is a root of the generalized eigenvalue problem det( A  w(j) B ) = 0 and m(j) = beta(j)/alpha(j) is a root of the essentially equivalent problem det( m(j) A  B ) = 0 DGEGV computes the generalized eigenvalues (alpha(1),beta(1)), ..., (alpha(n),beta(n)), the matrix L whose columns contain the generalized left eigenvectors l, and the matrix R whose columns contain the generalized right eigenvectors r for the pair (A,B). When DDRVGG is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified. For each size ("n") and each type of matrix, one matrix will be generated and used to test the nonsymmetric eigenroutines. For each matrix, 7 tests will be performed and compared with the threshhold THRESH: Results from DGEGS: T (1)  A  Q S Z  / ( A n ulp ) T (2)  B  Q T Z  / ( B n ulp ) T (3)  I  QQ  / ( n ulp ) T (4)  I  ZZ  / ( n ulp ) (5) maximum over j of D(j) where: if alpha(j) is real: alpha(j)  S(j,j) beta(j)  T(j,j) D(j) =  +  max(alpha(j),S(j,j)) max(beta(j),T(j,j)) if alpha(j) is complex:  det( s S  w T )  D(j) =  ulp max( s norm(S), w norm(T) )*norm( s S  w T ) and S and T are here the 2 x 2 diagonal blocks of S and T corresponding to the jth eigenvalue. Results from DGEGV: (6) max over all left eigenvalue/vector pairs (beta/alpha,l) of  l**H * (beta A  alpha B)  / ( ulp max( beta A, alpha B ) ) where l**H is the conjugate tranpose of l. (7) max over all right eigenvalue/vector pairs (beta/alpha,r) of  (beta A  alpha B) r  / ( ulp max( beta A, alpha B ) ) Test Matrices   The sizes of the test matrices are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) ( 0, 0 ) (a pair of zero matrices) (2) ( I, 0 ) (an identity and a zero matrix) (3) ( 0, I ) (an identity and a zero matrix) (4) ( I, I ) (a pair of identity matrices) t t (5) ( J , J ) (a pair of transposed Jordan blocks) t ( I 0 ) (6) ( X, Y ) where X = ( J 0 ) and Y = ( t ) ( 0 I ) ( 0 J ) and I is a k x k identity and J a (k+1)x(k+1) Jordan block; k=(N1)/2 (7) ( D, I ) where D is diag( 0, 1,..., N1 ) (a diagonal matrix with those diagonal entries.) (8) ( I, D ) (9) ( big*D, small*I ) where "big" is near overflow and small=1/big (10) ( small*D, big*I ) (11) ( big*I, small*D ) (12) ( small*I, big*D ) (13) ( big*D, big*I ) (14) ( small*D, small*I ) (15) ( D1, D2 ) where D1 is diag( 0, 0, 1, ..., N3, 0 ) and D2 is diag( 0, N3, N4,..., 1, 0, 0 ) t t (16) Q ( J , J ) Z where Q and Z are random orthogonal matrices. (17) Q ( T1, T2 ) Z where T1 and T2 are upper triangular matrices with random O(1) entries above the diagonal and diagonal entries diag(T1) = ( 0, 0, 1, ..., N3, 0 ) and diag(T2) = ( 0, N3, N4,..., 1, 0, 0 ) (18) Q ( T1, T2 ) Z diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 ) diag(T2) = ( 0, 1, 0, 1,..., 1, 0 ) s = machine precision. (19) Q ( T1, T2 ) Z diag(T1)=( 0,0,1,1, 1d, ..., 1(N5)*d=s, 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 ) N5 (20) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) (21) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, r1, r2, ..., r(N4), 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) where r1,..., r(N4) are random. (22) Q ( big*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (23) Q ( small*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (24) Q ( small*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (25) Q ( big*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (26) Q ( T1, T2 ) Z where T1 and T2 are random uppertriangular matrices.
[in]  NSIZES  NSIZES is INTEGER The number of sizes of matrices to use. If it is zero, DDRVGG does nothing. It must be at least zero. 
[in]  NN  NN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero. 
[in]  NTYPES  NTYPES is INTEGER The number of elements in DOTYPE. If it is zero, DDRVGG does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. . 
[in]  DOTYPE  DOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored. 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to DDRVGG to continue the same random number sequence. 
[in]  THRESH  THRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero. 
[in]  THRSHN  THRSHN is DOUBLE PRECISION Threshhold for reporting eigenvector normalization error. If the normalization of any eigenvector differs from 1 by more than THRSHN*ulp, then a special error message will be printed. (This is handled separately from the other tests, since only a compiler or programming error should cause an error message, at least if THRSHN is at least 510.) 
[in]  NOUNIT  NOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) 
[in,out]  A  A is DOUBLE PRECISION array, dimension (LDA, max(NN)) Used to hold the original A matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. 
[in]  LDA  LDA is INTEGER The leading dimension of A, B, S, T, S2, and T2. It must be at least 1 and at least max( NN ). 
[in,out]  B  B is DOUBLE PRECISION array, dimension (LDA, max(NN)) Used to hold the original B matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. 
[out]  S  S is DOUBLE PRECISION array, dimension (LDA, max(NN)) The Schur form matrix computed from A by DGEGS. On exit, S contains the Schur form matrix corresponding to the matrix in A. 
[out]  T  T is DOUBLE PRECISION array, dimension (LDA, max(NN)) The upper triangular matrix computed from B by DGEGS. 
[out]  S2  S2 is DOUBLE PRECISION array, dimension (LDA, max(NN)) The matrix computed from A by DGEGV. This will be the Schur form of some matrix related to A, but will not, in general, be the same as S. 
[out]  T2  T2 is DOUBLE PRECISION array, dimension (LDA, max(NN)) The matrix computed from B by DGEGV. This will be the Schur form of some matrix related to B, but will not, in general, be the same as T. 
[out]  Q  Q is DOUBLE PRECISION array, dimension (LDQ, max(NN)) The (left) orthogonal matrix computed by DGEGS. 
[in]  LDQ  LDQ is INTEGER The leading dimension of Q, Z, VL, and VR. It must be at least 1 and at least max( NN ). 
[out]  Z  Z is DOUBLE PRECISION array of dimension( LDQ, max(NN) ) The (right) orthogonal matrix computed by DGEGS. 
[out]  ALPHR1  ALPHR1 is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  ALPHI1  ALPHI1 is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  BETA1  BETA1 is DOUBLE PRECISION array, dimension (max(NN)) The generalized eigenvalues of (A,B) computed by DGEGS. ( ALPHR1(k)+ALPHI1(k)*i ) / BETA1(k) is the kth generalized eigenvalue of the matrices in A and B. 
[out]  ALPHR2  ALPHR2 is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  ALPHI2  ALPHI2 is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  BETA2  BETA2 is DOUBLE PRECISION array, dimension (max(NN)) The generalized eigenvalues of (A,B) computed by DGEGV. ( ALPHR2(k)+ALPHI2(k)*i ) / BETA2(k) is the kth generalized eigenvalue of the matrices in A and B. 
[out]  VL  VL is DOUBLE PRECISION array, dimension (LDQ, max(NN)) The (block lower triangular) left eigenvector matrix for the matrices in A and B. (See DTGEVC for the format.) 
[out]  VR  VR is DOUBLE PRECISION array, dimension (LDQ, max(NN)) The (block upper triangular) right eigenvector matrix for the matrices in A and B. (See DTGEVC for the format.) 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER The number of entries in WORK. This must be at least 2*N + MAX( 6*N, N*(NB+1), (k+1)*(2*k+N+1) ), where "k" is the sum of the blocksize and numberofshifts for DHGEQZ, and NB is the greatest of the blocksizes for DGEQRF, DORMQR, and DORGQR. (The blocksizes and the numberofshifts are retrieved through calls to ILAENV.) 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (15) The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value. > 0: A routine returned an error code. INFO is the absolute value of the INFO value returned. 
Definition at line 450 of file ddrvgg.f.
subroutine ddrvsg  (  integer  NSIZES, 
integer, dimension( * )  NN,  
integer  NTYPES,  
logical, dimension( * )  DOTYPE,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
integer  NOUNIT,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( ldb, * )  B,  
integer  LDB,  
double precision, dimension( * )  D,  
double precision, dimension( ldz, * )  Z,  
integer  LDZ,  
double precision, dimension( lda, * )  AB,  
double precision, dimension( ldb, * )  BB,  
double precision, dimension( * )  AP,  
double precision, dimension( * )  BP,  
double precision, dimension( * )  WORK,  
integer  NWORK,  
integer, dimension( * )  IWORK,  
integer  LIWORK,  
double precision, dimension( * )  RESULT,  
integer  INFO  
) 
DDRVSG
DDRVSG checks the real symmetric generalized eigenproblem drivers. DSYGV computes all eigenvalues and, optionally, eigenvectors of a real symmetricdefinite generalized eigenproblem. DSYGVD computes all eigenvalues and, optionally, eigenvectors of a real symmetricdefinite generalized eigenproblem using a divide and conquer algorithm. DSYGVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetricdefinite generalized eigenproblem. DSPGV computes all eigenvalues and, optionally, eigenvectors of a real symmetricdefinite generalized eigenproblem in packed storage. DSPGVD computes all eigenvalues and, optionally, eigenvectors of a real symmetricdefinite generalized eigenproblem in packed storage using a divide and conquer algorithm. DSPGVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetricdefinite generalized eigenproblem in packed storage. DSBGV computes all eigenvalues and, optionally, eigenvectors of a real symmetricdefinite banded generalized eigenproblem. DSBGVD computes all eigenvalues and, optionally, eigenvectors of a real symmetricdefinite banded generalized eigenproblem using a divide and conquer algorithm. DSBGVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetricdefinite banded generalized eigenproblem. When DDRVSG is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified. For each size ("n") and each type of matrix, one matrix A of the given type will be generated; a random wellconditioned matrix B is also generated and the pair (A,B) is used to test the drivers. For each pair (A,B), the following tests are performed: (1) DSYGV with ITYPE = 1 and UPLO ='U':  A Z  B Z D  / ( A Z n ulp ) (2) as (1) but calling DSPGV (3) as (1) but calling DSBGV (4) as (1) but with UPLO = 'L' (5) as (4) but calling DSPGV (6) as (4) but calling DSBGV (7) DSYGV with ITYPE = 2 and UPLO ='U':  A B Z  Z D  / ( A Z n ulp ) (8) as (7) but calling DSPGV (9) as (7) but with UPLO = 'L' (10) as (9) but calling DSPGV (11) DSYGV with ITYPE = 3 and UPLO ='U':  B A Z  Z D  / ( A Z n ulp ) (12) as (11) but calling DSPGV (13) as (11) but with UPLO = 'L' (14) as (13) but calling DSPGV DSYGVD, DSPGVD and DSBGVD performed the same 14 tests. DSYGVX, DSPGVX and DSBGVX performed the above 14 tests with the parameter RANGE = 'A', 'N' and 'I', respectively. The "sizes" are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. This type is used for the matrix A which has halfbandwidth KA. B is generated as a wellconditioned positive definite matrix with halfbandwidth KB (<= KA). Currently, the list of possible types for A is: (1) The zero matrix. (2) The identity matrix. (3) A diagonal matrix with evenly spaced entries 1, ..., ULP and random signs. (ULP = (first number larger than 1)  1 ) (4) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random signs. (5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP and random signs. (6) Same as (4), but multiplied by SQRT( overflow threshold ) (7) Same as (4), but multiplied by SQRT( underflow threshold ) (8) A matrix of the form U* D U, where U is orthogonal and D has evenly spaced entries 1, ..., ULP with random signs on the diagonal. (9) A matrix of the form U* D U, where U is orthogonal and D has geometrically spaced entries 1, ..., ULP with random signs on the diagonal. (10) A matrix of the form U* D U, where U is orthogonal and D has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal. (11) Same as (8), but multiplied by SQRT( overflow threshold ) (12) Same as (8), but multiplied by SQRT( underflow threshold ) (13) symmetric matrix with random entries chosen from (1,1). (14) Same as (13), but multiplied by SQRT( overflow threshold ) (15) Same as (13), but multiplied by SQRT( underflow threshold) (16) Same as (8), but with KA = 1 and KB = 1 (17) Same as (8), but with KA = 2 and KB = 1 (18) Same as (8), but with KA = 2 and KB = 2 (19) Same as (8), but with KA = 3 and KB = 1 (20) Same as (8), but with KA = 3 and KB = 2 (21) Same as (8), but with KA = 3 and KB = 3
NSIZES INTEGER The number of sizes of matrices to use. If it is zero, DDRVSG does nothing. It must be at least zero. Not modified. NN INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero. Not modified. NTYPES INTEGER The number of elements in DOTYPE. If it is zero, DDRVSG does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. . Not modified. DOTYPE LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored. Not modified. ISEED INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to DDRVSG to continue the same random number sequence. Modified. THRESH DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero. Not modified. NOUNIT INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) Not modified. A DOUBLE PRECISION array, dimension (LDA , max(NN)) Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used. Modified. LDA INTEGER The leading dimension of A and AB. It must be at least 1 and at least max( NN ). Not modified. B DOUBLE PRECISION array, dimension (LDB , max(NN)) Used to hold the symmetric positive definite matrix for the generailzed problem. On exit, B contains the last matrix actually used. Modified. LDB INTEGER The leading dimension of B and BB. It must be at least 1 and at least max( NN ). Not modified. D DOUBLE PRECISION array, dimension (max(NN)) The eigenvalues of A. On exit, the eigenvalues in D correspond with the matrix in A. Modified. Z DOUBLE PRECISION array, dimension (LDZ, max(NN)) The matrix of eigenvectors. Modified. LDZ INTEGER The leading dimension of Z. It must be at least 1 and at least max( NN ). Not modified. AB DOUBLE PRECISION array, dimension (LDA, max(NN)) Workspace. Modified. BB DOUBLE PRECISION array, dimension (LDB, max(NN)) Workspace. Modified. AP DOUBLE PRECISION array, dimension (max(NN)**2) Workspace. Modified. BP DOUBLE PRECISION array, dimension (max(NN)**2) Workspace. Modified. WORK DOUBLE PRECISION array, dimension (NWORK) Workspace. Modified. NWORK INTEGER The number of entries in WORK. This must be at least 1+5*N+2*N*lg(N)+3*N**2 where N = max( NN(j) ) and lg( N ) = smallest integer k such that 2**k >= N. Not modified. IWORK INTEGER array, dimension (LIWORK) Workspace. Modified. LIWORK INTEGER The number of entries in WORK. This must be at least 6*N. Not modified. RESULT DOUBLE PRECISION array, dimension (70) The values computed by the 70 tests described above. Modified. INFO INTEGER If 0, then everything ran OK. 1: NSIZES < 0 2: Some NN(j) < 0 3: NTYPES < 0 5: THRESH < 0 9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ). 16: LDZ < 1 or LDZ < NMAX. 21: NWORK too small. 23: LIWORK too small. If DLATMR, SLATMS, DSYGV, DSPGV, DSBGV, SSYGVD, SSPGVD, DSBGVD, DSYGVX, DSPGVX or SSBGVX returns an error code, the absolute value of it is returned. Modified.  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. NTEST The number of tests that have been run on this matrix. NTESTT The total number of tests for this call. NMAX Largest value in NN. NMATS The number of matrices generated so far. NERRS The number of tests which have exceeded THRESH so far (computed by DLAFTS). COND, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. ULP, ULPINV Finest relative precision and its inverse. RTOVFL, RTUNFL Square roots of the previous 2 values. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) )
Definition at line 354 of file ddrvsg.f.
subroutine ddrvst  (  integer  NSIZES, 
integer, dimension( * )  NN,  
integer  NTYPES,  
logical, dimension( * )  DOTYPE,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
integer  NOUNIT,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( * )  D1,  
double precision, dimension( * )  D2,  
double precision, dimension( * )  D3,  
double precision, dimension( * )  D4,  
double precision, dimension( * )  EVEIGS,  
double precision, dimension( * )  WA1,  
double precision, dimension( * )  WA2,  
double precision, dimension( * )  WA3,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( ldu, * )  V,  
double precision, dimension( * )  TAU,  
double precision, dimension( ldu, * )  Z,  
double precision, dimension( * )  WORK,  
integer  LWORK,  
integer, dimension( * )  IWORK,  
integer  LIWORK,  
double precision, dimension( * )  RESULT,  
integer  INFO  
) 
DDRVST
DDRVST checks the symmetric eigenvalue problem drivers. DSTEV computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix. DSTEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix. DSTEVR computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix using the Relatively Robust Representation where it can. DSYEV computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix. DSYEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix. DSYEVR computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix using the Relatively Robust Representation where it can. DSPEV computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix in packed storage. DSPEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix in packed storage. DSBEV computes all eigenvalues and, optionally, eigenvectors of a real symmetric band matrix. DSBEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix. DSYEVD computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix using a divide and conquer algorithm. DSPEVD computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix in packed storage, using a divide and conquer algorithm. DSBEVD computes all eigenvalues and, optionally, eigenvectors of a real symmetric band matrix, using a divide and conquer algorithm. When DDRVST is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified. For each size ("n") and each type of matrix, one matrix will be generated and used to test the appropriate drivers. For each matrix and each driver routine called, the following tests will be performed: (1)  A  Z D Z'  / ( A n ulp ) (2)  I  Z Z'  / ( n ulp ) (3)  D1  D2  / ( D1 ulp ) where Z is the matrix of eigenvectors returned when the eigenvector option is given and D1 and D2 are the eigenvalues returned with and without the eigenvector option. The "sizes" are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) The zero matrix. (2) The identity matrix. (3) A diagonal matrix with evenly spaced eigenvalues 1, ..., ULP and random signs. (ULP = (first number larger than 1)  1 ) (4) A diagonal matrix with geometrically spaced eigenvalues 1, ..., ULP and random signs. (5) A diagonal matrix with "clustered" eigenvalues 1, ULP, ..., ULP and random signs. (6) Same as (4), but multiplied by SQRT( overflow threshold ) (7) Same as (4), but multiplied by SQRT( underflow threshold ) (8) A matrix of the form U' D U, where U is orthogonal and D has evenly spaced entries 1, ..., ULP with random signs on the diagonal. (9) A matrix of the form U' D U, where U is orthogonal and D has geometrically spaced entries 1, ..., ULP with random signs on the diagonal. (10) A matrix of the form U' D U, where U is orthogonal and D has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal. (11) Same as (8), but multiplied by SQRT( overflow threshold ) (12) Same as (8), but multiplied by SQRT( underflow threshold ) (13) Symmetric matrix with random entries chosen from (1,1). (14) Same as (13), but multiplied by SQRT( overflow threshold ) (15) Same as (13), but multiplied by SQRT( underflow threshold ) (16) A band matrix with half bandwidth randomly chosen between 0 and N1, with evenly spaced eigenvalues 1, ..., ULP with random signs. (17) Same as (16), but multiplied by SQRT( overflow threshold ) (18) Same as (16), but multiplied by SQRT( underflow threshold )
NSIZES INTEGER The number of sizes of matrices to use. If it is zero, DDRVST does nothing. It must be at least zero. Not modified. NN INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero. Not modified. NTYPES INTEGER The number of elements in DOTYPE. If it is zero, DDRVST does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. . Not modified. DOTYPE LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored. Not modified. ISEED INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to DDRVST to continue the same random number sequence. Modified. THRESH DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero. Not modified. NOUNIT INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) Not modified. A DOUBLE PRECISION array, dimension (LDA , max(NN)) Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used. Modified. LDA INTEGER The leading dimension of A. It must be at least 1 and at least max( NN ). Not modified. D1 DOUBLE PRECISION array, dimension (max(NN)) The eigenvalues of A, as computed by DSTEQR simlutaneously with Z. On exit, the eigenvalues in D1 correspond with the matrix in A. Modified. D2 DOUBLE PRECISION array, dimension (max(NN)) The eigenvalues of A, as computed by DSTEQR if Z is not computed. On exit, the eigenvalues in D2 correspond with the matrix in A. Modified. D3 DOUBLE PRECISION array, dimension (max(NN)) The eigenvalues of A, as computed by DSTERF. On exit, the eigenvalues in D3 correspond with the matrix in A. Modified. D4 DOUBLE PRECISION array, dimension EVEIGS DOUBLE PRECISION array, dimension (max(NN)) The eigenvalues as computed by DSTEV('N', ... ) (I reserve the right to change this to the output of whichever algorithm computes the most accurate eigenvalues). WA1 DOUBLE PRECISION array, dimension WA2 DOUBLE PRECISION array, dimension WA3 DOUBLE PRECISION array, dimension U DOUBLE PRECISION array, dimension (LDU, max(NN)) The orthogonal matrix computed by DSYTRD + DORGTR. Modified. LDU INTEGER The leading dimension of U, Z, and V. It must be at least 1 and at least max( NN ). Not modified. V DOUBLE PRECISION array, dimension (LDU, max(NN)) The Housholder vectors computed by DSYTRD in reducing A to tridiagonal form. Modified. TAU DOUBLE PRECISION array, dimension (max(NN)) The Householder factors computed by DSYTRD in reducing A to tridiagonal form. Modified. Z DOUBLE PRECISION array, dimension (LDU, max(NN)) The orthogonal matrix of eigenvectors computed by DSTEQR, DPTEQR, and DSTEIN. Modified. WORK DOUBLE PRECISION array, dimension (LWORK) Workspace. Modified. LWORK INTEGER The number of entries in WORK. This must be at least 1 + 4 * Nmax + 2 * Nmax * lg Nmax + 4 * Nmax**2 where Nmax = max( NN(j), 2 ) and lg = log base 2. Not modified. IWORK INTEGER array, dimension (6 + 6*Nmax + 5 * Nmax * lg Nmax ) where Nmax = max( NN(j), 2 ) and lg = log base 2. Workspace. Modified. RESULT DOUBLE PRECISION array, dimension (105) The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. Modified. INFO INTEGER If 0, then everything ran OK. 1: NSIZES < 0 2: Some NN(j) < 0 3: NTYPES < 0 5: THRESH < 0 9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ). 16: LDU < 1 or LDU < NMAX. 21: LWORK too small. If DLATMR, DLATMS, DSYTRD, DORGTR, DSTEQR, DSTERF, or DORMTR returns an error code, the absolute value of it is returned. Modified.  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. NTEST The number of tests performed, or which can be performed so far, for the current matrix. NTESTT The total number of tests performed so far. NMAX Largest value in NN. NMATS The number of matrices generated so far. NERRS The number of tests which have exceeded THRESH so far (computed by DLAFTS). COND, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. ULP, ULPINV Finest relative precision and its inverse. RTOVFL, RTUNFL Square roots of the previous 2 values. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) ) The tests performed are: Routine tested 1=  A  U S U'  / ( A n ulp ) DSTEV('V', ... ) 2=  I  U U'  / ( n ulp ) DSTEV('V', ... ) 3= D(with Z)  D(w/o Z) / (D ulp) DSTEV('N', ... ) 4=  A  U S U'  / ( A n ulp ) DSTEVX('V','A', ... ) 5=  I  U U'  / ( n ulp ) DSTEVX('V','A', ... ) 6= D(with Z)  EVEIGS / (D ulp) DSTEVX('N','A', ... ) 7=  A  U S U'  / ( A n ulp ) DSTEVR('V','A', ... ) 8=  I  U U'  / ( n ulp ) DSTEVR('V','A', ... ) 9= D(with Z)  EVEIGS / (D ulp) DSTEVR('N','A', ... ) 10=  A  U S U'  / ( A n ulp ) DSTEVX('V','I', ... ) 11=  I  U U'  / ( n ulp ) DSTEVX('V','I', ... ) 12= D(with Z)  D(w/o Z) / (D ulp) DSTEVX('N','I', ... ) 13=  A  U S U'  / ( A n ulp ) DSTEVX('V','V', ... ) 14=  I  U U'  / ( n ulp ) DSTEVX('V','V', ... ) 15= D(with Z)  D(w/o Z) / (D ulp) DSTEVX('N','V', ... ) 16=  A  U S U'  / ( A n ulp ) DSTEVD('V', ... ) 17=  I  U U'  / ( n ulp ) DSTEVD('V', ... ) 18= D(with Z)  EVEIGS / (D ulp) DSTEVD('N', ... ) 19=  A  U S U'  / ( A n ulp ) DSTEVR('V','I', ... ) 20=  I  U U'  / ( n ulp ) DSTEVR('V','I', ... ) 21= D(with Z)  D(w/o Z) / (D ulp) DSTEVR('N','I', ... ) 22=  A  U S U'  / ( A n ulp ) DSTEVR('V','V', ... ) 23=  I  U U'  / ( n ulp ) DSTEVR('V','V', ... ) 24= D(with Z)  D(w/o Z) / (D ulp) DSTEVR('N','V', ... ) 25=  A  U S U'  / ( A n ulp ) DSYEV('L','V', ... ) 26=  I  U U'  / ( n ulp ) DSYEV('L','V', ... ) 27= D(with Z)  D(w/o Z) / (D ulp) DSYEV('L','N', ... ) 28=  A  U S U'  / ( A n ulp ) DSYEVX('L','V','A', ... ) 29=  I  U U'  / ( n ulp ) DSYEVX('L','V','A', ... ) 30= D(with Z)  D(w/o Z) / (D ulp) DSYEVX('L','N','A', ... ) 31=  A  U S U'  / ( A n ulp ) DSYEVX('L','V','I', ... ) 32=  I  U U'  / ( n ulp ) DSYEVX('L','V','I', ... ) 33= D(with Z)  D(w/o Z) / (D ulp) DSYEVX('L','N','I', ... ) 34=  A  U S U'  / ( A n ulp ) DSYEVX('L','V','V', ... ) 35=  I  U U'  / ( n ulp ) DSYEVX('L','V','V', ... ) 36= D(with Z)  D(w/o Z) / (D ulp) DSYEVX('L','N','V', ... ) 37=  A  U S U'  / ( A n ulp ) DSPEV('L','V', ... ) 38=  I  U U'  / ( n ulp ) DSPEV('L','V', ... ) 39= D(with Z)  D(w/o Z) / (D ulp) DSPEV('L','N', ... ) 40=  A  U S U'  / ( A n ulp ) DSPEVX('L','V','A', ... ) 41=  I  U U'  / ( n ulp ) DSPEVX('L','V','A', ... ) 42= D(with Z)  D(w/o Z) / (D ulp) DSPEVX('L','N','A', ... ) 43=  A  U S U'  / ( A n ulp ) DSPEVX('L','V','I', ... ) 44=  I  U U'  / ( n ulp ) DSPEVX('L','V','I', ... ) 45= D(with Z)  D(w/o Z) / (D ulp) DSPEVX('L','N','I', ... ) 46=  A  U S U'  / ( A n ulp ) DSPEVX('L','V','V', ... ) 47=  I  U U'  / ( n ulp ) DSPEVX('L','V','V', ... ) 48= D(with Z)  D(w/o Z) / (D ulp) DSPEVX('L','N','V', ... ) 49=  A  U S U'  / ( A n ulp ) DSBEV('L','V', ... ) 50=  I  U U'  / ( n ulp ) DSBEV('L','V', ... ) 51= D(with Z)  D(w/o Z) / (D ulp) DSBEV('L','N', ... ) 52=  A  U S U'  / ( A n ulp ) DSBEVX('L','V','A', ... ) 53=  I  U U'  / ( n ulp ) DSBEVX('L','V','A', ... ) 54= D(with Z)  D(w/o Z) / (D ulp) DSBEVX('L','N','A', ... ) 55=  A  U S U'  / ( A n ulp ) DSBEVX('L','V','I', ... ) 56=  I  U U'  / ( n ulp ) DSBEVX('L','V','I', ... ) 57= D(with Z)  D(w/o Z) / (D ulp) DSBEVX('L','N','I', ... ) 58=  A  U S U'  / ( A n ulp ) DSBEVX('L','V','V', ... ) 59=  I  U U'  / ( n ulp ) DSBEVX('L','V','V', ... ) 60= D(with Z)  D(w/o Z) / (D ulp) DSBEVX('L','N','V', ... ) 61=  A  U S U'  / ( A n ulp ) DSYEVD('L','V', ... ) 62=  I  U U'  / ( n ulp ) DSYEVD('L','V', ... ) 63= D(with Z)  D(w/o Z) / (D ulp) DSYEVD('L','N', ... ) 64=  A  U S U'  / ( A n ulp ) DSPEVD('L','V', ... ) 65=  I  U U'  / ( n ulp ) DSPEVD('L','V', ... ) 66= D(with Z)  D(w/o Z) / (D ulp) DSPEVD('L','N', ... ) 67=  A  U S U'  / ( A n ulp ) DSBEVD('L','V', ... ) 68=  I  U U'  / ( n ulp ) DSBEVD('L','V', ... ) 69= D(with Z)  D(w/o Z) / (D ulp) DSBEVD('L','N', ... ) 70=  A  U S U'  / ( A n ulp ) DSYEVR('L','V','A', ... ) 71=  I  U U'  / ( n ulp ) DSYEVR('L','V','A', ... ) 72= D(with Z)  D(w/o Z) / (D ulp) DSYEVR('L','N','A', ... ) 73=  A  U S U'  / ( A n ulp ) DSYEVR('L','V','I', ... ) 74=  I  U U'  / ( n ulp ) DSYEVR('L','V','I', ... ) 75= D(with Z)  D(w/o Z) / (D ulp) DSYEVR('L','N','I', ... ) 76=  A  U S U'  / ( A n ulp ) DSYEVR('L','V','V', ... ) 77=  I  U U'  / ( n ulp ) DSYEVR('L','V','V', ... ) 78= D(with Z)  D(w/o Z) / (D ulp) DSYEVR('L','N','V', ... ) Tests 25 through 78 are repeated (as tests 79 through 132) with UPLO='U' To be added in 1999 79=  A  U S U'  / ( A n ulp ) DSPEVR('L','V','A', ... ) 80=  I  U U'  / ( n ulp ) DSPEVR('L','V','A', ... ) 81= D(with Z)  D(w/o Z) / (D ulp) DSPEVR('L','N','A', ... ) 82=  A  U S U'  / ( A n ulp ) DSPEVR('L','V','I', ... ) 83=  I  U U'  / ( n ulp ) DSPEVR('L','V','I', ... ) 84= D(with Z)  D(w/o Z) / (D ulp) DSPEVR('L','N','I', ... ) 85=  A  U S U'  / ( A n ulp ) DSPEVR('L','V','V', ... ) 86=  I  U U'  / ( n ulp ) DSPEVR('L','V','V', ... ) 87= D(with Z)  D(w/o Z) / (D ulp) DSPEVR('L','N','V', ... ) 88=  A  U S U'  / ( A n ulp ) DSBEVR('L','V','A', ... ) 89=  I  U U'  / ( n ulp ) DSBEVR('L','V','A', ... ) 90= D(with Z)  D(w/o Z) / (D ulp) DSBEVR('L','N','A', ... ) 91=  A  U S U'  / ( A n ulp ) DSBEVR('L','V','I', ... ) 92=  I  U U'  / ( n ulp ) DSBEVR('L','V','I', ... ) 93= D(with Z)  D(w/o Z) / (D ulp) DSBEVR('L','N','I', ... ) 94=  A  U S U'  / ( A n ulp ) DSBEVR('L','V','V', ... ) 95=  I  U U'  / ( n ulp ) DSBEVR('L','V','V', ... ) 96= D(with Z)  D(w/o Z) / (D ulp) DSBEVR('L','N','V', ... )
Definition at line 451 of file ddrvst.f.
subroutine ddrvsx  (  integer  NSIZES, 
integer, dimension( * )  NN,  
integer  NTYPES,  
logical, dimension( * )  DOTYPE,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
integer  NIUNIT,  
integer  NOUNIT,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( lda, * )  H,  
double precision, dimension( lda, * )  HT,  
double precision, dimension( * )  WR,  
double precision, dimension( * )  WI,  
double precision, dimension( * )  WRT,  
double precision, dimension( * )  WIT,  
double precision, dimension( * )  WRTMP,  
double precision, dimension( * )  WITMP,  
double precision, dimension( ldvs, * )  VS,  
integer  LDVS,  
double precision, dimension( ldvs, * )  VS1,  
double precision, dimension( 17 )  RESULT,  
double precision, dimension( * )  WORK,  
integer  LWORK,  
integer, dimension( * )  IWORK,  
logical, dimension( * )  BWORK,  
integer  INFO  
) 
DDRVSX
DDRVSX checks the nonsymmetric eigenvalue (Schur form) problem expert driver DGEESX. DDRVSX uses both test matrices generated randomly depending on data supplied in the calling sequence, as well as on data read from an input file and including precomputed condition numbers to which it compares the ones it computes. When DDRVSX is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified. For each size ("n") and each type of matrix, one matrix will be generated and used to test the nonsymmetric eigenroutines. For each matrix, 15 tests will be performed: (1) 0 if T is in Schur form, 1/ulp otherwise (no sorting of eigenvalues) (2)  A  VS T VS'  / ( n A ulp ) Here VS is the matrix of Schur eigenvectors, and T is in Schur form (no sorting of eigenvalues). (3)  I  VS VS'  / ( n ulp ) (no sorting of eigenvalues). (4) 0 if WR+sqrt(1)*WI are eigenvalues of T 1/ulp otherwise (no sorting of eigenvalues) (5) 0 if T(with VS) = T(without VS), 1/ulp otherwise (no sorting of eigenvalues) (6) 0 if eigenvalues(with VS) = eigenvalues(without VS), 1/ulp otherwise (no sorting of eigenvalues) (7) 0 if T is in Schur form, 1/ulp otherwise (with sorting of eigenvalues) (8)  A  VS T VS'  / ( n A ulp ) Here VS is the matrix of Schur eigenvectors, and T is in Schur form (with sorting of eigenvalues). (9)  I  VS VS'  / ( n ulp ) (with sorting of eigenvalues). (10) 0 if WR+sqrt(1)*WI are eigenvalues of T 1/ulp otherwise If workspace sufficient, also compare WR, WI with and without reciprocal condition numbers (with sorting of eigenvalues) (11) 0 if T(with VS) = T(without VS), 1/ulp otherwise If workspace sufficient, also compare T with and without reciprocal condition numbers (with sorting of eigenvalues) (12) 0 if eigenvalues(with VS) = eigenvalues(without VS), 1/ulp otherwise If workspace sufficient, also compare VS with and without reciprocal condition numbers (with sorting of eigenvalues) (13) if sorting worked and SDIM is the number of eigenvalues which were SELECTed If workspace sufficient, also compare SDIM with and without reciprocal condition numbers (14) if RCONDE the same no matter if VS and/or RCONDV computed (15) if RCONDV the same no matter if VS and/or RCONDE computed The "sizes" are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) The zero matrix. (2) The identity matrix. (3) A (transposed) Jordan block, with 1's on the diagonal. (4) A diagonal matrix with evenly spaced entries 1, ..., ULP and random signs. (ULP = (first number larger than 1)  1 ) (5) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random signs. (6) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP and random signs. (7) Same as (4), but multiplied by a constant near the overflow threshold (8) Same as (4), but multiplied by a constant near the underflow threshold (9) A matrix of the form U' T U, where U is orthogonal and T has evenly spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (10) A matrix of the form U' T U, where U is orthogonal and T has geometrically spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (11) A matrix of the form U' T U, where U is orthogonal and T has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (12) A matrix of the form U' T U, where U is orthogonal and T has real or complex conjugate paired eigenvalues randomly chosen from ( ULP, 1 ) and random O(1) entries in the upper triangle. (13) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has evenly spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (14) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has geometrically spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (15) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (16) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has real or complex conjugate paired eigenvalues randomly chosen from ( ULP, 1 ) and random O(1) entries in the upper triangle. (17) Same as (16), but multiplied by a constant near the overflow threshold (18) Same as (16), but multiplied by a constant near the underflow threshold (19) Nonsymmetric matrix with random entries chosen from (1,1). If N is at least 4, all entries in first two rows and last row, and first column and last two columns are zero. (20) Same as (19), but multiplied by a constant near the overflow threshold (21) Same as (19), but multiplied by a constant near the underflow threshold In addition, an input file will be read from logical unit number NIUNIT. The file contains matrices along with precomputed eigenvalues and reciprocal condition numbers for the eigenvalue average and right invariant subspace. For these matrices, in addition to tests (1) to (15) we will compute the following two tests: (16) RCONDE  RCDEIN / cond(RCONDE) RCONDE is the reciprocal average eigenvalue condition number computed by DGEESX and RCDEIN (the precomputed true value) is supplied as input. cond(RCONDE) is the condition number of RCONDE, and takes errors in computing RCONDE into account, so that the resulting quantity should be O(ULP). cond(RCONDE) is essentially given by norm(A)/RCONDV. (17) RCONDV  RCDVIN / cond(RCONDV) RCONDV is the reciprocal right invariant subspace condition number computed by DGEESX and RCDVIN (the precomputed true value) is supplied as input. cond(RCONDV) is the condition number of RCONDV, and takes errors in computing RCONDV into account, so that the resulting quantity should be O(ULP). cond(RCONDV) is essentially given by norm(A)/RCONDE.
[in]  NSIZES  NSIZES is INTEGER The number of sizes of matrices to use. NSIZES must be at least zero. If it is zero, no randomly generated matrices are tested, but any test matrices read from NIUNIT will be tested. 
[in]  NN  NN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero. 
[in]  NTYPES  NTYPES is INTEGER The number of elements in DOTYPE. NTYPES must be at least zero. If it is zero, no randomly generated test matrices are tested, but and test matrices read from NIUNIT will be tested. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. . 
[in]  DOTYPE  DOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored. 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to DDRVSX to continue the same random number sequence. 
[in]  THRESH  THRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero. 
[in]  NIUNIT  NIUNIT is INTEGER The FORTRAN unit number for reading in the data file of problems to solve. 
[in]  NOUNIT  NOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns INFO not equal to 0.) 
[out]  A  A is DOUBLE PRECISION array, dimension (LDA, max(NN)) Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used. 
[in]  LDA  LDA is INTEGER The leading dimension of A, and H. LDA must be at least 1 and at least max( NN ). 
[out]  H  H is DOUBLE PRECISION array, dimension (LDA, max(NN)) Another copy of the test matrix A, modified by DGEESX. 
[out]  HT  HT is DOUBLE PRECISION array, dimension (LDA, max(NN)) Yet another copy of the test matrix A, modified by DGEESX. 
[out]  WR  WR is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  WI  WI is DOUBLE PRECISION array, dimension (max(NN)) The real and imaginary parts of the eigenvalues of A. On exit, WR + WI*i are the eigenvalues of the matrix in A. 
[out]  WRT  WRT is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  WIT  WIT is DOUBLE PRECISION array, dimension (max(NN)) Like WR, WI, these arrays contain the eigenvalues of A, but those computed when DGEESX only computes a partial eigendecomposition, i.e. not Schur vectors 
[out]  WRTMP  WRTMP is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  WITMP  WITMP is DOUBLE PRECISION array, dimension (max(NN)) More temporary storage for eigenvalues. 
[out]  VS  VS is DOUBLE PRECISION array, dimension (LDVS, max(NN)) VS holds the computed Schur vectors. 
[in]  LDVS  LDVS is INTEGER Leading dimension of VS. Must be at least max(1,max(NN)). 
[out]  VS1  VS1 is DOUBLE PRECISION array, dimension (LDVS, max(NN)) VS1 holds another copy of the computed Schur vectors. 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (17) The values computed by the 17 tests described above. The values are currently limited to 1/ulp, to avoid overflow. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER The number of entries in WORK. This must be at least max(3*NN(j),2*NN(j)**2) for all j. 
[out]  IWORK  IWORK is INTEGER array, dimension (max(NN)*max(NN)) 
[out]  BWORK  BWORK is LOGICAL array, dimension (max(NN)) 
[out]  INFO  INFO is INTEGER If 0, successful exit. <0, input parameter INFO is incorrect >0, DLATMR, SLATMS, SLATME or DGET24 returned an error code and INFO is its absolute value  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. NMAX Largest value in NN. NERRS The number of tests which have exceeded THRESH COND, CONDS, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. ULP, ULPINV Finest relative precision and its inverse. RTULP, RTULPI Square roots of the previous 4 values. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) ) KCONDS(j) Selectw whether CONDS is to be 1 or 1/sqrt(ulp). (0 means irrelevant.) 
Definition at line 452 of file ddrvsx.f.
subroutine ddrvvx  (  integer  NSIZES, 
integer, dimension( * )  NN,  
integer  NTYPES,  
logical, dimension( * )  DOTYPE,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
integer  NIUNIT,  
integer  NOUNIT,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( lda, * )  H,  
double precision, dimension( * )  WR,  
double precision, dimension( * )  WI,  
double precision, dimension( * )  WR1,  
double precision, dimension( * )  WI1,  
double precision, dimension( ldvl, * )  VL,  
integer  LDVL,  
double precision, dimension( ldvr, * )  VR,  
integer  LDVR,  
double precision, dimension( ldlre, * )  LRE,  
integer  LDLRE,  
double precision, dimension( * )  RCONDV,  
double precision, dimension( * )  RCNDV1,  
double precision, dimension( * )  RCDVIN,  
double precision, dimension( * )  RCONDE,  
double precision, dimension( * )  RCNDE1,  
double precision, dimension( * )  RCDEIN,  
double precision, dimension( * )  SCALE,  
double precision, dimension( * )  SCALE1,  
double precision, dimension( 11 )  RESULT,  
double precision, dimension( * )  WORK,  
integer  NWORK,  
integer, dimension( * )  IWORK,  
integer  INFO  
) 
DDRVVX
DDRVVX checks the nonsymmetric eigenvalue problem expert driver DGEEVX. DDRVVX uses both test matrices generated randomly depending on data supplied in the calling sequence, as well as on data read from an input file and including precomputed condition numbers to which it compares the ones it computes. When DDRVVX is called, a number of matrix "sizes" ("n's") and a number of matrix "types" are specified in the calling sequence. For each size ("n") and each type of matrix, one matrix will be generated and used to test the nonsymmetric eigenroutines. For each matrix, 9 tests will be performed: (1)  A * VR  VR * W  / ( n A ulp ) Here VR is the matrix of unit right eigenvectors. W is a block diagonal matrix, with a 1x1 block for each real eigenvalue and a 2x2 block for each complex conjugate pair. If eigenvalues j and j+1 are a complex conjugate pair, so WR(j) = WR(j+1) = wr and WI(j) =  WI(j+1) = wi, then the 2 x 2 block corresponding to the pair will be: ( wr wi ) ( wi wr ) Such a block multiplying an n x 2 matrix ( ur ui ) on the right will be the same as multiplying ur + i*ui by wr + i*wi. (2)  A**H * VL  VL * W**H  / ( n A ulp ) Here VL is the matrix of unit left eigenvectors, A**H is the conjugate transpose of A, and W is as above. (3)  VR(i)  1  / ulp and largest component real VR(i) denotes the ith column of VR. (4)  VL(i)  1  / ulp and largest component real VL(i) denotes the ith column of VL. (5) W(full) = W(partial) W(full) denotes the eigenvalues computed when VR, VL, RCONDV and RCONDE are also computed, and W(partial) denotes the eigenvalues computed when only some of VR, VL, RCONDV, and RCONDE are computed. (6) VR(full) = VR(partial) VR(full) denotes the right eigenvectors computed when VL, RCONDV and RCONDE are computed, and VR(partial) denotes the result when only some of VL and RCONDV are computed. (7) VL(full) = VL(partial) VL(full) denotes the left eigenvectors computed when VR, RCONDV and RCONDE are computed, and VL(partial) denotes the result when only some of VR and RCONDV are computed. (8) 0 if SCALE, ILO, IHI, ABNRM (full) = SCALE, ILO, IHI, ABNRM (partial) 1/ulp otherwise SCALE, ILO, IHI and ABNRM describe how the matrix is balanced. (full) is when VR, VL, RCONDE and RCONDV are also computed, and (partial) is when some are not computed. (9) RCONDV(full) = RCONDV(partial) RCONDV(full) denotes the reciprocal condition numbers of the right eigenvectors computed when VR, VL and RCONDE are also computed. RCONDV(partial) denotes the reciprocal condition numbers when only some of VR, VL and RCONDE are computed. The "sizes" are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. Currently, the list of possible types is: (1) The zero matrix. (2) The identity matrix. (3) A (transposed) Jordan block, with 1's on the diagonal. (4) A diagonal matrix with evenly spaced entries 1, ..., ULP and random signs. (ULP = (first number larger than 1)  1 ) (5) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random signs. (6) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP and random signs. (7) Same as (4), but multiplied by a constant near the overflow threshold (8) Same as (4), but multiplied by a constant near the underflow threshold (9) A matrix of the form U' T U, where U is orthogonal and T has evenly spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (10) A matrix of the form U' T U, where U is orthogonal and T has geometrically spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (11) A matrix of the form U' T U, where U is orthogonal and T has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (12) A matrix of the form U' T U, where U is orthogonal and T has real or complex conjugate paired eigenvalues randomly chosen from ( ULP, 1 ) and random O(1) entries in the upper triangle. (13) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has evenly spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (14) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has geometrically spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (15) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has "clustered" entries 1, ULP,..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle. (16) A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has real or complex conjugate paired eigenvalues randomly chosen from ( ULP, 1 ) and random O(1) entries in the upper triangle. (17) Same as (16), but multiplied by a constant near the overflow threshold (18) Same as (16), but multiplied by a constant near the underflow threshold (19) Nonsymmetric matrix with random entries chosen from (1,1). If N is at least 4, all entries in first two rows and last row, and first column and last two columns are zero. (20) Same as (19), but multiplied by a constant near the overflow threshold (21) Same as (19), but multiplied by a constant near the underflow threshold In addition, an input file will be read from logical unit number NIUNIT. The file contains matrices along with precomputed eigenvalues and reciprocal condition numbers for the eigenvalues and right eigenvectors. For these matrices, in addition to tests (1) to (9) we will compute the following two tests: (10) RCONDV  RCDVIN / cond(RCONDV) RCONDV is the reciprocal right eigenvector condition number computed by DGEEVX and RCDVIN (the precomputed true value) is supplied as input. cond(RCONDV) is the condition number of RCONDV, and takes errors in computing RCONDV into account, so that the resulting quantity should be O(ULP). cond(RCONDV) is essentially given by norm(A)/RCONDE. (11) RCONDE  RCDEIN / cond(RCONDE) RCONDE is the reciprocal eigenvalue condition number computed by DGEEVX and RCDEIN (the precomputed true value) is supplied as input. cond(RCONDE) is the condition number of RCONDE, and takes errors in computing RCONDE into account, so that the resulting quantity should be O(ULP). cond(RCONDE) is essentially given by norm(A)/RCONDV.
[in]  NSIZES  NSIZES is INTEGER The number of sizes of matrices to use. NSIZES must be at least zero. If it is zero, no randomly generated matrices are tested, but any test matrices read from NIUNIT will be tested. 
[in]  NN  NN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. The values must be at least zero. 
[in]  NTYPES  NTYPES is INTEGER The number of elements in DOTYPE. NTYPES must be at least zero. If it is zero, no randomly generated test matrices are tested, but and test matrices read from NIUNIT will be tested. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. . 
[in]  DOTYPE  DOTYPE is LOGICAL array, dimension (NTYPES) If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored. 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to DDRVVX to continue the same random number sequence. 
[in]  THRESH  THRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero. 
[in]  NIUNIT  NIUNIT is INTEGER The FORTRAN unit number for reading in the data file of problems to solve. 
[in]  NOUNIT  NOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns INFO not equal to 0.) 
[out]  A  A is DOUBLE PRECISION array, dimension (LDA, max(NN,12)) Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used. 
[in]  LDA  LDA is INTEGER The leading dimension of the arrays A and H. LDA >= max(NN,12), since 12 is the dimension of the largest matrix in the precomputed input file. 
[out]  H  H is DOUBLE PRECISION array, dimension (LDA, max(NN,12)) Another copy of the test matrix A, modified by DGEEVX. 
[out]  WR  WR is DOUBLE PRECISION array, dimension (max(NN)) 
[out]  WI  WI is DOUBLE PRECISION array, dimension (max(NN)) The real and imaginary parts of the eigenvalues of A. On exit, WR + WI*i are the eigenvalues of the matrix in A. 
[out]  WR1  WR1 is DOUBLE PRECISION array, dimension (max(NN,12)) 
[out]  WI1  WI1 is DOUBLE PRECISION array, dimension (max(NN,12)) Like WR, WI, these arrays contain the eigenvalues of A, but those computed when DGEEVX only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors. 
[out]  VL  VL is DOUBLE PRECISION array, dimension (LDVL, max(NN,12)) VL holds the computed left eigenvectors. 
[in]  LDVL  LDVL is INTEGER Leading dimension of VL. Must be at least max(1,max(NN,12)). 
[out]  VR  VR is DOUBLE PRECISION array, dimension (LDVR, max(NN,12)) VR holds the computed right eigenvectors. 
[in]  LDVR  LDVR is INTEGER Leading dimension of VR. Must be at least max(1,max(NN,12)). 
[out]  LRE  LRE is DOUBLE PRECISION array, dimension (LDLRE, max(NN,12)) LRE holds the computed right or left eigenvectors. 
[in]  LDLRE  LDLRE is INTEGER Leading dimension of LRE. Must be at least max(1,max(NN,12)) 
[out]  RCONDV  RCONDV is DOUBLE PRECISION array, dimension (N) RCONDV holds the computed reciprocal condition numbers for eigenvectors. 
[out]  RCNDV1  RCNDV1 is DOUBLE PRECISION array, dimension (N) RCNDV1 holds more computed reciprocal condition numbers for eigenvectors. 
[out]  RCDVIN  RCDVIN is DOUBLE PRECISION array, dimension (N) When COMP = .TRUE. RCDVIN holds the precomputed reciprocal condition numbers for eigenvectors to be compared with RCONDV. 
[out]  RCONDE  RCONDE is DOUBLE PRECISION array, dimension (N) RCONDE holds the computed reciprocal condition numbers for eigenvalues. 
[out]  RCNDE1  RCNDE1 is DOUBLE PRECISION array, dimension (N) RCNDE1 holds more computed reciprocal condition numbers for eigenvalues. 
[out]  RCDEIN  RCDEIN is DOUBLE PRECISION array, dimension (N) When COMP = .TRUE. RCDEIN holds the precomputed reciprocal condition numbers for eigenvalues to be compared with RCONDE. 
[out]  SCALE  SCALE is DOUBLE PRECISION array, dimension (N) Holds information describing balancing of matrix. 
[out]  SCALE1  SCALE1 is DOUBLE PRECISION array, dimension (N) Holds information describing balancing of matrix. 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (11) The values computed by the seven tests described above. The values are currently limited to 1/ulp, to avoid overflow. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (NWORK) 
[in]  NWORK  NWORK is INTEGER The number of entries in WORK. This must be at least max(6*12+2*12**2,6*NN(j)+2*NN(j)**2) = max( 360 ,6*NN(j)+2*NN(j)**2) for all j. 
[out]  IWORK  IWORK is INTEGER array, dimension (2*max(NN,12)) 
[out]  INFO  INFO is INTEGER If 0, then successful exit. If <0, then input paramter INFO is incorrect. If >0, DLATMR, SLATMS, SLATME or DGET23 returned an error code, and INFO is its absolute value.  Some Local Variables and Parameters:      ZERO, ONE Real 0 and 1. MAXTYP The number of types defined. NMAX Largest value in NN or 12. NERRS The number of tests which have exceeded THRESH COND, CONDS, IMODE Values to be passed to the matrix generators. ANORM Norm of A; passed to matrix generators. OVFL, UNFL Overflow and underflow thresholds. ULP, ULPINV Finest relative precision and its inverse. RTULP, RTULPI Square roots of the previous 4 values. The following four arrays decode JTYPE: KTYPE(j) The general type (110) for type "j". KMODE(j) The MODE value to be passed to the matrix generator for type "j". KMAGN(j) The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) ) KCONDS(j) Selectw whether CONDS is to be 1 or 1/sqrt(ulp). (0 means irrelevant.) 
Definition at line 518 of file ddrvvx.f.
subroutine derrbd  (  character*3  PATH, 
integer  NUNIT  
) 
DERRBD
DERRBD tests the error exits for DGEBRD, DORGBR, DORMBR, DBDSQR and DBDSDC.
[in]  PATH  PATH is CHARACTER*3 The LAPACK path name for the routines to be tested. 
[in]  NUNIT  NUNIT is INTEGER The unit number for output. 
Definition at line 56 of file derrbd.f.
subroutine derrec  (  character*3  PATH, 
integer  NUNIT  
) 
DERREC
DERREC tests the error exits for the routines for eigen condition estimation for DOUBLE PRECISION matrices: DTRSYL, STREXC, STRSNA and STRSEN.
[in]  PATH  PATH is CHARACTER*3 The LAPACK path name for the routines to be tested. 
[in]  NUNIT  NUNIT is INTEGER The unit number for output. 
Definition at line 57 of file derrec.f.
subroutine derred  (  character*3  PATH, 
integer  NUNIT  
) 
DERRED
DERRED tests the error exits for the eigenvalue driver routines for DOUBLE PRECISION matrices: PATH driver description    SEV DGEEV find eigenvalues/eigenvectors for nonsymmetric A SES DGEES find eigenvalues/Schur form for nonsymmetric A SVX DGEEVX SGEEV + balancing and condition estimation SSX DGEESX SGEES + balancing and condition estimation DBD DGESVD compute SVD of an MbyN matrix A DGESDD compute SVD of an MbyN matrix A (by divide and conquer) DGEJSV compute SVD of an MbyN matrix A where M >= N
[in]  PATH  PATH is CHARACTER*3 The LAPACK path name for the routines to be tested. 
[in]  NUNIT  NUNIT is INTEGER The unit number for output. 
Definition at line 67 of file derred.f.
subroutine derrgg  (  character*3  PATH, 
integer  NUNIT  
) 
DERRGG
DERRGG tests the error exits for DGGES, DGGESX, DGGEV, DGGEVX, DGGGLM, DGGHRD, DGGLSE, DGGQRF, DGGRQF, DGGSVD, DGGSVP, DHGEQZ, DORCSD, DTGEVC, DTGEXC, DTGSEN, DTGSJA, DTGSNA, and DTGSYL.
[in]  PATH  PATH is CHARACTER*3 The LAPACK path name for the routines to be tested. 
[in]  NUNIT  NUNIT is INTEGER The unit number for output. 
Definition at line 57 of file derrgg.f.
subroutine derrhs  (  character*3  PATH, 
integer  NUNIT  
) 
DERRHS
DERRHS tests the error exits for DGEBAK, SGEBAL, SGEHRD, DORGHR, DORMHR, DHSEQR, SHSEIN, and DTREVC.
[in]  PATH  PATH is CHARACTER*3 The LAPACK path name for the routines to be tested. 
[in]  NUNIT  NUNIT is INTEGER The unit number for output. 
Definition at line 56 of file derrhs.f.
subroutine derrst  (  character*3  PATH, 
integer  NUNIT  
) 
DERRST
DERRST tests the error exits for DSYTRD, DORGTR, DORMTR, DSPTRD, DOPGTR, DOPMTR, DSTEQR, SSTERF, SSTEBZ, SSTEIN, DPTEQR, DSBTRD, DSYEV, SSYEVX, SSYEVD, DSBEV, SSBEVX, SSBEVD, DSPEV, SSPEVX, SSPEVD, DSTEV, SSTEVX, SSTEVD, and SSTEDC.
[in]  PATH  PATH is CHARACTER*3 The LAPACK path name for the routines to be tested. 
[in]  NUNIT  NUNIT is INTEGER The unit number for output. 
Definition at line 58 of file derrst.f.
subroutine dget02  (  character  TRANS, 
integer  M,  
integer  N,  
integer  NRHS,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( ldx, * )  X,  
integer  LDX,  
double precision, dimension( ldb, * )  B,  
integer  LDB,  
double precision, dimension( * )  RWORK,  
double precision  RESID  
) 
DGET02
DGET02 computes the residual for a solution of a system of linear equations A*x = b or A'*x = b: RESID = norm(B  A*X) / ( norm(A) * norm(X) * EPS ), where EPS is the machine epsilon.
[in]  TRANS  TRANS is CHARACTER*1 Specifies the form of the system of equations: = 'N': A *x = b = 'T': A'*x = b, where A' is the transpose of A = 'C': A'*x = b, where A' is the transpose of A 
[in]  M  M is INTEGER The number of rows of the matrix A. M >= 0. 
[in]  N  N is INTEGER The number of columns of the matrix A. N >= 0. 
[in]  NRHS  NRHS is INTEGER The number of columns of B, the matrix of right hand sides. NRHS >= 0. 
[in]  A  A is DOUBLE PRECISION array, dimension (LDA,N) The original M x N matrix A. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). 
[in]  X  X is DOUBLE PRECISION array, dimension (LDX,NRHS) The computed solution vectors for the system of linear equations. 
[in]  LDX  LDX is INTEGER The leading dimension of the array X. If TRANS = 'N', LDX >= max(1,N); if TRANS = 'T' or 'C', LDX >= max(1,M). 
[in,out]  B  B is DOUBLE PRECISION array, dimension (LDB,NRHS) On entry, the right hand side vectors for the system of linear equations. On exit, B is overwritten with the difference B  A*X. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. IF TRANS = 'N', LDB >= max(1,M); if TRANS = 'T' or 'C', LDB >= max(1,N). 
[out]  RWORK  RWORK is DOUBLE PRECISION array, dimension (M) 
[out]  RESID  RESID is DOUBLE PRECISION The maximum over the number of right hand sides of norm(B  A*X) / ( norm(A) * norm(X) * EPS ). 
Definition at line 133 of file dget02.f.
subroutine dget10  (  integer  M, 
integer  N,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( ldb, * )  B,  
integer  LDB,  
double precision, dimension( * )  WORK,  
double precision  RESULT  
) 
DGET10
DGET10 compares two matrices A and B and computes the ratio RESULT = norm( A  B ) / ( norm(A) * M * EPS )
[in]  M  M is INTEGER The number of rows of the matrices A and B. 
[in]  N  N is INTEGER The number of columns of the matrices A and B. 
[in]  A  A is DOUBLE PRECISION array, dimension (LDA,N) The m by n matrix A. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). 
[in]  B  B is DOUBLE PRECISION array, dimension (LDB,N) The m by n matrix B. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,M). 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (M) 
[out]  RESULT  RESULT is DOUBLE PRECISION RESULT = norm( A  B ) / ( norm(A) * M * EPS ) 
Definition at line 94 of file dget10.f.
subroutine dget22  (  character  TRANSA, 
character  TRANSE,  
character  TRANSW,  
integer  N,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( lde, * )  E,  
integer  LDE,  
double precision, dimension( * )  WR,  
double precision, dimension( * )  WI,  
double precision, dimension( * )  WORK,  
double precision, dimension( 2 )  RESULT  
) 
DGET22
DGET22 does an eigenvector check. The basic test is: RESULT(1) =  A E  E W  / ( A E ulp ) using the 1norm. It also tests the normalization of E: RESULT(2) = max  mnorm(E(j))  1  / ( n ulp ) j where E(j) is the jth eigenvector, and mnorm is the maxnorm of a vector. If an eigenvector is complex, as determined from WI(j) nonzero, then the maxnorm of the vector ( er + i*ei ) is the maximum of er(1) + ei(1), ... , er(n) + ei(n) W is a block diagonal matrix, with a 1 by 1 block for each real eigenvalue and a 2 by 2 block for each complex conjugate pair. If eigenvalues j and j+1 are a complex conjugate pair, so that WR(j) = WR(j+1) = wr and WI(j) =  WI(j+1) = wi, then the 2 by 2 block corresponding to the pair will be: ( wr wi ) ( wi wr ) Such a block multiplying an n by 2 matrix ( ur ui ) on the right will be the same as multiplying ur + i*ui by wr + i*wi. To handle various schemes for storage of left eigenvectors, there are options to use Atranspose instead of A, Etranspose instead of E, and/or Wtranspose instead of W.
[in]  TRANSA  TRANSA is CHARACTER*1 Specifies whether or not A is transposed. = 'N': No transpose = 'T': Transpose = 'C': Conjugate transpose (= Transpose) 
[in]  TRANSE  TRANSE is CHARACTER*1 Specifies whether or not E is transposed. = 'N': No transpose, eigenvectors are in columns of E = 'T': Transpose, eigenvectors are in rows of E = 'C': Conjugate transpose (= Transpose) 
[in]  TRANSW  TRANSW is CHARACTER*1 Specifies whether or not W is transposed. = 'N': No transpose = 'T': Transpose, use WI(j) instead of WI(j) = 'C': Conjugate transpose, use WI(j) instead of WI(j) 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in]  A  A is DOUBLE PRECISION array, dimension (LDA,N) The matrix whose eigenvectors are in E. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in]  E  E is DOUBLE PRECISION array, dimension (LDE,N) The matrix of eigenvectors. If TRANSE = 'N', the eigenvectors are stored in the columns of E, if TRANSE = 'T' or 'C', the eigenvectors are stored in the rows of E. 
[in]  LDE  LDE is INTEGER The leading dimension of the array E. LDE >= max(1,N). 
[in]  WR  WR is DOUBLE PRECISION array, dimension (N) 
[in]  WI  WI is DOUBLE PRECISION array, dimension (N) The real and imaginary parts of the eigenvalues of A. Purely real eigenvalues are indicated by WI(j) = 0. Complex conjugate pairs are indicated by WR(j)=WR(j+1) and WI(j) =  WI(j+1) nonzero; the real part is assumed to be stored in the jth row/column and the imaginary part in the (j+1)th row/column. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (N*(N+1)) 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (2) RESULT(1) =  A E  E W  / ( A E ulp ) RESULT(2) = max  mnorm(E(j))  1  / ( n ulp ) 
Definition at line 167 of file dget22.f.
subroutine dget23  (  logical  COMP, 
character  BALANC,  
integer  JTYPE,  
double precision  THRESH,  
integer, dimension( 4 )  ISEED,  
integer  NOUNIT,  
integer  N,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( lda, * )  H,  
double precision, dimension( * )  WR,  
double precision, dimension( * )  WI,  
double precision, dimension( * )  WR1,  
double precision, dimension( * )  WI1,  
double precision, dimension( ldvl, * )  VL,  
integer  LDVL,  
double precision, dimension( ldvr, * )  VR,  
integer  LDVR,  
double precision, dimension( ldlre, * )  LRE,  
integer  LDLRE,  
double precision, dimension( * )  RCONDV,  
double precision, dimension( * )  RCNDV1,  
double precision, dimension( * )  RCDVIN,  
double precision, dimension( * )  RCONDE,  
double precision, dimension( * )  RCNDE1,  
double precision, dimension( * )  RCDEIN,  
double precision, dimension( * )  SCALE,  
double precision, dimension( * )  SCALE1,  
double precision, dimension( 11 )  RESULT,  
double precision, dimension( * )  WORK,  
integer  LWORK,  
integer, dimension( * )  IWORK,  
integer  INFO  
) 
DGET23
DGET23 checks the nonsymmetric eigenvalue problem driver SGEEVX. If COMP = .FALSE., the first 8 of the following tests will be performed on the input matrix A, and also test 9 if LWORK is sufficiently large. if COMP is .TRUE. all 11 tests will be performed. (1)  A * VR  VR * W  / ( n A ulp ) Here VR is the matrix of unit right eigenvectors. W is a block diagonal matrix, with a 1x1 block for each real eigenvalue and a 2x2 block for each complex conjugate pair. If eigenvalues j and j+1 are a complex conjugate pair, so WR(j) = WR(j+1) = wr and WI(j) =  WI(j+1) = wi, then the 2 x 2 block corresponding to the pair will be: ( wr wi ) ( wi wr ) Such a block multiplying an n x 2 matrix ( ur ui ) on the right will be the same as multiplying ur + i*ui by wr + i*wi. (2)  A**H * VL  VL * W**H  / ( n A ulp ) Here VL is the matrix of unit left eigenvectors, A**H is the conjugate transpose of A, and W is as above. (3)  VR(i)  1  / ulp and largest component real VR(i) denotes the ith column of VR. (4)  VL(i)  1  / ulp and largest component real VL(i) denotes the ith column of VL. (5) 0 if W(full) = W(partial), 1/ulp otherwise W(full) denotes the eigenvalues computed when VR, VL, RCONDV and RCONDE are also computed, and W(partial) denotes the eigenvalues computed when only some of VR, VL, RCONDV, and RCONDE are computed. (6) 0 if VR(full) = VR(partial), 1/ulp otherwise VR(full) denotes the right eigenvectors computed when VL, RCONDV and RCONDE are computed, and VR(partial) denotes the result when only some of VL and RCONDV are computed. (7) 0 if VL(full) = VL(partial), 1/ulp otherwise VL(full) denotes the left eigenvectors computed when VR, RCONDV and RCONDE are computed, and VL(partial) denotes the result when only some of VR and RCONDV are computed. (8) 0 if SCALE, ILO, IHI, ABNRM (full) = SCALE, ILO, IHI, ABNRM (partial) 1/ulp otherwise SCALE, ILO, IHI and ABNRM describe how the matrix is balanced. (full) is when VR, VL, RCONDE and RCONDV are also computed, and (partial) is when some are not computed. (9) 0 if RCONDV(full) = RCONDV(partial), 1/ulp otherwise RCONDV(full) denotes the reciprocal condition numbers of the right eigenvectors computed when VR, VL and RCONDE are also computed. RCONDV(partial) denotes the reciprocal condition numbers when only some of VR, VL and RCONDE are computed. (10) RCONDV  RCDVIN / cond(RCONDV) RCONDV is the reciprocal right eigenvector condition number computed by DGEEVX and RCDVIN (the precomputed true value) is supplied as input. cond(RCONDV) is the condition number of RCONDV, and takes errors in computing RCONDV into account, so that the resulting quantity should be O(ULP). cond(RCONDV) is essentially given by norm(A)/RCONDE. (11) RCONDE  RCDEIN / cond(RCONDE) RCONDE is the reciprocal eigenvalue condition number computed by DGEEVX and RCDEIN (the precomputed true value) is supplied as input. cond(RCONDE) is the condition number of RCONDE, and takes errors in computing RCONDE into account, so that the resulting quantity should be O(ULP). cond(RCONDE) is essentially given by norm(A)/RCONDV.
[in]  COMP  COMP is LOGICAL COMP describes which input tests to perform: = .FALSE. if the computed condition numbers are not to be tested against RCDVIN and RCDEIN = .TRUE. if they are to be compared 
[in]  BALANC  BALANC is CHARACTER Describes the balancing option to be tested. = 'N' for no permuting or diagonal scaling = 'P' for permuting but no diagonal scaling = 'S' for no permuting but diagonal scaling = 'B' for permuting and diagonal scaling 
[in]  JTYPE  JTYPE is INTEGER Type of input matrix. Used to label output if error occurs. 
[in]  THRESH  THRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero. 
[in]  ISEED  ISEED is INTEGER array, dimension (4) If COMP = .FALSE., the random number generator seed used to produce matrix. If COMP = .TRUE., ISEED(1) = the number of the example. Used to label output if error occurs. 
[in]  NOUNIT  NOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns INFO not equal to 0.) 
[in]  N  N is INTEGER The dimension of A. N must be at least 0. 
[in,out]  A  A is DOUBLE PRECISION array, dimension (LDA,N) Used to hold the matrix whose eigenvalues are to be computed. 
[in]  LDA  LDA is INTEGER The leading dimension of A, and H. LDA must be at least 1 and at least N. 
[out]  H  H is DOUBLE PRECISION array, dimension (LDA,N) Another copy of the test matrix A, modified by DGEEVX. 
[out]  WR  WR is DOUBLE PRECISION array, dimension (N) 
[out]  WI  WI is DOUBLE PRECISION array, dimension (N) The real and imaginary parts of the eigenvalues of A. On exit, WR + WI*i are the eigenvalues of the matrix in A. 
[out]  WR1  WR1 is DOUBLE PRECISION array, dimension (N) 
[out]  WI1  WI1 is DOUBLE PRECISION array, dimension (N) Like WR, WI, these arrays contain the eigenvalues of A, but those computed when DGEEVX only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors. 
[out]  VL  VL is DOUBLE PRECISION array, dimension (LDVL,N) VL holds the computed left eigenvectors. 
[in]  LDVL  LDVL is INTEGER Leading dimension of VL. Must be at least max(1,N). 
[out]  VR  VR is DOUBLE PRECISION array, dimension (LDVR,N) VR holds the computed right eigenvectors. 
[in]  LDVR  LDVR is INTEGER Leading dimension of VR. Must be at least max(1,N). 
[out]  LRE  LRE is DOUBLE PRECISION array, dimension (LDLRE,N) LRE holds the computed right or left eigenvectors. 
[in]  LDLRE  LDLRE is INTEGER Leading dimension of LRE. Must be at least max(1,N). 
[out]  RCONDV  RCONDV is DOUBLE PRECISION array, dimension (N) RCONDV holds the computed reciprocal condition numbers for eigenvectors. 
[out]  RCNDV1  RCNDV1 is DOUBLE PRECISION array, dimension (N) RCNDV1 holds more computed reciprocal condition numbers for eigenvectors. 
[in]  RCDVIN  RCDVIN is DOUBLE PRECISION array, dimension (N) When COMP = .TRUE. RCDVIN holds the precomputed reciprocal condition numbers for eigenvectors to be compared with RCONDV. 
[out]  RCONDE  RCONDE is DOUBLE PRECISION array, dimension (N) RCONDE holds the computed reciprocal condition numbers for eigenvalues. 
[out]  RCNDE1  RCNDE1 is DOUBLE PRECISION array, dimension (N) RCNDE1 holds more computed reciprocal condition numbers for eigenvalues. 
[in]  RCDEIN  RCDEIN is DOUBLE PRECISION array, dimension (N) When COMP = .TRUE. RCDEIN holds the precomputed reciprocal condition numbers for eigenvalues to be compared with RCONDE. 
[out]  SCALE  SCALE is DOUBLE PRECISION array, dimension (N) Holds information describing balancing of matrix. 
[out]  SCALE1  SCALE1 is DOUBLE PRECISION array, dimension (N) Holds information describing balancing of matrix. 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (11) The values computed by the 11 tests described above. The values are currently limited to 1/ulp, to avoid overflow. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER The number of entries in WORK. This must be at least 3*N, and 6*N+N**2 if tests 9, 10 or 11 are to be performed. 
[out]  IWORK  IWORK is INTEGER array, dimension (2*N) 
[out]  INFO  INFO is INTEGER If 0, successful exit. If <0, input parameter INFO had an incorrect value. If >0, DGEEVX returned an error code, the absolute value of which is returned. 
Definition at line 375 of file dget23.f.
subroutine dget24  (  logical  COMP, 
integer  JTYPE,  
double precision  THRESH,  
integer, dimension( 4 )  ISEED,  
integer  NOUNIT,  
integer  N,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( lda, * )  H,  
double precision, dimension( lda, * )  HT,  
double precision, dimension( * )  WR,  
double precision, dimension( * )  WI,  
double precision, dimension( * )  WRT,  
double precision, dimension( * )  WIT,  
double precision, dimension( * )  WRTMP,  
double precision, dimension( * )  WITMP,  
double precision, dimension( ldvs, * )  VS,  
integer  LDVS,  
double precision, dimension( ldvs, * )  VS1,  
double precision  RCDEIN,  
double precision  RCDVIN,  
integer  NSLCT,  
integer, dimension( * )  ISLCT,  
double precision, dimension( 17 )  RESULT,  
double precision, dimension( * )  WORK,  
integer  LWORK,  
integer, dimension( * )  IWORK,  
logical, dimension( * )  BWORK,  
integer  INFO  
) 
DGET24
DGET24 checks the nonsymmetric eigenvalue (Schur form) problem expert driver DGEESX. If COMP = .FALSE., the first 13 of the following tests will be be performed on the input matrix A, and also tests 14 and 15 if LWORK is sufficiently large. If COMP = .TRUE., all 17 test will be performed. (1) 0 if T is in Schur form, 1/ulp otherwise (no sorting of eigenvalues) (2)  A  VS T VS'  / ( n A ulp ) Here VS is the matrix of Schur eigenvectors, and T is in Schur form (no sorting of eigenvalues). (3)  I  VS VS'  / ( n ulp ) (no sorting of eigenvalues). (4) 0 if WR+sqrt(1)*WI are eigenvalues of T 1/ulp otherwise (no sorting of eigenvalues) (5) 0 if T(with VS) = T(without VS), 1/ulp otherwise (no sorting of eigenvalues) (6) 0 if eigenvalues(with VS) = eigenvalues(without VS), 1/ulp otherwise (no sorting of eigenvalues) (7) 0 if T is in Schur form, 1/ulp otherwise (with sorting of eigenvalues) (8)  A  VS T VS'  / ( n A ulp ) Here VS is the matrix of Schur eigenvectors, and T is in Schur form (with sorting of eigenvalues). (9)  I  VS VS'  / ( n ulp ) (with sorting of eigenvalues). (10) 0 if WR+sqrt(1)*WI are eigenvalues of T 1/ulp otherwise If workspace sufficient, also compare WR, WI with and without reciprocal condition numbers (with sorting of eigenvalues) (11) 0 if T(with VS) = T(without VS), 1/ulp otherwise If workspace sufficient, also compare T with and without reciprocal condition numbers (with sorting of eigenvalues) (12) 0 if eigenvalues(with VS) = eigenvalues(without VS), 1/ulp otherwise If workspace sufficient, also compare VS with and without reciprocal condition numbers (with sorting of eigenvalues) (13) if sorting worked and SDIM is the number of eigenvalues which were SELECTed If workspace sufficient, also compare SDIM with and without reciprocal condition numbers (14) if RCONDE the same no matter if VS and/or RCONDV computed (15) if RCONDV the same no matter if VS and/or RCONDE computed (16) RCONDE  RCDEIN / cond(RCONDE) RCONDE is the reciprocal average eigenvalue condition number computed by DGEESX and RCDEIN (the precomputed true value) is supplied as input. cond(RCONDE) is the condition number of RCONDE, and takes errors in computing RCONDE into account, so that the resulting quantity should be O(ULP). cond(RCONDE) is essentially given by norm(A)/RCONDV. (17) RCONDV  RCDVIN / cond(RCONDV) RCONDV is the reciprocal right invariant subspace condition number computed by DGEESX and RCDVIN (the precomputed true value) is supplied as input. cond(RCONDV) is the condition number of RCONDV, and takes errors in computing RCONDV into account, so that the resulting quantity should be O(ULP). cond(RCONDV) is essentially given by norm(A)/RCONDE.
[in]  COMP  COMP is LOGICAL COMP describes which input tests to perform: = .FALSE. if the computed condition numbers are not to be tested against RCDVIN and RCDEIN = .TRUE. if they are to be compared 
[in]  JTYPE  JTYPE is INTEGER Type of input matrix. Used to label output if error occurs. 
[in]  ISEED  ISEED is INTEGER array, dimension (4) If COMP = .FALSE., the random number generator seed used to produce matrix. If COMP = .TRUE., ISEED(1) = the number of the example. Used to label output if error occurs. 
[in]  THRESH  THRESH is DOUBLE PRECISION A test will count as "failed" if the "error", computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero. 
[in]  NOUNIT  NOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns INFO not equal to 0.) 
[in]  N  N is INTEGER The dimension of A. N must be at least 0. 
[in,out]  A  A is DOUBLE PRECISION array, dimension (LDA, N) Used to hold the matrix whose eigenvalues are to be computed. 
[in]  LDA  LDA is INTEGER The leading dimension of A, and H. LDA must be at least 1 and at least N. 
[out]  H  H is DOUBLE PRECISION array, dimension (LDA, N) Another copy of the test matrix A, modified by DGEESX. 
[out]  HT  HT is DOUBLE PRECISION array, dimension (LDA, N) Yet another copy of the test matrix A, modified by DGEESX. 
[out]  WR  WR is DOUBLE PRECISION array, dimension (N) 
[out]  WI  WI is DOUBLE PRECISION array, dimension (N) The real and imaginary parts of the eigenvalues of A. On exit, WR + WI*i are the eigenvalues of the matrix in A. 
[out]  WRT  WRT is DOUBLE PRECISION array, dimension (N) 
[out]  WIT  WIT is DOUBLE PRECISION array, dimension (N) Like WR, WI, these arrays contain the eigenvalues of A, but those computed when DGEESX only computes a partial eigendecomposition, i.e. not Schur vectors 
[out]  WRTMP  WRTMP is DOUBLE PRECISION array, dimension (N) 
[out]  WITMP  WITMP is DOUBLE PRECISION array, dimension (N) Like WR, WI, these arrays contain the eigenvalues of A, but sorted by increasing real part. 
[out]  VS  VS is DOUBLE PRECISION array, dimension (LDVS, N) VS holds the computed Schur vectors. 
[in]  LDVS  LDVS is INTEGER Leading dimension of VS. Must be at least max(1, N). 
[out]  VS1  VS1 is DOUBLE PRECISION array, dimension (LDVS, N) VS1 holds another copy of the computed Schur vectors. 
[in]  RCDEIN  RCDEIN is DOUBLE PRECISION When COMP = .TRUE. RCDEIN holds the precomputed reciprocal condition number for the average of selected eigenvalues. 
[in]  RCDVIN  RCDVIN is DOUBLE PRECISION When COMP = .TRUE. RCDVIN holds the precomputed reciprocal condition number for the selected right invariant subspace. 
[in]  NSLCT  NSLCT is INTEGER When COMP = .TRUE. the number of selected eigenvalues corresponding to the precomputed values RCDEIN and RCDVIN. 
[in]  ISLCT  ISLCT is INTEGER array, dimension (NSLCT) When COMP = .TRUE. ISLCT selects the eigenvalues of the input matrix corresponding to the precomputed values RCDEIN and RCDVIN. For I=1, ... ,NSLCT, if ISLCT(I) = J, then the eigenvalue with the Jth largest real part is selected. Not referenced if COMP = .FALSE. 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (17) The values computed by the 17 tests described above. The values are currently limited to 1/ulp, to avoid overflow. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER The number of entries in WORK to be passed to DGEESX. This must be at least 3*N, and N+N**2 if tests 1416 are to be performed. 
[out]  IWORK  IWORK is INTEGER array, dimension (N*N) 
[out]  BWORK  BWORK is LOGICAL array, dimension (N) 
[out]  INFO  INFO is INTEGER If 0, successful exit. If <0, input parameter INFO had an incorrect value. If >0, DGEESX returned an error code, the absolute value of which is returned. 
Definition at line 341 of file dget24.f.
subroutine dget31  (  double precision  RMAX, 
integer  LMAX,  
integer, dimension( 2 )  NINFO,  
integer  KNT  
) 
DGET31
DGET31 tests DLALN2, a routine for solving (ca A  w D)X = sB where A is an NA by NA matrix (NA=1 or 2 only), w is a real (NW=1) or complex (NW=2) constant, ca is a real constant, D is an NA by NA real diagonal matrix, and B is an NA by NW matrix (when NW=2 the second column of B contains the imaginary part of the solution). The code returns X and s, where s is a scale factor, less than or equal to 1, which is chosen to avoid overflow in X. If any singular values of ca Aw D are less than another input parameter SMIN, they are perturbed up to SMIN. The test condition is that the scaled residual norm( (ca Aw D)*X  s*B ) / ( max( ulp*norm(ca Aw D), SMIN )*norm(X) ) should be on the order of 1. Here, ulp is the machine precision. Also, it is verified that SCALE is less than or equal to 1, and that XNORM = infinitynorm(X).
[out]  RMAX  RMAX is DOUBLE PRECISION Value of the largest test ratio. 
[out]  LMAX  LMAX is INTEGER Example number where largest test ratio achieved. 
[out]  NINFO  NINFO is INTEGER array, dimension (3) NINFO(1) = number of examples with INFO less than 0 NINFO(2) = number of examples with INFO greater than 0 
[out]  KNT  KNT is INTEGER Total number of examples tested. 
Definition at line 92 of file dget31.f.
subroutine dget32  (  double precision  RMAX, 
integer  LMAX,  
integer  NINFO,  
integer  KNT  
) 
DGET32
DGET32 tests DLASY2, a routine for solving op(TL)*X + ISGN*X*op(TR) = SCALE*B where TL is N1 by N1, TR is N2 by N2, and N1,N2 =1 or 2 only. X and B are N1 by N2, op() is an optional transpose, an ISGN = 1 or 1. SCALE is chosen less than or equal to 1 to avoid overflow in X. The test condition is that the scaled residual norm( op(TL)*X + ISGN*X*op(TR) = SCALE*B ) / ( max( ulp*norm(TL), ulp*norm(TR)) * norm(X), SMLNUM ) should be on the order of 1. Here, ulp is the machine precision. Also, it is verified that SCALE is less than or equal to 1, and that XNORM = infinitynorm(X).
[out]  RMAX  RMAX is DOUBLE PRECISION Value of the largest test ratio. 
[out]  LMAX  LMAX is INTEGER Example number where largest test ratio achieved. 
[out]  NINFO  NINFO is INTEGER Number of examples returned with INFO.NE.0. 
[out]  KNT  KNT is INTEGER Total number of examples tested. 
Definition at line 83 of file dget32.f.
subroutine dget33  (  double precision  RMAX, 
integer  LMAX,  
integer  NINFO,  
integer  KNT  
) 
DGET33
DGET33 tests DLANV2, a routine for putting 2 by 2 blocks into standard form. In other words, it computes a two by two rotation [[C,S];[S,C]] where in [ C S ][T(1,1) T(1,2)][ C S ] = [ T11 T12 ] [S C ][T(2,1) T(2,2)][ S C ] [ T21 T22 ] either 1) T21=0 (real eigenvalues), or 2) T11=T22 and T21*T12<0 (complex conjugate eigenvalues). We also verify that the residual is small.
[out]  RMAX  RMAX is DOUBLE PRECISION Value of the largest test ratio. 
[out]  LMAX  LMAX is INTEGER Example number where largest test ratio achieved. 
[out]  NINFO  NINFO is INTEGER Number of examples returned with INFO .NE. 0. 
[out]  KNT  KNT is INTEGER Total number of examples tested. 
Definition at line 77 of file dget33.f.
subroutine dget34  (  double precision  RMAX, 
integer  LMAX,  
integer, dimension( 2 )  NINFO,  
integer  KNT  
) 
DGET34
DGET34 tests DLAEXC, a routine for swapping adjacent blocks (either 1 by 1 or 2 by 2) on the diagonal of a matrix in real Schur form. Thus, DLAEXC computes an orthogonal matrix Q such that Q' * [ A B ] * Q = [ C1 B1 ] [ 0 C ] [ 0 A1 ] where C1 is similar to C and A1 is similar to A. Both A and C are assumed to be in standard form (equal diagonal entries and offdiagonal with differing signs) and A1 and C1 are returned with the same properties. The test code verifies these last last assertions, as well as that the residual in the above equation is small.
[out]  RMAX  RMAX is DOUBLE PRECISION Value of the largest test ratio. 
[out]  LMAX  LMAX is INTEGER Example number where largest test ratio achieved. 
[out]  NINFO  NINFO is INTEGER array, dimension (2) NINFO(J) is the number of examples where INFO=J occurred. 
[out]  KNT  KNT is INTEGER Total number of examples tested. 
Definition at line 83 of file dget34.f.
subroutine dget35  (  double precision  RMAX, 
integer  LMAX,  
integer  NINFO,  
integer  KNT  
) 
DGET35
DGET35 tests DTRSYL, a routine for solving the Sylvester matrix equation op(A)*X + ISGN*X*op(B) = scale*C, A and B are assumed to be in Schur canonical form, op() represents an optional transpose, and ISGN can be 1 or +1. Scale is an output less than or equal to 1, chosen to avoid overflow in X. The test code verifies that the following residual is order 1: norm(op(A)*X + ISGN*X*op(B)  scale*C) / (EPS*max(norm(A),norm(B))*norm(X))
[out]  RMAX  RMAX is DOUBLE PRECISION Value of the largest test ratio. 
[out]  LMAX  LMAX is INTEGER Example number where largest test ratio achieved. 
[out]  NINFO  NINFO is INTEGER Number of examples where INFO is nonzero. 
[out]  KNT  KNT is INTEGER Total number of examples tested. 
Definition at line 79 of file dget35.f.
subroutine dget36  (  double precision  RMAX, 
integer  LMAX,  
integer, dimension( 3 )  NINFO,  
integer  KNT,  
integer  NIN  
) 
DGET36
DGET36 tests DTREXC, a routine for moving blocks (either 1 by 1 or 2 by 2) on the diagonal of a matrix in real Schur form. Thus, DLAEXC computes an orthogonal matrix Q such that Q' * T1 * Q = T2 and where one of the diagonal blocks of T1 (the one at row IFST) has been moved to position ILST. The test code verifies that the residual Q'*T1*QT2 is small, that T2 is in Schur form, and that the final position of the IFST block is ILST (within +1). The test matrices are read from a file with logical unit number NIN.
[out]  RMAX  RMAX is DOUBLE PRECISION Value of the largest test ratio. 
[out]  LMAX  LMAX is INTEGER Example number where largest test ratio achieved. 
[out]  NINFO  NINFO is INTEGER array, dimension (3) NINFO(J) is the number of examples where INFO=J. 
[out]  KNT  KNT is INTEGER Total number of examples tested. 
[in]  NIN  NIN is INTEGER Input logical unit number. 
Definition at line 89 of file dget36.f.
subroutine dget37  (  double precision, dimension( 3 )  RMAX, 
integer, dimension( 3 )  LMAX,  
integer, dimension( 3 )  NINFO,  
integer  KNT,  
integer  NIN  
) 
DGET37
DGET37 tests DTRSNA, a routine for estimating condition numbers of eigenvalues and/or right eigenvectors of a matrix. The test matrices are read from a file with logical unit number NIN.
[out]  RMAX  RMAX is DOUBLE PRECISION array, dimension (3) Value of the largest test ratio. RMAX(1) = largest ratio comparing different calls to DTRSNA RMAX(2) = largest error in reciprocal condition numbers taking their conditioning into account RMAX(3) = largest error in reciprocal condition numbers not taking their conditioning into account (may be larger than RMAX(2)) 
[out]  LMAX  LMAX is INTEGER array, dimension (3) LMAX(i) is example number where largest test ratio RMAX(i) is achieved. Also: If DGEHRD returns INFO nonzero on example i, LMAX(1)=i If DHSEQR returns INFO nonzero on example i, LMAX(2)=i If DTRSNA returns INFO nonzero on example i, LMAX(3)=i 
[out]  NINFO  NINFO is INTEGER array, dimension (3) NINFO(1) = No. of times DGEHRD returned INFO nonzero NINFO(2) = No. of times DHSEQR returned INFO nonzero NINFO(3) = No. of times DTRSNA returned INFO nonzero 
[out]  KNT  KNT is INTEGER Total number of examples tested. 
[in]  NIN  NIN is INTEGER Input logical unit number 
Definition at line 91 of file dget37.f.
subroutine dget38  (  double precision, dimension( 3 )  RMAX, 
integer, dimension( 3 )  LMAX,  
integer, dimension( 3 )  NINFO,  
integer  KNT,  
integer  NIN  
) 
DGET38
DGET38 tests DTRSEN, a routine for estimating condition numbers of a cluster of eigenvalues and/or its associated right invariant subspace The test matrices are read from a file with logical unit number NIN.
[out]  RMAX  RMAX is DOUBLE PRECISION array, dimension (3) Values of the largest test ratios. RMAX(1) = largest residuals from DHST01 or comparing different calls to DTRSEN RMAX(2) = largest error in reciprocal condition numbers taking their conditioning into account RMAX(3) = largest error in reciprocal condition numbers not taking their conditioning into account (may be larger than RMAX(2)) 
[out]  LMAX  LMAX is INTEGER array, dimension (3) LMAX(i) is example number where largest test ratio RMAX(i) is achieved. Also: If DGEHRD returns INFO nonzero on example i, LMAX(1)=i If DHSEQR returns INFO nonzero on example i, LMAX(2)=i If DTRSEN returns INFO nonzero on example i, LMAX(3)=i 
[out]  NINFO  NINFO is INTEGER array, dimension (3) NINFO(1) = No. of times DGEHRD returned INFO nonzero NINFO(2) = No. of times DHSEQR returned INFO nonzero NINFO(3) = No. of times DTRSEN returned INFO nonzero 
[out]  KNT  KNT is INTEGER Total number of examples tested. 
[in]  NIN  NIN is INTEGER Input logical unit number. 
Definition at line 92 of file dget38.f.
subroutine dget39  (  double precision  RMAX, 
integer  LMAX,  
integer  NINFO,  
integer  KNT  
) 
DGET39
DGET39 tests DLAQTR, a routine for solving the real or special complex quasi upper triangular system op(T)*p = scale*c, or op(T + iB)*(p+iq) = scale*(c+id), in real arithmetic. T is upper quasitriangular. If it is complex, then the first diagonal block of T must be 1 by 1, B has the special structure B = [ b(1) b(2) ... b(n) ] [ w ] [ w ] [ . ] [ w ] op(A) = A or A', where A' denotes the conjugate transpose of the matrix A. On input, X = [ c ]. On output, X = [ p ]. [ d ] [ q ] Scale is an output less than or equal to 1, chosen to avoid overflow in X. This subroutine is specially designed for the condition number estimation in the eigenproblem routine DTRSNA. The test code verifies that the following residual is order 1: (T+i*B)*(x1+i*x2)  scale*(d1+i*d2)  max(ulp*(T+B)*(x1+x2), (T+B)*smlnum/ulp, smlnum) (The (T+B)*smlnum/ulp term accounts for possible (gradual or nongradual) underflow in x1 and x2.)
[out]  RMAX  RMAX is DOUBLE PRECISION Value of the largest test ratio. 
[out]  LMAX  LMAX is INTEGER Example number where largest test ratio achieved. 
[out]  NINFO  NINFO is INTEGER Number of examples where INFO is nonzero. 
[out]  KNT  KNT is INTEGER Total number of examples tested. 
Definition at line 104 of file dget39.f.
subroutine dget51  (  integer  ITYPE, 
integer  N,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( ldb, * )  B,  
integer  LDB,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( ldv, * )  V,  
integer  LDV,  
double precision, dimension( * )  WORK,  
double precision  RESULT  
) 
DGET51
DGET51 generally checks a decomposition of the form A = U B V' where ' means transpose and U and V are orthogonal. Specifically, if ITYPE=1 RESULT =  A  U B V'  / ( A n ulp ) If ITYPE=2, then: RESULT =  A  B  / ( A n ulp ) If ITYPE=3, then: RESULT =  I  UU'  / ( n ulp )
[in]  ITYPE  ITYPE is INTEGER Specifies the type of tests to be performed. =1: RESULT =  A  U B V'  / ( A n ulp ) =2: RESULT =  A  B  / ( A n ulp ) =3: RESULT =  I  UU'  / ( n ulp ) 
[in]  N  N is INTEGER The size of the matrix. If it is zero, DGET51 does nothing. It must be at least zero. 
[in]  A  A is DOUBLE PRECISION array, dimension (LDA, N) The original (unfactored) matrix. 
[in]  LDA  LDA is INTEGER The leading dimension of A. It must be at least 1 and at least N. 
[in]  B  B is DOUBLE PRECISION array, dimension (LDB, N) The factored matrix. 
[in]  LDB  LDB is INTEGER The leading dimension of B. It must be at least 1 and at least N. 
[in]  U  U is DOUBLE PRECISION array, dimension (LDU, N) The orthogonal matrix on the lefthand side in the decomposition. Not referenced if ITYPE=2 
[in]  LDU  LDU is INTEGER The leading dimension of U. LDU must be at least N and at least 1. 
[in]  V  V is DOUBLE PRECISION array, dimension (LDV, N) The orthogonal matrix on the lefthand side in the decomposition. Not referenced if ITYPE=2 
[in]  LDV  LDV is INTEGER The leading dimension of V. LDV must be at least N and at least 1. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (2*N**2) 
[out]  RESULT  RESULT is DOUBLE PRECISION The values computed by the test specified by ITYPE. The value is currently limited to 1/ulp, to avoid overflow. Errors are flagged by RESULT=10/ulp. 
Definition at line 149 of file dget51.f.
subroutine dget52  (  logical  LEFT, 
integer  N,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( ldb, * )  B,  
integer  LDB,  
double precision, dimension( lde, * )  E,  
integer  LDE,  
double precision, dimension( * )  ALPHAR,  
double precision, dimension( * )  ALPHAI,  
double precision, dimension( * )  BETA,  
double precision, dimension( * )  WORK,  
double precision, dimension( 2 )  RESULT  
) 
DGET52
DGET52 does an eigenvector check for the generalized eigenvalue problem. The basic test for right eigenvectors is:  b(j) A E(j)  a(j) B E(j)  RESULT(1) = max  j n ulp max( b(j) A, a(j) B ) using the 1norm. Here, a(j)/b(j) = w is the jth generalized eigenvalue of A  w B, or, equivalently, b(j)/a(j) = m is the jth generalized eigenvalue of m A  B. For real eigenvalues, the test is straightforward. For complex eigenvalues, E(j) and a(j) are complex, represented by Er(j) + i*Ei(j) and ar(j) + i*ai(j), resp., so the test for that eigenvector becomes max( Wr, Wi )  n ulp max( b(j) A, (ar(j)+ai(j)) B ) where Wr = b(j) A Er(j)  ar(j) B Er(j) + ai(j) B Ei(j) Wi = b(j) A Ei(j)  ai(j) B Er(j)  ar(j) B Ei(j) T T _ For left eigenvectors, A , B , a, and b are used. DGET52 also tests the normalization of E. Each eigenvector is supposed to be normalized so that the maximum "absolute value" of its elements is 1, where in this case, "absolute value" of a complex value x is Re(x) + Im(x) ; let us call this maximum "absolute value" norm of a vector v M(v). if a(j)=b(j)=0, then the eigenvector is set to be the jth coordinate vector. The normalization test is: RESULT(2) = max  M(v(j))  1  / ( n ulp ) eigenvectors v(j)
[in]  LEFT  LEFT is LOGICAL =.TRUE.: The eigenvectors in the columns of E are assumed to be *left* eigenvectors. =.FALSE.: The eigenvectors in the columns of E are assumed to be *right* eigenvectors. 
[in]  N  N is INTEGER The size of the matrices. If it is zero, DGET52 does nothing. It must be at least zero. 
[in]  A  A is DOUBLE PRECISION array, dimension (LDA, N) The matrix A. 
[in]  LDA  LDA is INTEGER The leading dimension of A. It must be at least 1 and at least N. 
[in]  B  B is DOUBLE PRECISION array, dimension (LDB, N) The matrix B. 
[in]  LDB  LDB is INTEGER The leading dimension of B. It must be at least 1 and at least N. 
[in]  E  E is DOUBLE PRECISION array, dimension (LDE, N) The matrix of eigenvectors. It must be O( 1 ). Complex eigenvalues and eigenvectors always come in pairs, the eigenvalue and its conjugate being stored in adjacent elements of ALPHAR, ALPHAI, and BETA. Thus, if a(j)/b(j) and a(j+1)/b(j+1) are a complex conjugate pair of generalized eigenvalues, then E(,j) contains the real part of the eigenvector and E(,j+1) contains the imaginary part. Note that whether E(,j) is a real eigenvector or part of a complex one is specified by whether ALPHAI(j) is zero or not. 
[in]  LDE  LDE is INTEGER The leading dimension of E. It must be at least 1 and at least N. 
[in]  ALPHAR  ALPHAR is DOUBLE PRECISION array, dimension (N) The real parts of the values a(j) as described above, which, along with b(j), define the generalized eigenvalues. Complex eigenvalues always come in complex conjugate pairs a(j)/b(j) and a(j+1)/b(j+1), which are stored in adjacent elements in ALPHAR, ALPHAI, and BETA. Thus, if the jth and (j+1)st eigenvalues form a pair, ALPHAR(j+1)/BETA(j+1) is assumed to be equal to ALPHAR(j)/BETA(j). 
[in]  ALPHAI  ALPHAI is DOUBLE PRECISION array, dimension (N) The imaginary parts of the values a(j) as described above, which, along with b(j), define the generalized eigenvalues. If ALPHAI(j)=0, then the eigenvalue is real, otherwise it is part of a complex conjugate pair. Complex eigenvalues always come in complex conjugate pairs a(j)/b(j) and a(j+1)/b(j+1), which are stored in adjacent elements in ALPHAR, ALPHAI, and BETA. Thus, if the jth and (j+1)st eigenvalues form a pair, ALPHAI(j+1)/BETA(j+1) is assumed to be equal to ALPHAI(j)/BETA(j). Also, nonzero values in ALPHAI are assumed to always come in adjacent pairs. 
[in]  BETA  BETA is DOUBLE PRECISION array, dimension (N) The values b(j) as described above, which, along with a(j), define the generalized eigenvalues. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (N**2+N) 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (2) The values computed by the test described above. If A E or B E is likely to overflow, then RESULT(1:2) is set to 10 / ulp. 
Definition at line 199 of file dget52.f.
subroutine dget53  (  double precision, dimension( lda, * )  A, 
integer  LDA,  
double precision, dimension( ldb, * )  B,  
integer  LDB,  
double precision  SCALE,  
double precision  WR,  
double precision  WI,  
double precision  RESULT,  
integer  INFO  
) 
DGET53
DGET53 checks the generalized eigenvalues computed by DLAG2. The basic test for an eigenvalue is:  det( s A  w B )  RESULT =  ulp max( s norm(A), w norm(B) )*norm( s A  w B ) Two "safety checks" are performed: (1) ulp*max( s*norm(A), w*norm(B) ) must be at least safe_minimum. This insures that the test performed is not essentially det(0*A + 0*B)=0. (2) s*norm(A) + w*norm(B) must be less than 1/safe_minimum. This insures that s*A  w*B will not overflow. If these tests are not passed, then s and w are scaled and tested anyway, if this is possible.
[in]  A  A is DOUBLE PRECISION array, dimension (LDA, 2) The 2x2 matrix A. 
[in]  LDA  LDA is INTEGER The leading dimension of A. It must be at least 2. 
[in]  B  B is DOUBLE PRECISION array, dimension (LDB, N) The 2x2 uppertriangular matrix B. 
[in]  LDB  LDB is INTEGER The leading dimension of B. It must be at least 2. 
[in]  SCALE  SCALE is DOUBLE PRECISION The "scale factor" s in the formula s A  w B . It is assumed to be nonnegative. 
[in]  WR  WR is DOUBLE PRECISION The real part of the eigenvalue w in the formula s A  w B . 
[in]  WI  WI is DOUBLE PRECISION The imaginary part of the eigenvalue w in the formula s A  w B . 
[out]  RESULT  RESULT is DOUBLE PRECISION If INFO is 2 or less, the value computed by the test described above. If INFO=3, this will just be 1/ulp. 
[out]  INFO  INFO is INTEGER =0: The input data pass the "safety checks". =1: s*norm(A) + w*norm(B) > 1/safe_minimum. =2: ulp*max( s*norm(A), w*norm(B) ) < safe_minimum =3: same as INFO=2, but s and w could not be scaled so as to compute the test. 
Definition at line 127 of file dget53.f.
subroutine dget54  (  integer  N, 
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( ldb, * )  B,  
integer  LDB,  
double precision, dimension( lds, * )  S,  
integer  LDS,  
double precision, dimension( ldt, * )  T,  
integer  LDT,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( ldv, * )  V,  
integer  LDV,  
double precision, dimension( * )  WORK,  
double precision  RESULT  
) 
DGET54
DGET54 checks a generalized decomposition of the form A = U*S*V' and B = U*T* V' where ' means transpose and U and V are orthogonal. Specifically, RESULT = ( A  U*S*V', B  U*T*V' ) / (( A, B )*n*ulp )
[in]  N  N is INTEGER The size of the matrix. If it is zero, DGET54 does nothing. It must be at least zero. 
[in]  A  A is DOUBLE PRECISION array, dimension (LDA, N) The original (unfactored) matrix A. 
[in]  LDA  LDA is INTEGER The leading dimension of A. It must be at least 1 and at least N. 
[in]  B  B is DOUBLE PRECISION array, dimension (LDB, N) The original (unfactored) matrix B. 
[in]  LDB  LDB is INTEGER The leading dimension of B. It must be at least 1 and at least N. 
[in]  S  S is DOUBLE PRECISION array, dimension (LDS, N) The factored matrix S. 
[in]  LDS  LDS is INTEGER The leading dimension of S. It must be at least 1 and at least N. 
[in]  T  T is DOUBLE PRECISION array, dimension (LDT, N) The factored matrix T. 
[in]  LDT  LDT is INTEGER The leading dimension of T. It must be at least 1 and at least N. 
[in]  U  U is DOUBLE PRECISION array, dimension (LDU, N) The orthogonal matrix on the lefthand side in the decomposition. 
[in]  LDU  LDU is INTEGER The leading dimension of U. LDU must be at least N and at least 1. 
[in]  V  V is DOUBLE PRECISION array, dimension (LDV, N) The orthogonal matrix on the lefthand side in the decomposition. 
[in]  LDV  LDV is INTEGER The leading dimension of V. LDV must be at least N and at least 1. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (3*N**2) 
[out]  RESULT  RESULT is DOUBLE PRECISION The value RESULT, It is currently limited to 1/ulp, to avoid overflow. Errors are flagged by RESULT=10/ulp. 
Definition at line 156 of file dget54.f.
subroutine dglmts  (  integer  N, 
integer  M,  
integer  P,  
double precision, dimension( lda, * )  A,  
double precision, dimension( lda, * )  AF,  
integer  LDA,  
double precision, dimension( ldb, * )  B,  
double precision, dimension( ldb, * )  BF,  
integer  LDB,  
double precision, dimension( * )  D,  
double precision, dimension( * )  DF,  
double precision, dimension( * )  X,  
double precision, dimension( * )  U,  
double precision, dimension( lwork )  WORK,  
integer  LWORK,  
double precision, dimension( * )  RWORK,  
double precision  RESULT  
) 
DGLMTS
DGLMTS tests DGGGLM  a subroutine for solving the generalized linear model problem.
[in]  N  N is INTEGER The number of rows of the matrices A and B. N >= 0. 
[in]  M  M is INTEGER The number of columns of the matrix A. M >= 0. 
[in]  P  P is INTEGER The number of columns of the matrix B. P >= 0. 
[in]  A  A is DOUBLE PRECISION array, dimension (LDA,M) The NbyM matrix A. 
[out]  AF  AF is DOUBLE PRECISION array, dimension (LDA,M) 
[in]  LDA  LDA is INTEGER The leading dimension of the arrays A, AF. LDA >= max(M,N). 
[in]  B  B is DOUBLE PRECISION array, dimension (LDB,P) The NbyP matrix A. 
[out]  BF  BF is DOUBLE PRECISION array, dimension (LDB,P) 
[in]  LDB  LDB is INTEGER The leading dimension of the arrays B, BF. LDB >= max(P,N). 
[in]  D  D is DOUBLE PRECISION array, dimension( N ) On input, the left hand side of the GLM. 
[out]  DF  DF is DOUBLE PRECISION array, dimension( N ) 
[out]  X  X is DOUBLE PRECISION array, dimension( M ) solution vector X in the GLM problem. 
[out]  U  U is DOUBLE PRECISION array, dimension( P ) solution vector U in the GLM problem. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER The dimension of the array WORK. 
[out]  RWORK  RWORK is DOUBLE PRECISION array, dimension (M) 
[out]  RESULT  RESULT is DOUBLE PRECISION The test ratio: norm( d  A*x  B*u ) RESULT =  (norm(A)+norm(B))*(norm(x)+norm(u))*EPS 
Definition at line 146 of file dglmts.f.
subroutine dgqrts  (  integer  N, 
integer  M,  
integer  P,  
double precision, dimension( lda, * )  A,  
double precision, dimension( lda, * )  AF,  
double precision, dimension( lda, * )  Q,  
double precision, dimension( lda, * )  R,  
integer  LDA,  
double precision, dimension( * )  TAUA,  
double precision, dimension( ldb, * )  B,  
double precision, dimension( ldb, * )  BF,  
double precision, dimension( ldb, * )  Z,  
double precision, dimension( ldb, * )  T,  
double precision, dimension( ldb, * )  BWK,  
integer  LDB,  
double precision, dimension( * )  TAUB,  
double precision, dimension( lwork )  WORK,  
integer  LWORK,  
double precision, dimension( * )  RWORK,  
double precision, dimension( 4 )  RESULT  
) 
DGQRTS
DGQRTS tests DGGQRF, which computes the GQR factorization of an NbyM matrix A and a NbyP matrix B: A = Q*R and B = Q*T*Z.
[in]  N  N is INTEGER The number of rows of the matrices A and B. N >= 0. 
[in]  M  M is INTEGER The number of columns of the matrix A. M >= 0. 
[in]  P  P is INTEGER The number of columns of the matrix B. P >= 0. 
[in]  A  A is DOUBLE PRECISION array, dimension (LDA,M) The NbyM matrix A. 
[out]  AF  AF is DOUBLE PRECISION array, dimension (LDA,N) Details of the GQR factorization of A and B, as returned by DGGQRF, see SGGQRF for further details. 
[out]  Q  Q is DOUBLE PRECISION array, dimension (LDA,N) The MbyM orthogonal matrix Q. 
[out]  R  R is DOUBLE PRECISION array, dimension (LDA,MAX(M,N)) 
[in]  LDA  LDA is INTEGER The leading dimension of the arrays A, AF, R and Q. LDA >= max(M,N). 
[out]  TAUA  TAUA is DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors of the elementary reflectors, as returned by DGGQRF. 
[in]  B  B is DOUBLE PRECISION array, dimension (LDB,P) On entry, the NbyP matrix A. 
[out]  BF  BF is DOUBLE PRECISION array, dimension (LDB,N) Details of the GQR factorization of A and B, as returned by DGGQRF, see SGGQRF for further details. 
[out]  Z  Z is DOUBLE PRECISION array, dimension (LDB,P) The PbyP orthogonal matrix Z. 
[out]  T  T is DOUBLE PRECISION array, dimension (LDB,max(P,N)) 
[out]  BWK  BWK is DOUBLE PRECISION array, dimension (LDB,N) 
[in]  LDB  LDB is INTEGER The leading dimension of the arrays B, BF, Z and T. LDB >= max(P,N). 
[out]  TAUB  TAUB is DOUBLE PRECISION array, dimension (min(P,N)) The scalar factors of the elementary reflectors, as returned by DGGRQF. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER The dimension of the array WORK, LWORK >= max(N,M,P)**2. 
[out]  RWORK  RWORK is DOUBLE PRECISION array, dimension (max(N,M,P)) 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (4) The test ratios: RESULT(1) = norm( R  Q'*A ) / ( MAX(M,N)*norm(A)*ULP) RESULT(2) = norm( T*Z  Q'*B ) / (MAX(P,N)*norm(B)*ULP) RESULT(3) = norm( I  Q'*Q ) / ( M*ULP ) RESULT(4) = norm( I  Z'*Z ) / ( P*ULP ) 
Definition at line 176 of file dgqrts.f.
subroutine dgrqts  (  integer  M, 
integer  P,  
integer  N,  
double precision, dimension( lda, * )  A,  
double precision, dimension( lda, * )  AF,  
double precision, dimension( lda, * )  Q,  
double precision, dimension( lda, * )  R,  
integer  LDA,  
double precision, dimension( * )  TAUA,  
double precision, dimension( ldb, * )  B,  
double precision, dimension( ldb, * )  BF,  
double precision, dimension( ldb, * )  Z,  
double precision, dimension( ldb, * )  T,  
double precision, dimension( ldb, * )  BWK,  
integer  LDB,  
double precision, dimension( * )  TAUB,  
double precision, dimension( lwork )  WORK,  
integer  LWORK,  
double precision, dimension( * )  RWORK,  
double precision, dimension( 4 )  RESULT  
) 
DGRQTS
DGRQTS tests DGGRQF, which computes the GRQ factorization of an MbyN matrix A and a PbyN matrix B: A = R*Q and B = Z*T*Q.
[in]  M  M is INTEGER The number of rows of the matrix A. M >= 0. 
[in]  P  P is INTEGER The number of rows of the matrix B. P >= 0. 
[in]  N  N is INTEGER The number of columns of the matrices A and B. N >= 0. 
[in]  A  A is DOUBLE PRECISION array, dimension (LDA,N) The MbyN matrix A. 
[out]  AF  AF is DOUBLE PRECISION array, dimension (LDA,N) Details of the GRQ factorization of A and B, as returned by DGGRQF, see SGGRQF for further details. 
[out]  Q  Q is DOUBLE PRECISION array, dimension (LDA,N) The NbyN orthogonal matrix Q. 
[out]  R  R is DOUBLE PRECISION array, dimension (LDA,MAX(M,N)) 
[in]  LDA  LDA is INTEGER The leading dimension of the arrays A, AF, R and Q. LDA >= max(M,N). 
[out]  TAUA  TAUA is DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors of the elementary reflectors, as returned by DGGQRC. 
[in]  B  B is DOUBLE PRECISION array, dimension (LDB,N) On entry, the PbyN matrix A. 
[out]  BF  BF is DOUBLE PRECISION array, dimension (LDB,N) Details of the GQR factorization of A and B, as returned by DGGRQF, see SGGRQF for further details. 
[out]  Z  Z is DOUBLE PRECISION array, dimension (LDB,P) The PbyP orthogonal matrix Z. 
[out]  T  T is DOUBLE PRECISION array, dimension (LDB,max(P,N)) 
[out]  BWK  BWK is DOUBLE PRECISION array, dimension (LDB,N) 
[in]  LDB  LDB is INTEGER The leading dimension of the arrays B, BF, Z and T. LDB >= max(P,N). 
[out]  TAUB  TAUB is DOUBLE PRECISION array, dimension (min(P,N)) The scalar factors of the elementary reflectors, as returned by DGGRQF. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER The dimension of the array WORK, LWORK >= max(M,P,N)**2. 
[out]  RWORK  RWORK is DOUBLE PRECISION array, dimension (M) 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (4) The test ratios: RESULT(1) = norm( R  A*Q' ) / ( MAX(M,N)*norm(A)*ULP) RESULT(2) = norm( T*Q  Z'*B ) / (MAX(P,N)*norm(B)*ULP) RESULT(3) = norm( I  Q'*Q ) / ( N*ULP ) RESULT(4) = norm( I  Z'*Z ) / ( P*ULP ) 
Definition at line 176 of file dgrqts.f.
subroutine dgsvts  (  integer  M, 
integer  P,  
integer  N,  
double precision, dimension( lda, * )  A,  
double precision, dimension( lda, * )  AF,  
integer  LDA,  
double precision, dimension( ldb, * )  B,  
double precision, dimension( ldb, * )  BF,  
integer  LDB,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( ldv, * )  V,  
integer  LDV,  
double precision, dimension( ldq, * )  Q,  
integer  LDQ,  
double precision, dimension( * )  ALPHA,  
double precision, dimension( * )  BETA,  
double precision, dimension( ldr, * )  R,  
integer  LDR,  
integer, dimension( * )  IWORK,  
double precision, dimension( lwork )  WORK,  
integer  LWORK,  
double precision, dimension( * )  RWORK,  
double precision, dimension( 6 )  RESULT  
) 
DGSVTS
DGSVTS tests DGGSVD, which computes the GSVD of an MbyN matrix A and a PbyN matrix B: U'*A*Q = D1*R and V'*B*Q = D2*R.
[in]  M  M is INTEGER The number of rows of the matrix A. M >= 0. 
[in]  P  P is INTEGER The number of rows of the matrix B. P >= 0. 
[in]  N  N is INTEGER The number of columns of the matrices A and B. N >= 0. 
[in]  A  A is DOUBLE PRECISION array, dimension (LDA,M) The MbyN matrix A. 
[out]  AF  AF is DOUBLE PRECISION array, dimension (LDA,N) Details of the GSVD of A and B, as returned by DGGSVD, see DGGSVD for further details. 
[in]  LDA  LDA is INTEGER The leading dimension of the arrays A and AF. LDA >= max( 1,M ). 
[in]  B  B is DOUBLE PRECISION array, dimension (LDB,P) On entry, the PbyN matrix B. 
[out]  BF  BF is DOUBLE PRECISION array, dimension (LDB,N) Details of the GSVD of A and B, as returned by DGGSVD, see DGGSVD for further details. 
[in]  LDB  LDB is INTEGER The leading dimension of the arrays B and BF. LDB >= max(1,P). 
[out]  U  U is DOUBLE PRECISION array, dimension(LDU,M) The M by M orthogonal matrix U. 
[in]  LDU  LDU is INTEGER The leading dimension of the array U. LDU >= max(1,M). 
[out]  V  V is DOUBLE PRECISION array, dimension(LDV,M) The P by P orthogonal matrix V. 
[in]  LDV  LDV is INTEGER The leading dimension of the array V. LDV >= max(1,P). 
[out]  Q  Q is DOUBLE PRECISION array, dimension(LDQ,N) The N by N orthogonal matrix Q. 
[in]  LDQ  LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N). 
[out]  ALPHA  ALPHA is DOUBLE PRECISION array, dimension (N) 
[out]  BETA  BETA is DOUBLE PRECISION array, dimension (N) The generalized singular value pairs of A and B, the ``diagonal'' matrices D1 and D2 are constructed from ALPHA and BETA, see subroutine DGGSVD for details. 
[out]  R  R is DOUBLE PRECISION array, dimension(LDQ,N) The upper triangular matrix R. 
[in]  LDR  LDR is INTEGER The leading dimension of the array R. LDR >= max(1,N). 
[out]  IWORK  IWORK is INTEGER array, dimension (N) 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER The dimension of the array WORK, LWORK >= max(M,P,N)*max(M,P,N). 
[out]  RWORK  RWORK is DOUBLE PRECISION array, dimension (max(M,P,N)) 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (6) The test ratios: RESULT(1) = norm( U'*A*Q  D1*R ) / ( MAX(M,N)*norm(A)*ULP) RESULT(2) = norm( V'*B*Q  D2*R ) / ( MAX(P,N)*norm(B)*ULP) RESULT(3) = norm( I  U'*U ) / ( M*ULP ) RESULT(4) = norm( I  V'*V ) / ( P*ULP ) RESULT(5) = norm( I  Q'*Q ) / ( N*ULP ) RESULT(6) = 0 if ALPHA is in decreasing order; = ULPINV otherwise. 
Definition at line 209 of file dgsvts.f.
subroutine dhst01  (  integer  N, 
integer  ILO,  
integer  IHI,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( ldh, * )  H,  
integer  LDH,  
double precision, dimension( ldq, * )  Q,  
integer  LDQ,  
double precision, dimension( lwork )  WORK,  
integer  LWORK,  
double precision, dimension( 2 )  RESULT  
) 
DHST01
DHST01 tests the reduction of a general matrix A to upper Hessenberg form: A = Q*H*Q'. Two test ratios are computed; RESULT(1) = norm( A  Q*H*Q' ) / ( norm(A) * N * EPS ) RESULT(2) = norm( I  Q'*Q ) / ( N * EPS ) The matrix Q is assumed to be given explicitly as it would be following DGEHRD + DORGHR. In this version, ILO and IHI are not used and are assumed to be 1 and N, respectively.
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in]  ILO  ILO is INTEGER 
[in]  IHI  IHI is INTEGER A is assumed to be upper triangular in rows and columns 1:ILO1 and IHI+1:N, so Q differs from the identity only in rows and columns ILO+1:IHI. 
[in]  A  A is DOUBLE PRECISION array, dimension (LDA,N) The original n by n matrix A. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in]  H  H is DOUBLE PRECISION array, dimension (LDH,N) The upper Hessenberg matrix H from the reduction A = Q*H*Q' as computed by DGEHRD. H is assumed to be zero below the first subdiagonal. 
[in]  LDH  LDH is INTEGER The leading dimension of the array H. LDH >= max(1,N). 
[in]  Q  Q is DOUBLE PRECISION array, dimension (LDQ,N) The orthogonal matrix Q from the reduction A = Q*H*Q' as computed by DGEHRD + DORGHR. 
[in]  LDQ  LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N). 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER The length of the array WORK. LWORK >= 2*N*N. 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (2) RESULT(1) = norm( A  Q*H*Q' ) / ( norm(A) * N * EPS ) RESULT(2) = norm( I  Q'*Q ) / ( N * EPS ) 
Definition at line 134 of file dhst01.f.
subroutine dlafts  (  character*3  TYPE, 
integer  M,  
integer  N,  
integer  IMAT,  
integer  NTESTS,  
double precision, dimension( * )  RESULT,  
integer, dimension( 4 )  ISEED,  
double precision  THRESH,  
integer  IOUNIT,  
integer  IE  
) 
DLAFTS
DLAFTS tests the result vector against the threshold value to see which tests for this matrix type failed to pass the threshold. Output is to the file given by unit IOUNIT.
TYPE  CHARACTER*3 On entry, TYPE specifies the matrix type to be used in the printed messages. Not modified. N  INTEGER On entry, N specifies the order of the test matrix. Not modified. IMAT  INTEGER On entry, IMAT specifies the type of the test matrix. A listing of the different types is printed by DLAHD2 to the output file if a test fails to pass the threshold. Not modified. NTESTS  INTEGER On entry, NTESTS is the number of tests performed on the subroutines in the path given by TYPE. Not modified. RESULT  DOUBLE PRECISION array of dimension( NTESTS ) On entry, RESULT contains the test ratios from the tests performed in the calling program. Not modified. ISEED  INTEGER array of dimension( 4 ) Contains the random seed that generated the matrix used for the tests whose ratios are in RESULT. Not modified. THRESH  DOUBLE PRECISION On entry, THRESH specifies the acceptable threshold of the test ratios. If RESULT( K ) > THRESH, then the Kth test did not pass the threshold and a message will be printed. Not modified. IOUNIT  INTEGER On entry, IOUNIT specifies the unit number of the file to which the messages are printed. Not modified. IE  INTEGER On entry, IE contains the number of tests which have failed to pass the threshold so far. Updated on exit if any of the ratios in RESULT also fail.
Definition at line 99 of file dlafts.f.
subroutine dlahd2  (  integer  IOUNIT, 
character*3  PATH  
) 
DLAHD2
DLAHD2 prints header information for the different test paths.
[in]  IOUNIT  IOUNIT is INTEGER. On entry, IOUNIT specifies the unit number to which the header information should be printed. 
[in]  PATH  PATH is CHARACTER*3. On entry, PATH contains the name of the path for which the header information is to be printed. Current paths are DHS, ZHS: Nonsymmetric eigenproblem. DST, ZST: Symmetric eigenproblem. DSG, ZSG: Symmetric Generalized eigenproblem. DBD, ZBD: Singular Value Decomposition (SVD) DBB, ZBB: General Banded reduction to bidiagonal form These paths also are supplied in double precision (replace leading S by D and leading C by Z in path names). 
Definition at line 66 of file dlahd2.f.
subroutine dlarfy  (  character  UPLO, 
integer  N,  
double precision, dimension( * )  V,  
integer  INCV,  
double precision  TAU,  
double precision, dimension( ldc, * )  C,  
integer  LDC,  
double precision, dimension( * )  WORK  
) 
DLARFY
DLARFY applies an elementary reflector, or Householder matrix, H, to an n x n symmetric matrix C, from both the left and the right. H is represented in the form H = I  tau * v * v' where tau is a scalar and v is a vector. If tau is zero, then H is taken to be the unit matrix.
[in]  UPLO  UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix C is stored. = 'U': Upper triangle = 'L': Lower triangle 
[in]  N  N is INTEGER The number of rows and columns of the matrix C. N >= 0. 
[in]  V  V is DOUBLE PRECISION array, dimension (1 + (N1)*abs(INCV)) The vector v as described above. 
[in]  INCV  INCV is INTEGER The increment between successive elements of v. INCV must not be zero. 
[in]  TAU  TAU is DOUBLE PRECISION The value tau as described above. 
[in,out]  C  C is DOUBLE PRECISION array, dimension (LDC, N) On entry, the matrix C. On exit, C is overwritten by H * C * H'. 
[in]  LDC  LDC is INTEGER The leading dimension of the array C. LDC >= max( 1, N ). 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (N) 
Definition at line 109 of file dlarfy.f.
subroutine dlarhs  (  character*3  PATH, 
character  XTYPE,  
character  UPLO,  
character  TRANS,  
integer  M,  
integer  N,  
integer  KL,  
integer  KU,  
integer  NRHS,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( ldx, * )  X,  
integer  LDX,  
double precision, dimension( ldb, * )  B,  
integer  LDB,  
integer, dimension( 4 )  ISEED,  
integer  INFO  
) 
DLARHS
DLARHS chooses a set of NRHS random solution vectors and sets up the right hand sides for the linear system op( A ) * X = B, where op( A ) may be A or A' (transpose of A).
[in]  PATH  PATH is CHARACTER*3 The type of the real matrix A. PATH may be given in any combination of upper and lower case. Valid types include xGE: General m x n matrix xGB: General banded matrix xPO: Symmetric positive definite, 2D storage xPP: Symmetric positive definite packed xPB: Symmetric positive definite banded xSY: Symmetric indefinite, 2D storage xSP: Symmetric indefinite packed xSB: Symmetric indefinite banded xTR: Triangular xTP: Triangular packed xTB: Triangular banded xQR: General m x n matrix xLQ: General m x n matrix xQL: General m x n matrix xRQ: General m x n matrix where the leading character indicates the precision. 
[in]  XTYPE  XTYPE is CHARACTER*1 Specifies how the exact solution X will be determined: = 'N': New solution; generate a random X. = 'C': Computed; use value of X on entry. 
[in]  UPLO  UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the matrix A is stored, if A is symmetric. = 'U': Upper triangular = 'L': Lower triangular 
[in]  TRANS  TRANS is CHARACTER*1 Specifies the operation applied to the matrix A. = 'N': System is A * x = b = 'T': System is A'* x = b = 'C': System is A'* x = b 
[in]  M  M is INTEGER The number or rows of the matrix A. M >= 0. 
[in]  N  N is INTEGER The number of columns of the matrix A. N >= 0. 
[in]  KL  KL is INTEGER Used only if A is a band matrix; specifies the number of subdiagonals of A if A is a general band matrix or if A is symmetric or triangular and UPLO = 'L'; specifies the number of superdiagonals of A if A is symmetric or triangular and UPLO = 'U'. 0 <= KL <= M1. 
[in]  KU  KU is INTEGER Used only if A is a general band matrix or if A is triangular. If PATH = xGB, specifies the number of superdiagonals of A, and 0 <= KU <= N1. If PATH = xTR, xTP, or xTB, specifies whether or not the matrix has unit diagonal: = 1: matrix has nonunit diagonal (default) = 2: matrix has unit diagonal 
[in]  NRHS  NRHS is INTEGER The number of right hand side vectors in the system A*X = B. 
[in]  A  A is DOUBLE PRECISION array, dimension (LDA,N) The test matrix whose type is given by PATH. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. If PATH = xGB, LDA >= KL+KU+1. If PATH = xPB, xSB, xHB, or xTB, LDA >= KL+1. Otherwise, LDA >= max(1,M). 
[in,out]  X  X is or output) DOUBLE PRECISION array, dimension(LDX,NRHS) On entry, if XTYPE = 'C' (for 'Computed'), then X contains the exact solution to the system of linear equations. On exit, if XTYPE = 'N' (for 'New'), then X is initialized with random values. 
[in]  LDX  LDX is INTEGER The leading dimension of the array X. If TRANS = 'N', LDX >= max(1,N); if TRANS = 'T', LDX >= max(1,M). 
[out]  B  B is DOUBLE PRECISION array, dimension (LDB,NRHS) The right hand side vector(s) for the system of equations, computed from B = op(A) * X, where op(A) is determined by TRANS. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. If TRANS = 'N', LDB >= max(1,M); if TRANS = 'T', LDB >= max(1,N). 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) The seed vector for the random number generator (used in DLATMS). Modified on exit. 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value 
Definition at line 204 of file dlarhs.f.
subroutine dlasum  (  character*3  TYPE, 
integer  IOUNIT,  
integer  IE,  
integer  NRUN  
) 
subroutine dlatb9  (  character*3  PATH, 
integer  IMAT,  
integer  M,  
integer  P,  
integer  N,  
character  TYPE,  
integer  KLA,  
integer  KUA,  
integer  KLB,  
integer  KUB,  
double precision  ANORM,  
double precision  BNORM,  
integer  MODEA,  
integer  MODEB,  
double precision  CNDNMA,  
double precision  CNDNMB,  
character  DISTA,  
character  DISTB  
) 
DLATB9
DLATB9 sets parameters for the matrix generator based on the type of matrix to be generated.
[in]  PATH  PATH is CHARACTER*3 The LAPACK path name. 
[in]  IMAT  IMAT is INTEGER An integer key describing which matrix to generate for this path. = 1: A: diagonal, B: upper triangular = 2: A: upper triangular, B: upper triangular = 3: A: lower triangular, B: upper triangular Else: A: general dense, B: general dense 
[in]  M  M is INTEGER The number of rows in the matrix to be generated. 
[in]  P  P is INTEGER 
[in]  N  N is INTEGER The number of columns in the matrix to be generated. 
[out]  TYPE  TYPE is CHARACTER*1 The type of the matrix to be generated: = 'S': symmetric matrix; = 'P': symmetric positive (semi)definite matrix; = 'N': nonsymmetric matrix. 
[out]  KLA  KLA is INTEGER The lower band width of the matrix to be generated. 
[out]  KUA  KUA is INTEGER The upper band width of the matrix to be generated. 
[out]  KLB  KLB is INTEGER The lower band width of the matrix to be generated. 
[out]  KUB  KUA is INTEGER The upper band width of the matrix to be generated. 
[out]  ANORM  ANORM is DOUBLE PRECISION The desired norm of the matrix to be generated. The diagonal matrix of singular values or eigenvalues is scaled by this value. 
[out]  BNORM  BNORM is DOUBLE PRECISION The desired norm of the matrix to be generated. The diagonal matrix of singular values or eigenvalues is scaled by this value. 
[out]  MODEA  MODEA is INTEGER A key indicating how to choose the vector of eigenvalues. 
[out]  MODEB  MODEB is INTEGER A key indicating how to choose the vector of eigenvalues. 
[out]  CNDNMA  CNDNMA is DOUBLE PRECISION The desired condition number. 
[out]  CNDNMB  CNDNMB is DOUBLE PRECISION The desired condition number. 
[out]  DISTA  DISTA is CHARACTER*1 The type of distribution to be used by the random number generator. 
[out]  DISTB  DISTB is CHARACTER*1 The type of distribution to be used by the random number generator. 
Definition at line 169 of file dlatb9.f.
subroutine dlatm4  (  integer  ITYPE, 
integer  N,  
integer  NZ1,  
integer  NZ2,  
integer  ISIGN,  
double precision  AMAGN,  
double precision  RCOND,  
double precision  TRIANG,  
integer  IDIST,  
integer, dimension( 4 )  ISEED,  
double precision, dimension( lda, * )  A,  
integer  LDA  
) 
DLATM4
DLATM4 generates basic square matrices, which may later be multiplied by others in order to produce test matrices. It is intended mainly to be used to test the generalized eigenvalue routines. It first generates the diagonal and (possibly) subdiagonal, according to the value of ITYPE, NZ1, NZ2, ISIGN, AMAGN, and RCOND. It then fills in the upper triangle with random numbers, if TRIANG is nonzero.
[in]  ITYPE  ITYPE is INTEGER The "type" of matrix on the diagonal and subdiagonal. If ITYPE < 0, then type abs(ITYPE) is generated and then swapped end for end (A(I,J) := A'(NJ,NI).) See also the description of AMAGN and ISIGN. Special types: = 0: the zero matrix. = 1: the identity. = 2: a transposed Jordan block. = 3: If N is odd, then a k+1 x k+1 transposed Jordan block followed by a k x k identity block, where k=(N1)/2. If N is even, then k=(N2)/2, and a zero diagonal entry is tacked onto the end. Diagonal types. The diagonal consists of NZ1 zeros, then k=NNZ1NZ2 nonzeros. The subdiagonal is zero. ITYPE specifies the nonzero diagonal entries as follows: = 4: 1, ..., k = 5: 1, RCOND, ..., RCOND = 6: 1, ..., 1, RCOND = 7: 1, a, a^2, ..., a^(k1)=RCOND = 8: 1, 1d, 12*d, ..., 1(k1)*d=RCOND = 9: random numbers chosen from (RCOND,1) = 10: random numbers with distribution IDIST (see DLARND.) 
[in]  N  N is INTEGER The order of the matrix. 
[in]  NZ1  NZ1 is INTEGER If abs(ITYPE) > 3, then the first NZ1 diagonal entries will be zero. 
[in]  NZ2  NZ2 is INTEGER If abs(ITYPE) > 3, then the last NZ2 diagonal entries will be zero. 
[in]  ISIGN  ISIGN is INTEGER = 0: The sign of the diagonal and subdiagonal entries will be left unchanged. = 1: The diagonal and subdiagonal entries will have their sign changed at random. = 2: If ITYPE is 2 or 3, then the same as ISIGN=1. Otherwise, with probability 0.5, oddeven pairs of diagonal entries A(2*j1,2*j1), A(2*j,2*j) will be converted to a 2x2 block by pre and postmultiplying by distinct random orthogonal rotations. The remaining diagonal entries will have their sign changed at random. 
[in]  AMAGN  AMAGN is DOUBLE PRECISION The diagonal and subdiagonal entries will be multiplied by AMAGN. 
[in]  RCOND  RCOND is DOUBLE PRECISION If abs(ITYPE) > 4, then the smallest diagonal entry will be entry will be RCOND. RCOND must be between 0 and 1. 
[in]  TRIANG  TRIANG is DOUBLE PRECISION The entries above the diagonal will be random numbers with magnitude bounded by TRIANG (i.e., random numbers multiplied by TRIANG.) 
[in]  IDIST  IDIST is INTEGER Specifies the type of distribution to be used to generate a random matrix. = 1: UNIFORM( 0, 1 ) = 2: UNIFORM( 1, 1 ) = 3: NORMAL ( 0, 1 ) 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry ISEED specifies the seed of the random number generator. The values of ISEED are changed on exit, and can be used in the next call to DLATM4 to continue the same random number sequence. Note: ISEED(4) should be odd, for the random number generator used at present. 
[out]  A  A is DOUBLE PRECISION array, dimension (LDA, N) Array to be computed. 
[in]  LDA  LDA is INTEGER Leading dimension of A. Must be at least 1 and at least N. 
Definition at line 175 of file dlatm4.f.
LOGICAL function dlctes  (  double precision  ZR, 
double precision  ZI,  
double precision  D  
) 
DLCTES
DLCTES returns .TRUE. if the eigenvalue (ZR/D) + sqrt(1)*(ZI/D) is to be selected (specifically, in this subroutine, if the real part of the eigenvalue is negative), and otherwise it returns .FALSE.. It is used by the test routine DDRGES to test whether the driver routine DGGES succesfully sorts eigenvalues.
[in]  ZR  ZR is DOUBLE PRECISION The numerator of the real part of a complex eigenvalue (ZR/D) + i*(ZI/D). 
[in]  ZI  ZI is DOUBLE PRECISION The numerator of the imaginary part of a complex eigenvalue (ZR/D) + i*(ZI). 
[in]  D  D is DOUBLE PRECISION The denominator part of a complex eigenvalue (ZR/D) + i*(ZI/D). 
Definition at line 69 of file dlctes.f.
LOGICAL function dlctsx  (  double precision  AR, 
double precision  AI,  
double precision  BETA  
) 
DLCTSX
This function is used to determine what eigenvalues will be selected. If this is part of the test driver DDRGSX, do not change the code UNLESS you are testing input examples and not using the builtin examples.
[in]  AR  AR is DOUBLE PRECISION The numerator of the real part of a complex eigenvalue (AR/BETA) + i*(AI/BETA). 
[in]  AI  AI is DOUBLE PRECISION The numerator of the imaginary part of a complex eigenvalue (AR/BETA) + i*(AI). 
[in]  BETA  BETA is DOUBLE PRECISION The denominator part of a complex eigenvalue (AR/BETA) + i*(AI/BETA). 
Definition at line 66 of file dlctsx.f.
subroutine dlsets  (  integer  M, 
integer  P,  
integer  N,  
double precision, dimension( lda, * )  A,  
double precision, dimension( lda, * )  AF,  
integer  LDA,  
double precision, dimension( ldb, * )  B,  
double precision, dimension( ldb, * )  BF,  
integer  LDB,  
double precision, dimension( * )  C,  
double precision, dimension( * )  CF,  
double precision, dimension( * )  D,  
double precision, dimension( * )  DF,  
double precision, dimension( * )  X,  
double precision, dimension( lwork )  WORK,  
integer  LWORK,  
double precision, dimension( * )  RWORK,  
double precision, dimension( 2 )  RESULT  
) 
DLSETS
DLSETS tests DGGLSE  a subroutine for solving linear equality constrained least square problem (LSE).
[in]  M  M is INTEGER The number of rows of the matrix A. M >= 0. 
[in]  P  P is INTEGER The number of rows of the matrix B. P >= 0. 
[in]  N  N is INTEGER The number of columns of the matrices A and B. N >= 0. 
[in]  A  A is DOUBLE PRECISION array, dimension (LDA,N) The MbyN matrix A. 
[out]  AF  AF is DOUBLE PRECISION array, dimension (LDA,N) 
[in]  LDA  LDA is INTEGER The leading dimension of the arrays A, AF, Q and R. LDA >= max(M,N). 
[in]  B  B is DOUBLE PRECISION array, dimension (LDB,N) The PbyN matrix A. 
[out]  BF  BF is DOUBLE PRECISION array, dimension (LDB,N) 
[in]  LDB  LDB is INTEGER The leading dimension of the arrays B, BF, V and S. LDB >= max(P,N). 
[in]  C  C is DOUBLE PRECISION array, dimension( M ) the vector C in the LSE problem. 
[out]  CF  CF is DOUBLE PRECISION array, dimension( M ) 
[in]  D  D is DOUBLE PRECISION array, dimension( P ) the vector D in the LSE problem. 
[out]  DF  DF is DOUBLE PRECISION array, dimension( P ) 
[out]  X  X is DOUBLE PRECISION array, dimension( N ) solution vector X in the LSE problem. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER The dimension of the array WORK. 
[out]  RWORK  RWORK is DOUBLE PRECISION array, dimension (M) 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (2) The test ratios: RESULT(1) = norm( A*x  c )/ norm(A)*norm(X)*EPS RESULT(2) = norm( B*x  d )/ norm(B)*norm(X)*EPS 
Definition at line 151 of file dlsets.f.
subroutine dort01  (  character  ROWCOL, 
integer  M,  
integer  N,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( * )  WORK,  
integer  LWORK,  
double precision  RESID  
) 
DORT01
DORT01 checks that the matrix U is orthogonal by computing the ratio RESID = norm( I  U*U' ) / ( n * EPS ), if ROWCOL = 'R', or RESID = norm( I  U'*U ) / ( m * EPS ), if ROWCOL = 'C'. Alternatively, if there isn't sufficient workspace to form I  U*U' or I  U'*U, the ratio is computed as RESID = abs( I  U*U' ) / ( n * EPS ), if ROWCOL = 'R', or RESID = abs( I  U'*U ) / ( m * EPS ), if ROWCOL = 'C'. where EPS is the machine precision. ROWCOL is used only if m = n; if m > n, ROWCOL is assumed to be 'C', and if m < n, ROWCOL is assumed to be 'R'.
[in]  ROWCOL  ROWCOL is CHARACTER Specifies whether the rows or columns of U should be checked for orthogonality. Used only if M = N. = 'R': Check for orthogonal rows of U = 'C': Check for orthogonal columns of U 
[in]  M  M is INTEGER The number of rows of the matrix U. 
[in]  N  N is INTEGER The number of columns of the matrix U. 
[in]  U  U is DOUBLE PRECISION array, dimension (LDU,N) The orthogonal matrix U. U is checked for orthogonal columns if m > n or if m = n and ROWCOL = 'C'. U is checked for orthogonal rows if m < n or if m = n and ROWCOL = 'R'. 
[in]  LDU  LDU is INTEGER The leading dimension of the array U. LDU >= max(1,M). 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER The length of the array WORK. For best performance, LWORK should be at least N*(N+1) if ROWCOL = 'C' or M*(M+1) if ROWCOL = 'R', but the test will be done even if LWORK is 0. 
[out]  RESID  RESID is DOUBLE PRECISION RESID = norm( I  U * U' ) / ( n * EPS ), if ROWCOL = 'R', or RESID = norm( I  U' * U ) / ( m * EPS ), if ROWCOL = 'C'. 
Definition at line 117 of file dort01.f.
subroutine dort03  (  character*( * )  RC, 
integer  MU,  
integer  MV,  
integer  N,  
integer  K,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( ldv, * )  V,  
integer  LDV,  
double precision, dimension( * )  WORK,  
integer  LWORK,  
double precision  RESULT,  
integer  INFO  
) 
DORT03
DORT03 compares two orthogonal matrices U and V to see if their corresponding rows or columns span the same spaces. The rows are checked if RC = 'R', and the columns are checked if RC = 'C'. RESULT is the maximum of  V*V'  I  / ( MV ulp ), if RC = 'R', or  V'*V  I  / ( MV ulp ), if RC = 'C', and the maximum over rows (or columns) 1 to K of  U(i)  S*V(i) / ( N ulp ) where S is +1 (chosen to minimize the expression), U(i) is the ith row (column) of U, and V(i) is the ith row (column) of V.
[in]  RC  RC is CHARACTER*1 If RC = 'R' the rows of U and V are to be compared. If RC = 'C' the columns of U and V are to be compared. 
[in]  MU  MU is INTEGER The number of rows of U if RC = 'R', and the number of columns if RC = 'C'. If MU = 0 DORT03 does nothing. MU must be at least zero. 
[in]  MV  MV is INTEGER The number of rows of V if RC = 'R', and the number of columns if RC = 'C'. If MV = 0 DORT03 does nothing. MV must be at least zero. 
[in]  N  N is INTEGER If RC = 'R', the number of columns in the matrices U and V, and if RC = 'C', the number of rows in U and V. If N = 0 DORT03 does nothing. N must be at least zero. 
[in]  K  K is INTEGER The number of rows or columns of U and V to compare. 0 <= K <= max(MU,MV). 
[in]  U  U is DOUBLE PRECISION array, dimension (LDU,N) The first matrix to compare. If RC = 'R', U is MU by N, and if RC = 'C', U is N by MU. 
[in]  LDU  LDU is INTEGER The leading dimension of U. If RC = 'R', LDU >= max(1,MU), and if RC = 'C', LDU >= max(1,N). 
[in]  V  V is DOUBLE PRECISION array, dimension (LDV,N) The second matrix to compare. If RC = 'R', V is MV by N, and if RC = 'C', V is N by MV. 
[in]  LDV  LDV is INTEGER The leading dimension of V. If RC = 'R', LDV >= max(1,MV), and if RC = 'C', LDV >= max(1,N). 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LWORK) 
[in]  LWORK  LWORK is INTEGER The length of the array WORK. For best performance, LWORK should be at least N*N if RC = 'C' or M*M if RC = 'R', but the tests will be done even if LWORK is 0. 
[out]  RESULT  RESULT is DOUBLE PRECISION The value computed by the test described above. RESULT is limited to 1/ulp to avoid overflow. 
[out]  INFO  INFO is INTEGER 0 indicates a successful exit k indicates the kth parameter had an illegal value 
Definition at line 156 of file dort03.f.
subroutine dsbt21  (  character  UPLO, 
integer  N,  
integer  KA,  
integer  KS,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( * )  D,  
double precision, dimension( * )  E,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( * )  WORK,  
double precision, dimension( 2 )  RESULT  
) 
DSBT21
DSBT21 generally checks a decomposition of the form A = U S U' where ' means transpose, A is symmetric banded, U is orthogonal, and S is diagonal (if KS=0) or symmetric tridiagonal (if KS=1). Specifically: RESULT(1) =  A  U S U'  / ( A n ulp ) *andC> RESULT(2) =  I  UU'  / ( n ulp )
[in]  UPLO  UPLO is CHARACTER If UPLO='U', the upper triangle of A and V will be used and the (strictly) lower triangle will not be referenced. If UPLO='L', the lower triangle of A and V will be used and the (strictly) upper triangle will not be referenced. 
[in]  N  N is INTEGER The size of the matrix. If it is zero, DSBT21 does nothing. It must be at least zero. 
[in]  KA  KA is INTEGER The bandwidth of the matrix A. It must be at least zero. If it is larger than N1, then max( 0, N1 ) will be used. 
[in]  KS  KS is INTEGER The bandwidth of the matrix S. It may only be zero or one. If zero, then S is diagonal, and E is not referenced. If one, then S is symmetric tridiagonal. 
[in]  A  A is DOUBLE PRECISION array, dimension (LDA, N) The original (unfactored) matrix. It is assumed to be symmetric, and only the upper (UPLO='U') or only the lower (UPLO='L') will be referenced. 
[in]  LDA  LDA is INTEGER The leading dimension of A. It must be at least 1 and at least min( KA, N1 ). 
[in]  D  D is DOUBLE PRECISION array, dimension (N) The diagonal of the (symmetric tri) diagonal matrix S. 
[in]  E  E is DOUBLE PRECISION array, dimension (N1) The offdiagonal of the (symmetric tri) diagonal matrix S. E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and (3,2) element, etc. Not referenced if KS=0. 
[in]  U  U is DOUBLE PRECISION array, dimension (LDU, N) The orthogonal matrix in the decomposition, expressed as a dense matrix (i.e., not as a product of Householder transformations, Givens transformations, etc.) 
[in]  LDU  LDU is INTEGER The leading dimension of U. LDU must be at least N and at least 1. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (N**2+N) 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (2) The values computed by the two tests described above. The values are currently limited to 1/ulp, to avoid overflow. 
Definition at line 146 of file dsbt21.f.
subroutine dsgt01  (  integer  ITYPE, 
character  UPLO,  
integer  N,  
integer  M,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( ldb, * )  B,  
integer  LDB,  
double precision, dimension( ldz, * )  Z,  
integer  LDZ,  
double precision, dimension( * )  D,  
double precision, dimension( * )  WORK,  
double precision, dimension( * )  RESULT  
) 
DSGT01
DDGT01 checks a decomposition of the form A Z = B Z D or A B Z = Z D or B A Z = Z D where A is a symmetric matrix, B is symmetric positive definite, Z is orthogonal, and D is diagonal. One of the following test ratios is computed: ITYPE = 1: RESULT(1) =  A Z  B Z D  / ( A Z n ulp ) ITYPE = 2: RESULT(1) =  A B Z  Z D  / ( A Z n ulp ) ITYPE = 3: RESULT(1) =  B A Z  Z D  / ( A Z n ulp )
[in]  ITYPE  ITYPE is INTEGER The form of the symmetric generalized eigenproblem. = 1: A*z = (lambda)*B*z = 2: A*B*z = (lambda)*z = 3: B*A*z = (lambda)*z 
[in]  UPLO  UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrices A and B is stored. = 'U': Upper triangular = 'L': Lower triangular 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. 
[in]  M  M is INTEGER The number of eigenvalues found. 0 <= M <= N. 
[in]  A  A is DOUBLE PRECISION array, dimension (LDA, N) The original symmetric matrix A. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[in]  B  B is DOUBLE PRECISION array, dimension (LDB, N) The original symmetric positive definite matrix B. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). 
[in]  Z  Z is DOUBLE PRECISION array, dimension (LDZ, M) The computed eigenvectors of the generalized eigenproblem. 
[in]  LDZ  LDZ is INTEGER The leading dimension of the array Z. LDZ >= max(1,N). 
[in]  D  D is DOUBLE PRECISION array, dimension (M) The computed eigenvalues of the generalized eigenproblem. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (N*N) 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (1) The test ratio as described above. 
Definition at line 146 of file dsgt01.f.
LOGICAL function dslect  (  double precision  ZR, 
double precision  ZI  
) 
DSLECT
DSLECT returns .TRUE. if the eigenvalue ZR+sqrt(1)*ZI is to be selected, and otherwise it returns .FALSE. It is used by DCHK41 to test if DGEES succesfully sorts eigenvalues, and by DCHK43 to test if DGEESX succesfully sorts eigenvalues. The common block /SSLCT/ controls how eigenvalues are selected. If SELOPT = 0, then DSLECT return .TRUE. when ZR is less than zero, and .FALSE. otherwise. If SELOPT is at least 1, DSLECT returns SELVAL(SELOPT) and adds 1 to SELOPT, cycling back to 1 at SELMAX.
[in]  ZR  ZR is DOUBLE PRECISION The real part of a complex eigenvalue ZR + i*ZI. 
[in]  ZI  ZI is DOUBLE PRECISION The imaginary part of a complex eigenvalue ZR + i*ZI. 
Definition at line 63 of file dslect.f.
subroutine dspt21  (  integer  ITYPE, 
character  UPLO,  
integer  N,  
integer  KBAND,  
double precision, dimension( * )  AP,  
double precision, dimension( * )  D,  
double precision, dimension( * )  E,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( * )  VP,  
double precision, dimension( * )  TAU,  
double precision, dimension( * )  WORK,  
double precision, dimension( 2 )  RESULT  
) 
DSPT21
DSPT21 generally checks a decomposition of the form A = U S U' where ' means transpose, A is symmetric (stored in packed format), U is orthogonal, and S is diagonal (if KBAND=0) or symmetric tridiagonal (if KBAND=1). If ITYPE=1, then U is represented as a dense matrix, otherwise the U is expressed as a product of Householder transformations, whose vectors are stored in the array "V" and whose scaling constants are in "TAU"; we shall use the letter "V" to refer to the product of Householder transformations (which should be equal to U). Specifically, if ITYPE=1, then: RESULT(1) =  A  U S U'  / ( A n ulp ) *andC> RESULT(2) =  I  UU'  / ( n ulp ) If ITYPE=2, then: RESULT(1) =  A  V S V'  / ( A n ulp ) If ITYPE=3, then: RESULT(1) =  I  VU'  / ( n ulp ) Packed storage means that, for example, if UPLO='U', then the columns of the upper triangle of A are stored one after another, so that A(1,j+1) immediately follows A(j,j) in the array AP. Similarly, if UPLO='L', then the columns of the lower triangle of A are stored one after another in AP, so that A(j+1,j+1) immediately follows A(n,j) in the array AP. This means that A(i,j) is stored in: AP( i + j*(j1)/2 ) if UPLO='U' AP( i + (2*nj)*(j1)/2 ) if UPLO='L' The array VP bears the same relation to the matrix V that A does to AP. For ITYPE > 1, the transformation U is expressed as a product of Householder transformations: If UPLO='U', then V = H(n1)...H(1), where H(j) = I  tau(j) v(j) v(j)' and the first j1 elements of v(j) are stored in V(1:j1,j+1), (i.e., VP( j*(j+1)/2 + 1 : j*(j+1)/2 + j1 ) ), the jth element is 1, and the last nj elements are 0. If UPLO='L', then V = H(1)...H(n1), where H(j) = I  tau(j) v(j) v(j)' and the first j elements of v(j) are 0, the (j+1)st is 1, and the (j+2)nd through nth elements are stored in V(j+2:n,j) (i.e., in VP( (2*nj)*(j1)/2 + j+2 : (2*nj)*(j1)/2 + n ) .)
[in]  ITYPE  ITYPE is INTEGER Specifies the type of tests to be performed. 1: U expressed as a dense orthogonal matrix: RESULT(1) =  A  U S U'  / ( A n ulp ) *andC> RESULT(2) =  I  UU'  / ( n ulp ) 2: U expressed as a product V of Housholder transformations: RESULT(1) =  A  V S V'  / ( A n ulp ) 3: U expressed both as a dense orthogonal matrix and as a product of Housholder transformations: RESULT(1) =  I  VU'  / ( n ulp ) 
[in]  UPLO  UPLO is CHARACTER If UPLO='U', AP and VP are considered to contain the upper triangle of A and V. If UPLO='L', AP and VP are considered to contain the lower triangle of A and V. 
[in]  N  N is INTEGER The size of the matrix. If it is zero, DSPT21 does nothing. It must be at least zero. 
[in]  KBAND  KBAND is INTEGER The bandwidth of the matrix. It may only be zero or one. If zero, then S is diagonal, and E is not referenced. If one, then S is symmetric tridiagonal. 
[in]  AP  AP is DOUBLE PRECISION array, dimension (N*(N+1)/2) The original (unfactored) matrix. It is assumed to be symmetric, and contains the columns of just the upper triangle (UPLO='U') or only the lower triangle (UPLO='L'), packed one after another. 
[in]  D  D is DOUBLE PRECISION array, dimension (N) The diagonal of the (symmetric tri) diagonal matrix. 
[in]  E  E is DOUBLE PRECISION array, dimension (N1) The offdiagonal of the (symmetric tri) diagonal matrix. E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and (3,2) element, etc. Not referenced if KBAND=0. 
[in]  U  U is DOUBLE PRECISION array, dimension (LDU, N) If ITYPE=1 or 3, this contains the orthogonal matrix in the decomposition, expressed as a dense matrix. If ITYPE=2, then it is not referenced. 
[in]  LDU  LDU is INTEGER The leading dimension of U. LDU must be at least N and at least 1. 
[in]  VP  VP is DOUBLE PRECISION array, dimension (N*(N+1)/2) If ITYPE=2 or 3, the columns of this array contain the Householder vectors used to describe the orthogonal matrix in the decomposition, as described in purpose. *NOTE* If ITYPE=2 or 3, V is modified and restored. The subdiagonal (if UPLO='L') or the superdiagonal (if UPLO='U') is set to one, and later reset to its original value, during the course of the calculation. If ITYPE=1, then it is neither referenced nor modified. 
[in]  TAU  TAU is DOUBLE PRECISION array, dimension (N) If ITYPE >= 2, then TAU(j) is the scalar factor of v(j) v(j)' in the Householder transformation H(j) of the product U = H(1)...H(n2) If ITYPE < 2, then TAU is not referenced. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (N**2+N) Workspace. 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (2) The values computed by the two tests described above. The values are currently limited to 1/ulp, to avoid overflow. RESULT(1) is always modified. RESULT(2) is modified only if ITYPE=1. 
Definition at line 219 of file dspt21.f.
subroutine dstech  (  integer  N, 
double precision, dimension( * )  A,  
double precision, dimension( * )  B,  
double precision, dimension( * )  EIG,  
double precision  TOL,  
double precision, dimension( * )  WORK,  
integer  INFO  
) 
DSTECH
Let T be the tridiagonal matrix with diagonal entries A(1) ,..., A(N) and offdiagonal entries B(1) ,..., B(N1)). DSTECH checks to see if EIG(1) ,..., EIG(N) are indeed accurate eigenvalues of T. It does this by expanding each EIG(I) into an interval [SVD(I)  EPS, SVD(I) + EPS], merging overlapping intervals if any, and using Sturm sequences to count and verify whether each resulting interval has the correct number of eigenvalues (using DSTECT). Here EPS = TOL*MAZHEPS*MAXEIG, where MACHEPS is the machine precision and MAXEIG is the absolute value of the largest eigenvalue. If each interval contains the correct number of eigenvalues, INFO = 0 is returned, otherwise INFO is the index of the first eigenvalue in the first bad interval.
[in]  N  N is INTEGER The dimension of the tridiagonal matrix T. 
[in]  A  A is DOUBLE PRECISION array, dimension (N) The diagonal entries of the tridiagonal matrix T. 
[in]  B  B is DOUBLE PRECISION array, dimension (N1) The offdiagonal entries of the tridiagonal matrix T. 
[in]  EIG  EIG is DOUBLE PRECISION array, dimension (N) The purported eigenvalues to be checked. 
[in]  TOL  TOL is DOUBLE PRECISION Error tolerance for checking, a multiple of the machine precision. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (N) 
[out]  INFO  INFO is INTEGER 0 if the eigenvalues are all correct (to within 1 + TOL*MAZHEPS*MAXEIG) >0 if the interval containing the INFOth eigenvalue contains the incorrect number of eigenvalues. 
Definition at line 102 of file dstech.f.
subroutine dstect  (  integer  N, 
double precision, dimension( * )  A,  
double precision, dimension( * )  B,  
double precision  SHIFT,  
integer  NUM  
) 
DSTECT
DSTECT counts the number NUM of eigenvalues of a tridiagonal matrix T which are less than or equal to SHIFT. T has diagonal entries A(1), ... , A(N), and offdiagonal entries B(1), ..., B(N1). See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966
[in]  N  N is INTEGER The dimension of the tridiagonal matrix T. 
[in]  A  A is DOUBLE PRECISION array, dimension (N) The diagonal entries of the tridiagonal matrix T. 
[in]  B  B is DOUBLE PRECISION array, dimension (N1) The offdiagonal entries of the tridiagonal matrix T. 
[in]  SHIFT  SHIFT is DOUBLE PRECISION The shift, used as described under Purpose. 
[out]  NUM  NUM is INTEGER The number of eigenvalues of T less than or equal to SHIFT. 
Definition at line 83 of file dstect.f.
subroutine dstt21  (  integer  N, 
integer  KBAND,  
double precision, dimension( * )  AD,  
double precision, dimension( * )  AE,  
double precision, dimension( * )  SD,  
double precision, dimension( * )  SE,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( * )  WORK,  
double precision, dimension( 2 )  RESULT  
) 
DSTT21
DSTT21 checks a decomposition of the form A = U S U' where ' means transpose, A is symmetric tridiagonal, U is orthogonal, and S is diagonal (if KBAND=0) or symmetric tridiagonal (if KBAND=1). Two tests are performed: RESULT(1) =  A  U S U'  / ( A n ulp ) RESULT(2) =  I  UU'  / ( n ulp )
[in]  N  N is INTEGER The size of the matrix. If it is zero, DSTT21 does nothing. It must be at least zero. 
[in]  KBAND  KBAND is INTEGER The bandwidth of the matrix S. It may only be zero or one. If zero, then S is diagonal, and SE is not referenced. If one, then S is symmetric tridiagonal. 
[in]  AD  AD is DOUBLE PRECISION array, dimension (N) The diagonal of the original (unfactored) matrix A. A is assumed to be symmetric tridiagonal. 
[in]  AE  AE is DOUBLE PRECISION array, dimension (N1) The offdiagonal of the original (unfactored) matrix A. A is assumed to be symmetric tridiagonal. AE(1) is the (1,2) and (2,1) element, AE(2) is the (2,3) and (3,2) element, etc. 
[in]  SD  SD is DOUBLE PRECISION array, dimension (N) The diagonal of the (symmetric tri) diagonal matrix S. 
[in]  SE  SE is DOUBLE PRECISION array, dimension (N1) The offdiagonal of the (symmetric tri) diagonal matrix S. Not referenced if KBSND=0. If KBAND=1, then AE(1) is the (1,2) and (2,1) element, SE(2) is the (2,3) and (3,2) element, etc. 
[in]  U  U is DOUBLE PRECISION array, dimension (LDU, N) The orthogonal matrix in the decomposition. 
[in]  LDU  LDU is INTEGER The leading dimension of U. LDU must be at least N. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (N*(N+1)) 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (2) The values computed by the two tests described above. The values are currently limited to 1/ulp, to avoid overflow. RESULT(1) is always modified. 
Definition at line 127 of file dstt21.f.
subroutine dstt22  (  integer  N, 
integer  M,  
integer  KBAND,  
double precision, dimension( * )  AD,  
double precision, dimension( * )  AE,  
double precision, dimension( * )  SD,  
double precision, dimension( * )  SE,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( ldwork, * )  WORK,  
integer  LDWORK,  
double precision, dimension( 2 )  RESULT  
) 
DSTT22
DSTT22 checks a set of M eigenvalues and eigenvectors, A U = U S where A is symmetric tridiagonal, the columns of U are orthogonal, and S is diagonal (if KBAND=0) or symmetric tridiagonal (if KBAND=1). Two tests are performed: RESULT(1) =  U' A U  S  / ( A m ulp ) RESULT(2) =  I  U'U  / ( m ulp )
[in]  N  N is INTEGER The size of the matrix. If it is zero, DSTT22 does nothing. It must be at least zero. 
[in]  M  M is INTEGER The number of eigenpairs to check. If it is zero, DSTT22 does nothing. It must be at least zero. 
[in]  KBAND  KBAND is INTEGER The bandwidth of the matrix S. It may only be zero or one. If zero, then S is diagonal, and SE is not referenced. If one, then S is symmetric tridiagonal. 
[in]  AD  AD is DOUBLE PRECISION array, dimension (N) The diagonal of the original (unfactored) matrix A. A is assumed to be symmetric tridiagonal. 
[in]  AE  AE is DOUBLE PRECISION array, dimension (N) The offdiagonal of the original (unfactored) matrix A. A is assumed to be symmetric tridiagonal. AE(1) is ignored, AE(2) is the (1,2) and (2,1) element, etc. 
[in]  SD  SD is DOUBLE PRECISION array, dimension (N) The diagonal of the (symmetric tri) diagonal matrix S. 
[in]  SE  SE is DOUBLE PRECISION array, dimension (N) The offdiagonal of the (symmetric tri) diagonal matrix S. Not referenced if KBSND=0. If KBAND=1, then AE(1) is ignored, SE(2) is the (1,2) and (2,1) element, etc. 
[in]  U  U is DOUBLE PRECISION array, dimension (LDU, N) The orthogonal matrix in the decomposition. 
[in]  LDU  LDU is INTEGER The leading dimension of U. LDU must be at least N. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (LDWORK, M+1) 
[in]  LDWORK  LDWORK is INTEGER The leading dimension of WORK. LDWORK must be at least max(1,M). 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (2) The values computed by the two tests described above. The values are currently limited to 1/ulp, to avoid overflow. 
Definition at line 139 of file dstt22.f.
subroutine dsvdch  (  integer  N, 
double precision, dimension( * )  S,  
double precision, dimension( * )  E,  
double precision, dimension( * )  SVD,  
double precision  TOL,  
integer  INFO  
) 
DSVDCH
DSVDCH checks to see if SVD(1) ,..., SVD(N) are accurate singular values of the bidiagonal matrix B with diagonal entries S(1) ,..., S(N) and superdiagonal entries E(1) ,..., E(N1)). It does this by expanding each SVD(I) into an interval [SVD(I) * (1EPS) , SVD(I) * (1+EPS)], merging overlapping intervals if any, and using Sturm sequences to count and verify whether each resulting interval has the correct number of singular values (using DSVDCT). Here EPS=TOL*MAX(N/10,1)*MAZHEP, where MACHEP is the machine precision. The routine assumes the singular values are sorted with SVD(1) the largest and SVD(N) smallest. If each interval contains the correct number of singular values, INFO = 0 is returned, otherwise INFO is the index of the first singular value in the first bad interval.
[in]  N  N is INTEGER The dimension of the bidiagonal matrix B. 
[in]  S  S is DOUBLE PRECISION array, dimension (N) The diagonal entries of the bidiagonal matrix B. 
[in]  E  E is DOUBLE PRECISION array, dimension (N1) The superdiagonal entries of the bidiagonal matrix B. 
[in]  SVD  SVD is DOUBLE PRECISION array, dimension (N) The computed singular values to be checked. 
[in]  TOL  TOL is DOUBLE PRECISION Error tolerance for checking, a multiplier of the machine precision. 
[out]  INFO  INFO is INTEGER =0 if the singular values are all correct (to within 1 + TOL*MAZHEPS) >0 if the interval containing the INFOth singular value contains the incorrect number of singular values. 
Definition at line 98 of file dsvdch.f.
subroutine dsvdct  (  integer  N, 
double precision, dimension( * )  S,  
double precision, dimension( * )  E,  
double precision  SHIFT,  
integer  NUM  
) 
DSVDCT
DSVDCT counts the number NUM of eigenvalues of a 2*N by 2*N tridiagonal matrix T which are less than or equal to SHIFT. T is formed by putting zeros on the diagonal and making the offdiagonals equal to S(1), E(1), S(2), E(2), ... , E(N1), S(N). If SHIFT is positive, NUM is equal to N plus the number of singular values of a bidiagonal matrix B less than or equal to SHIFT. Here B has diagonal entries S(1), ..., S(N) and superdiagonal entries E(1), ... E(N1). If SHIFT is negative, NUM is equal to the number of singular values of B greater than or equal to SHIFT. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966
[in]  N  N is INTEGER The dimension of the bidiagonal matrix B. 
[in]  S  S is DOUBLE PRECISION array, dimension (N) The diagonal entries of the bidiagonal matrix B. 
[in]  E  E is DOUBLE PRECISION array of dimension (N1) The superdiagonal entries of the bidiagonal matrix B. 
[in]  SHIFT  SHIFT is DOUBLE PRECISION The shift, used as described under Purpose. 
[out]  NUM  NUM is INTEGER The number of eigenvalues of T less than or equal to SHIFT. 
Definition at line 88 of file dsvdct.f.
DOUBLE PRECISION function dsxt1  (  integer  IJOB, 
double precision, dimension( * )  D1,  
integer  N1,  
double precision, dimension( * )  D2,  
integer  N2,  
double precision  ABSTOL,  
double precision  ULP,  
double precision  UNFL  
) 
DSXT1
DSXT1 computes the difference between a set of eigenvalues. The result is returned as the function value. IJOB = 1: Computes max { min  D1(i)D2(j)  } i j IJOB = 2: Computes max { min  D1(i)D2(j)  / i j ( ABSTOL + D1(i)*ULP ) }
[in]  IJOB  IJOB is INTEGER Specifies the type of tests to be performed. (See above.) 
[in]  D1  D1 is DOUBLE PRECISION array, dimension (N1) The first array. D1 should be in increasing order, i.e., D1(j) <= D1(j+1). 
[in]  N1  N1 is INTEGER The length of D1. 
[in]  D2  D2 is DOUBLE PRECISION array, dimension (N2) The second array. D2 should be in increasing order, i.e., D2(j) <= D2(j+1). 
[in]  N2  N2 is INTEGER The length of D2. 
[in]  ABSTOL  ABSTOL is DOUBLE PRECISION The absolute tolerance, used as a measure of the error. 
[in]  ULP  ULP is DOUBLE PRECISION Machine precision. 
[in]  UNFL  UNFL is DOUBLE PRECISION The smallest positive number whose reciprocal does not overflow. 
Definition at line 106 of file dsxt1.f.
subroutine dsyt21  (  integer  ITYPE, 
character  UPLO,  
integer  N,  
integer  KBAND,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( * )  D,  
double precision, dimension( * )  E,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( ldv, * )  V,  
integer  LDV,  
double precision, dimension( * )  TAU,  
double precision, dimension( * )  WORK,  
double precision, dimension( 2 )  RESULT  
) 
DSYT21
DSYT21 generally checks a decomposition of the form A = U S U' where ' means transpose, A is symmetric, U is orthogonal, and S is diagonal (if KBAND=0) or symmetric tridiagonal (if KBAND=1). If ITYPE=1, then U is represented as a dense matrix; otherwise U is expressed as a product of Householder transformations, whose vectors are stored in the array "V" and whose scaling constants are in "TAU". We shall use the letter "V" to refer to the product of Householder transformations (which should be equal to U). Specifically, if ITYPE=1, then: RESULT(1) =  A  U S U'  / ( A n ulp ) *andC> RESULT(2) =  I  UU'  / ( n ulp ) If ITYPE=2, then: RESULT(1) =  A  V S V'  / ( A n ulp ) If ITYPE=3, then: RESULT(1) =  I  VU'  / ( n ulp ) For ITYPE > 1, the transformation U is expressed as a product V = H(1)...H(n2), where H(j) = I  tau(j) v(j) v(j)' and each vector v(j) has its first j elements 0 and the remaining nj elements stored in V(j+1:n,j).
[in]  ITYPE  ITYPE is INTEGER Specifies the type of tests to be performed. 1: U expressed as a dense orthogonal matrix: RESULT(1) =  A  U S U'  / ( A n ulp ) *andC> RESULT(2) =  I  UU'  / ( n ulp ) 2: U expressed as a product V of Housholder transformations: RESULT(1) =  A  V S V'  / ( A n ulp ) 3: U expressed both as a dense orthogonal matrix and as a product of Housholder transformations: RESULT(1) =  I  VU'  / ( n ulp ) 
[in]  UPLO  UPLO is CHARACTER If UPLO='U', the upper triangle of A and V will be used and the (strictly) lower triangle will not be referenced. If UPLO='L', the lower triangle of A and V will be used and the (strictly) upper triangle will not be referenced. 
[in]  N  N is INTEGER The size of the matrix. If it is zero, DSYT21 does nothing. It must be at least zero. 
[in]  KBAND  KBAND is INTEGER The bandwidth of the matrix. It may only be zero or one. If zero, then S is diagonal, and E is not referenced. If one, then S is symmetric tridiagonal. 
[in]  A  A is DOUBLE PRECISION array, dimension (LDA, N) The original (unfactored) matrix. It is assumed to be symmetric, and only the upper (UPLO='U') or only the lower (UPLO='L') will be referenced. 
[in]  LDA  LDA is INTEGER The leading dimension of A. It must be at least 1 and at least N. 
[in]  D  D is DOUBLE PRECISION array, dimension (N) The diagonal of the (symmetric tri) diagonal matrix. 
[in]  E  E is DOUBLE PRECISION array, dimension (N1) The offdiagonal of the (symmetric tri) diagonal matrix. E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and (3,2) element, etc. Not referenced if KBAND=0. 
[in]  U  U is DOUBLE PRECISION array, dimension (LDU, N) If ITYPE=1 or 3, this contains the orthogonal matrix in the decomposition, expressed as a dense matrix. If ITYPE=2, then it is not referenced. 
[in]  LDU  LDU is INTEGER The leading dimension of U. LDU must be at least N and at least 1. 
[in]  V  V is DOUBLE PRECISION array, dimension (LDV, N) If ITYPE=2 or 3, the columns of this array contain the Householder vectors used to describe the orthogonal matrix in the decomposition. If UPLO='L', then the vectors are in the lower triangle, if UPLO='U', then in the upper triangle. *NOTE* If ITYPE=2 or 3, V is modified and restored. The subdiagonal (if UPLO='L') or the superdiagonal (if UPLO='U') is set to one, and later reset to its original value, during the course of the calculation. If ITYPE=1, then it is neither referenced nor modified. 
[in]  LDV  LDV is INTEGER The leading dimension of V. LDV must be at least N and at least 1. 
[in]  TAU  TAU is DOUBLE PRECISION array, dimension (N) If ITYPE >= 2, then TAU(j) is the scalar factor of v(j) v(j)' in the Householder transformation H(j) of the product U = H(1)...H(n2) If ITYPE < 2, then TAU is not referenced. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (2*N**2) 
[out]  RESULT  RESULT is DOUBLE PRECISION array, dimension (2) The values computed by the two tests described above. The values are currently limited to 1/ulp, to avoid overflow. RESULT(1) is always modified. RESULT(2) is modified only if ITYPE=1. 
Definition at line 205 of file dsyt21.f.
subroutine dsyt22  (  integer  ITYPE, 
character  UPLO,  
integer  N,  
integer  M,  
integer  KBAND,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( * )  D,  
double precision, dimension( * )  E,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( ldv, * )  V,  
integer  LDV,  
double precision, dimension( * )  TAU,  
double precision, dimension( * )  WORK,  
double precision, dimension( 2 )  RESULT  
) 
DSYT22
DSYT22 generally checks a decomposition of the form A U = U S where A is symmetric, the columns of U are orthonormal, and S is diagonal (if KBAND=0) or symmetric tridiagonal (if KBAND=1). If ITYPE=1, then U is represented as a dense matrix, otherwise the U is expressed as a product of Householder transformations, whose vectors are stored in the array "V" and whose scaling constants are in "TAU"; we shall use the letter "V" to refer to the product of Householder transformations (which should be equal to U). Specifically, if ITYPE=1, then: RESULT(1) =  U' A U  S  / ( A m ulp ) *andC> RESULT(2) =  I  U'U  / ( m ulp )
ITYPE INTEGER Specifies the type of tests to be performed. 1: U expressed as a dense orthogonal matrix: RESULT(1) =  A  U S U'  / ( A n ulp ) *andC> RESULT(2) =  I  UU'  / ( n ulp ) UPLO CHARACTER If UPLO='U', the upper triangle of A will be used and the (strictly) lower triangle will not be referenced. If UPLO='L', the lower triangle of A will be used and the (strictly) upper triangle will not be referenced. Not modified. N INTEGER The size of the matrix. If it is zero, DSYT22 does nothing. It must be at least zero. Not modified. M INTEGER The number of columns of U. If it is zero, DSYT22 does nothing. It must be at least zero. Not modified. KBAND INTEGER The bandwidth of the matrix. It may only be zero or one. If zero, then S is diagonal, and E is not referenced. If one, then S is symmetric tridiagonal. Not modified. A DOUBLE PRECISION array, dimension (LDA , N) The original (unfactored) matrix. It is assumed to be symmetric, and only the upper (UPLO='U') or only the lower (UPLO='L') will be referenced. Not modified. LDA INTEGER The leading dimension of A. It must be at least 1 and at least N. Not modified. D DOUBLE PRECISION array, dimension (N) The diagonal of the (symmetric tri) diagonal matrix. Not modified. E DOUBLE PRECISION array, dimension (N) The offdiagonal of the (symmetric tri) diagonal matrix. E(1) is ignored, E(2) is the (1,2) and (2,1) element, etc. Not referenced if KBAND=0. Not modified. U DOUBLE PRECISION array, dimension (LDU, N) If ITYPE=1 or 3, this contains the orthogonal matrix in the decomposition, expressed as a dense matrix. If ITYPE=2, then it is not referenced. Not modified. LDU INTEGER The leading dimension of U. LDU must be at least N and at least 1. Not modified. V DOUBLE PRECISION array, dimension (LDV, N) If ITYPE=2 or 3, the lower triangle of this array contains the Householder vectors used to describe the orthogonal matrix in the decomposition. If ITYPE=1, then it is not referenced. Not modified. LDV INTEGER The leading dimension of V. LDV must be at least N and at least 1. Not modified. TAU DOUBLE PRECISION array, dimension (N) If ITYPE >= 2, then TAU(j) is the scalar factor of v(j) v(j)' in the Householder transformation H(j) of the product U = H(1)...H(n2) If ITYPE < 2, then TAU is not referenced. Not modified. WORK DOUBLE PRECISION array, dimension (2*N**2) Workspace. Modified. RESULT DOUBLE PRECISION array, dimension (2) The values computed by the two tests described above. The values are currently limited to 1/ulp, to avoid overflow. RESULT(1) is always modified. RESULT(2) is modified only if LDU is at least N. Modified.
Definition at line 155 of file dsyt22.f.