 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

◆ dchkgg()

 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

Purpose:
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 threshold 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 threshold 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=(N-1)/2

(7)  ( D, I )         where D is diag( 0, 1,..., N-1 ) (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, ..., N-3, 0 ) and
D2 is diag( 0, N-3, N-4,..., 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, ..., N-3, 0 ) and diag(T2) =
( 0, N-3, N-4,..., 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, 1-d, ..., 1-(N-5)*d=s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 )

N-5
(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(N-4), 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
where r1,..., r(N-4) are random.

(22) U ( big*T1, small*T2 ) V    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(23) U ( small*T1, big*T2 ) V    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(24) U ( small*T1, small*T2 ) V  diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(25) U ( big*T1, big*T2 ) V      diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(26) U ( T1, T2 ) V     where T1 and T2 are random upper-triangular
matrices.
Parameters
 [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 13-15 will be computed and compared with THRESH. = .FALSE.: Only test ratios 1-12 will be computed and tested. Ratios 13-15 will be set to zero. = .TRUE.: All the test ratios 1-15 will be computed and tested. [in] THRSHN THRSHN is DOUBLE PRECISION Threshold 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 5--10.) [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 k-th 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 i-th 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 506 of file dchkgg.f.

511 *
512 * -- LAPACK test routine --
513 * -- LAPACK is a software package provided by Univ. of Tennessee, --
514 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
515 *
516 * .. Scalar Arguments ..
517  LOGICAL TSTDIF
518  INTEGER INFO, LDA, LDU, LWORK, NOUNIT, NSIZES, NTYPES
519  DOUBLE PRECISION THRESH, THRSHN
520 * ..
521 * .. Array Arguments ..
522  LOGICAL DOTYPE( * ), LLWORK( * )
523  INTEGER ISEED( 4 ), NN( * )
524  DOUBLE PRECISION A( LDA, * ), ALPHI1( * ), ALPHI3( * ),
525  \$ ALPHR1( * ), ALPHR3( * ), B( LDA, * ),
526  \$ BETA1( * ), BETA3( * ), EVECTL( LDU, * ),
527  \$ EVECTR( LDU, * ), H( LDA, * ), P1( LDA, * ),
528  \$ P2( LDA, * ), Q( LDU, * ), RESULT( 15 ),
529  \$ S1( LDA, * ), S2( LDA, * ), T( LDA, * ),
530  \$ U( LDU, * ), V( LDU, * ), WORK( * ),
531  \$ Z( LDU, * )
532 * ..
533 *
534 * =====================================================================
535 *
536 * .. Parameters ..
537  DOUBLE PRECISION ZERO, ONE
538  parameter( zero = 0.0d0, one = 1.0d0 )
539  INTEGER MAXTYP
540  parameter( maxtyp = 26 )
541 * ..
542 * .. Local Scalars ..
544  INTEGER I1, IADD, IINFO, IN, J, JC, JR, JSIZE, JTYPE,
545  \$ LWKOPT, MTYPES, N, N1, NERRS, NMATS, NMAX,
546  \$ NTEST, NTESTT
547  DOUBLE PRECISION ANORM, BNORM, SAFMAX, SAFMIN, TEMP1, TEMP2,
548  \$ ULP, ULPINV
549 * ..
550 * .. Local Arrays ..
551  INTEGER IASIGN( MAXTYP ), IBSIGN( MAXTYP ),
552  \$ IOLDSD( 4 ), KADD( 6 ), KAMAGN( MAXTYP ),
553  \$ KATYPE( MAXTYP ), KAZERO( MAXTYP ),
554  \$ KBMAGN( MAXTYP ), KBTYPE( MAXTYP ),
555  \$ KBZERO( MAXTYP ), KCLASS( MAXTYP ),
556  \$ KTRIAN( MAXTYP ), KZ1( 6 ), KZ2( 6 )
557  DOUBLE PRECISION DUMMA( 4 ), RMAGN( 0: 3 )
558 * ..
559 * .. External Functions ..
560  DOUBLE PRECISION DLAMCH, DLANGE, DLARND
561  EXTERNAL dlamch, dlange, dlarnd
562 * ..
563 * .. External Subroutines ..
564  EXTERNAL dgeqr2, dget51, dget52, dgghrd, dhgeqz, dlabad,
566  \$ dtgevc, xerbla
567 * ..
568 * .. Intrinsic Functions ..
569  INTRINSIC abs, dble, max, min, sign
570 * ..
571 * .. Data statements ..
572  DATA kclass / 15*1, 10*2, 1*3 /
573  DATA kz1 / 0, 1, 2, 1, 3, 3 /
574  DATA kz2 / 0, 0, 1, 2, 1, 1 /
575  DATA kadd / 0, 0, 0, 0, 3, 2 /
576  DATA katype / 0, 1, 0, 1, 2, 3, 4, 1, 4, 4, 1, 1, 4,
577  \$ 4, 4, 2, 4, 5, 8, 7, 9, 4*4, 0 /
578  DATA kbtype / 0, 0, 1, 1, 2, -3, 1, 4, 1, 1, 4, 4,
579  \$ 1, 1, -4, 2, -4, 8*8, 0 /
580  DATA kazero / 6*1, 2, 1, 2*2, 2*1, 2*2, 3, 1, 3,
581  \$ 4*5, 4*3, 1 /
582  DATA kbzero / 6*1, 1, 2, 2*1, 2*2, 2*1, 4, 1, 4,
583  \$ 4*6, 4*4, 1 /
584  DATA kamagn / 8*1, 2, 3, 2, 3, 2, 3, 7*1, 2, 3, 3,
585  \$ 2, 1 /
586  DATA kbmagn / 8*1, 3, 2, 3, 2, 2, 3, 7*1, 3, 2, 3,
587  \$ 2, 1 /
588  DATA ktrian / 16*0, 10*1 /
589  DATA iasign / 6*0, 2, 0, 2*2, 2*0, 3*2, 0, 2, 3*0,
590  \$ 5*2, 0 /
591  DATA ibsign / 7*0, 2, 2*0, 2*2, 2*0, 2, 0, 2, 9*0 /
592 * ..
593 * .. Executable Statements ..
594 *
595 * Check for errors
596 *
597  info = 0
598 *
600  nmax = 1
601  DO 10 j = 1, nsizes
602  nmax = max( nmax, nn( j ) )
603  IF( nn( j ).LT.0 )
605  10 CONTINUE
606 *
607 * Maximum blocksize and shift -- we assume that blocksize and number
608 * of shifts are monotone increasing functions of N.
609 *
610  lwkopt = max( 6*nmax, 2*nmax*nmax, 1 )
611 *
612 * Check for errors
613 *
614  IF( nsizes.LT.0 ) THEN
615  info = -1
616  ELSE IF( badnn ) THEN
617  info = -2
618  ELSE IF( ntypes.LT.0 ) THEN
619  info = -3
620  ELSE IF( thresh.LT.zero ) THEN
621  info = -6
622  ELSE IF( lda.LE.1 .OR. lda.LT.nmax ) THEN
623  info = -10
624  ELSE IF( ldu.LE.1 .OR. ldu.LT.nmax ) THEN
625  info = -19
626  ELSE IF( lwkopt.GT.lwork ) THEN
627  info = -30
628  END IF
629 *
630  IF( info.NE.0 ) THEN
631  CALL xerbla( 'DCHKGG', -info )
632  RETURN
633  END IF
634 *
635 * Quick return if possible
636 *
637  IF( nsizes.EQ.0 .OR. ntypes.EQ.0 )
638  \$ RETURN
639 *
640  safmin = dlamch( 'Safe minimum' )
641  ulp = dlamch( 'Epsilon' )*dlamch( 'Base' )
642  safmin = safmin / ulp
643  safmax = one / safmin
644  CALL dlabad( safmin, safmax )
645  ulpinv = one / ulp
646 *
647 * The values RMAGN(2:3) depend on N, see below.
648 *
649  rmagn( 0 ) = zero
650  rmagn( 1 ) = one
651 *
652 * Loop over sizes, types
653 *
654  ntestt = 0
655  nerrs = 0
656  nmats = 0
657 *
658  DO 240 jsize = 1, nsizes
659  n = nn( jsize )
660  n1 = max( 1, n )
661  rmagn( 2 ) = safmax*ulp / dble( n1 )
662  rmagn( 3 ) = safmin*ulpinv*n1
663 *
664  IF( nsizes.NE.1 ) THEN
665  mtypes = min( maxtyp, ntypes )
666  ELSE
667  mtypes = min( maxtyp+1, ntypes )
668  END IF
669 *
670  DO 230 jtype = 1, mtypes
671  IF( .NOT.dotype( jtype ) )
672  \$ GO TO 230
673  nmats = nmats + 1
674  ntest = 0
675 *
676 * Save ISEED in case of an error.
677 *
678  DO 20 j = 1, 4
679  ioldsd( j ) = iseed( j )
680  20 CONTINUE
681 *
682 * Initialize RESULT
683 *
684  DO 30 j = 1, 15
685  result( j ) = zero
686  30 CONTINUE
687 *
688 * Compute A and B
689 *
690 * Description of control parameters:
691 *
692 * KZLASS: =1 means w/o rotation, =2 means w/ rotation,
693 * =3 means random.
694 * KATYPE: the "type" to be passed to DLATM4 for computing A.
695 * KAZERO: the pattern of zeros on the diagonal for A:
696 * =1: ( xxx ), =2: (0, xxx ) =3: ( 0, 0, xxx, 0 ),
697 * =4: ( 0, xxx, 0, 0 ), =5: ( 0, 0, 1, xxx, 0 ),
698 * =6: ( 0, 1, 0, xxx, 0 ). (xxx means a string of
699 * non-zero entries.)
700 * KAMAGN: the magnitude of the matrix: =0: zero, =1: O(1),
701 * =2: large, =3: small.
702 * IASIGN: 1 if the diagonal elements of A are to be
703 * multiplied by a random magnitude 1 number, =2 if
704 * randomly chosen diagonal blocks are to be rotated
705 * to form 2x2 blocks.
706 * KBTYPE, KBZERO, KBMAGN, IBSIGN: the same, but for B.
707 * KTRIAN: =0: don't fill in the upper triangle, =1: do.
708 * KZ1, KZ2, KADD: used to implement KAZERO and KBZERO.
709 * RMAGN: used to implement KAMAGN and KBMAGN.
710 *
711  IF( mtypes.GT.maxtyp )
712  \$ GO TO 110
713  iinfo = 0
714  IF( kclass( jtype ).LT.3 ) THEN
715 *
716 * Generate A (w/o rotation)
717 *
718  IF( abs( katype( jtype ) ).EQ.3 ) THEN
719  in = 2*( ( n-1 ) / 2 ) + 1
720  IF( in.NE.n )
721  \$ CALL dlaset( 'Full', n, n, zero, zero, a, lda )
722  ELSE
723  in = n
724  END IF
725  CALL dlatm4( katype( jtype ), in, kz1( kazero( jtype ) ),
726  \$ kz2( kazero( jtype ) ), iasign( jtype ),
727  \$ rmagn( kamagn( jtype ) ), ulp,
728  \$ rmagn( ktrian( jtype )*kamagn( jtype ) ), 2,
729  \$ iseed, a, lda )
732  \$ a( iadd, iadd ) = rmagn( kamagn( jtype ) )
733 *
734 * Generate B (w/o rotation)
735 *
736  IF( abs( kbtype( jtype ) ).EQ.3 ) THEN
737  in = 2*( ( n-1 ) / 2 ) + 1
738  IF( in.NE.n )
739  \$ CALL dlaset( 'Full', n, n, zero, zero, b, lda )
740  ELSE
741  in = n
742  END IF
743  CALL dlatm4( kbtype( jtype ), in, kz1( kbzero( jtype ) ),
744  \$ kz2( kbzero( jtype ) ), ibsign( jtype ),
745  \$ rmagn( kbmagn( jtype ) ), one,
746  \$ rmagn( ktrian( jtype )*kbmagn( jtype ) ), 2,
747  \$ iseed, b, lda )
750  \$ b( iadd, iadd ) = rmagn( kbmagn( jtype ) )
751 *
752  IF( kclass( jtype ).EQ.2 .AND. n.GT.0 ) THEN
753 *
754 * Include rotations
755 *
756 * Generate U, V as Householder transformations times
757 * a diagonal matrix.
758 *
759  DO 50 jc = 1, n - 1
760  DO 40 jr = jc, n
761  u( jr, jc ) = dlarnd( 3, iseed )
762  v( jr, jc ) = dlarnd( 3, iseed )
763  40 CONTINUE
764  CALL dlarfg( n+1-jc, u( jc, jc ), u( jc+1, jc ), 1,
765  \$ work( jc ) )
766  work( 2*n+jc ) = sign( one, u( jc, jc ) )
767  u( jc, jc ) = one
768  CALL dlarfg( n+1-jc, v( jc, jc ), v( jc+1, jc ), 1,
769  \$ work( n+jc ) )
770  work( 3*n+jc ) = sign( one, v( jc, jc ) )
771  v( jc, jc ) = one
772  50 CONTINUE
773  u( n, n ) = one
774  work( n ) = zero
775  work( 3*n ) = sign( one, dlarnd( 2, iseed ) )
776  v( n, n ) = one
777  work( 2*n ) = zero
778  work( 4*n ) = sign( one, dlarnd( 2, iseed ) )
779 *
780 * Apply the diagonal matrices
781 *
782  DO 70 jc = 1, n
783  DO 60 jr = 1, n
784  a( jr, jc ) = work( 2*n+jr )*work( 3*n+jc )*
785  \$ a( jr, jc )
786  b( jr, jc ) = work( 2*n+jr )*work( 3*n+jc )*
787  \$ b( jr, jc )
788  60 CONTINUE
789  70 CONTINUE
790  CALL dorm2r( 'L', 'N', n, n, n-1, u, ldu, work, a,
791  \$ lda, work( 2*n+1 ), iinfo )
792  IF( iinfo.NE.0 )
793  \$ GO TO 100
794  CALL dorm2r( 'R', 'T', n, n, n-1, v, ldu, work( n+1 ),
795  \$ a, lda, work( 2*n+1 ), iinfo )
796  IF( iinfo.NE.0 )
797  \$ GO TO 100
798  CALL dorm2r( 'L', 'N', n, n, n-1, u, ldu, work, b,
799  \$ lda, work( 2*n+1 ), iinfo )
800  IF( iinfo.NE.0 )
801  \$ GO TO 100
802  CALL dorm2r( 'R', 'T', n, n, n-1, v, ldu, work( n+1 ),
803  \$ b, lda, work( 2*n+1 ), iinfo )
804  IF( iinfo.NE.0 )
805  \$ GO TO 100
806  END IF
807  ELSE
808 *
809 * Random matrices
810 *
811  DO 90 jc = 1, n
812  DO 80 jr = 1, n
813  a( jr, jc ) = rmagn( kamagn( jtype ) )*
814  \$ dlarnd( 2, iseed )
815  b( jr, jc ) = rmagn( kbmagn( jtype ) )*
816  \$ dlarnd( 2, iseed )
817  80 CONTINUE
818  90 CONTINUE
819  END IF
820 *
821  anorm = dlange( '1', n, n, a, lda, work )
822  bnorm = dlange( '1', n, n, b, lda, work )
823 *
824  100 CONTINUE
825 *
826  IF( iinfo.NE.0 ) THEN
827  WRITE( nounit, fmt = 9999 )'Generator', iinfo, n, jtype,
828  \$ ioldsd
829  info = abs( iinfo )
830  RETURN
831  END IF
832 *
833  110 CONTINUE
834 *
835 * Call DGEQR2, DORM2R, and DGGHRD to compute H, T, U, and V
836 *
837  CALL dlacpy( ' ', n, n, a, lda, h, lda )
838  CALL dlacpy( ' ', n, n, b, lda, t, lda )
839  ntest = 1
840  result( 1 ) = ulpinv
841 *
842  CALL dgeqr2( n, n, t, lda, work, work( n+1 ), iinfo )
843  IF( iinfo.NE.0 ) THEN
844  WRITE( nounit, fmt = 9999 )'DGEQR2', iinfo, n, jtype,
845  \$ ioldsd
846  info = abs( iinfo )
847  GO TO 210
848  END IF
849 *
850  CALL dorm2r( 'L', 'T', n, n, n, t, lda, work, h, lda,
851  \$ work( n+1 ), iinfo )
852  IF( iinfo.NE.0 ) THEN
853  WRITE( nounit, fmt = 9999 )'DORM2R', iinfo, n, jtype,
854  \$ ioldsd
855  info = abs( iinfo )
856  GO TO 210
857  END IF
858 *
859  CALL dlaset( 'Full', n, n, zero, one, u, ldu )
860  CALL dorm2r( 'R', 'N', n, n, n, t, lda, work, u, ldu,
861  \$ work( n+1 ), iinfo )
862  IF( iinfo.NE.0 ) THEN
863  WRITE( nounit, fmt = 9999 )'DORM2R', iinfo, n, jtype,
864  \$ ioldsd
865  info = abs( iinfo )
866  GO TO 210
867  END IF
868 *
869  CALL dgghrd( 'V', 'I', n, 1, n, h, lda, t, lda, u, ldu, v,
870  \$ ldu, iinfo )
871  IF( iinfo.NE.0 ) THEN
872  WRITE( nounit, fmt = 9999 )'DGGHRD', iinfo, n, jtype,
873  \$ ioldsd
874  info = abs( iinfo )
875  GO TO 210
876  END IF
877  ntest = 4
878 *
879 * Do tests 1--4
880 *
881  CALL dget51( 1, n, a, lda, h, lda, u, ldu, v, ldu, work,
882  \$ result( 1 ) )
883  CALL dget51( 1, n, b, lda, t, lda, u, ldu, v, ldu, work,
884  \$ result( 2 ) )
885  CALL dget51( 3, n, b, lda, t, lda, u, ldu, u, ldu, work,
886  \$ result( 3 ) )
887  CALL dget51( 3, n, b, lda, t, lda, v, ldu, v, ldu, work,
888  \$ result( 4 ) )
889 *
890 * Call DHGEQZ to compute S1, P1, S2, P2, Q, and Z, do tests.
891 *
892 * Compute T1 and UZ
893 *
894 * Eigenvalues only
895 *
896  CALL dlacpy( ' ', n, n, h, lda, s2, lda )
897  CALL dlacpy( ' ', n, n, t, lda, p2, lda )
898  ntest = 5
899  result( 5 ) = ulpinv
900 *
901  CALL dhgeqz( 'E', 'N', 'N', n, 1, n, s2, lda, p2, lda,
902  \$ alphr3, alphi3, beta3, q, ldu, z, ldu, work,
903  \$ lwork, iinfo )
904  IF( iinfo.NE.0 ) THEN
905  WRITE( nounit, fmt = 9999 )'DHGEQZ(E)', iinfo, n, jtype,
906  \$ ioldsd
907  info = abs( iinfo )
908  GO TO 210
909  END IF
910 *
911 * Eigenvalues and Full Schur Form
912 *
913  CALL dlacpy( ' ', n, n, h, lda, s2, lda )
914  CALL dlacpy( ' ', n, n, t, lda, p2, lda )
915 *
916  CALL dhgeqz( 'S', 'N', 'N', n, 1, n, s2, lda, p2, lda,
917  \$ alphr1, alphi1, beta1, q, ldu, z, ldu, work,
918  \$ lwork, iinfo )
919  IF( iinfo.NE.0 ) THEN
920  WRITE( nounit, fmt = 9999 )'DHGEQZ(S)', iinfo, n, jtype,
921  \$ ioldsd
922  info = abs( iinfo )
923  GO TO 210
924  END IF
925 *
926 * Eigenvalues, Schur Form, and Schur Vectors
927 *
928  CALL dlacpy( ' ', n, n, h, lda, s1, lda )
929  CALL dlacpy( ' ', n, n, t, lda, p1, lda )
930 *
931  CALL dhgeqz( 'S', 'I', 'I', n, 1, n, s1, lda, p1, lda,
932  \$ alphr1, alphi1, beta1, q, ldu, z, ldu, work,
933  \$ lwork, iinfo )
934  IF( iinfo.NE.0 ) THEN
935  WRITE( nounit, fmt = 9999 )'DHGEQZ(V)', iinfo, n, jtype,
936  \$ ioldsd
937  info = abs( iinfo )
938  GO TO 210
939  END IF
940 *
941  ntest = 8
942 *
943 * Do Tests 5--8
944 *
945  CALL dget51( 1, n, h, lda, s1, lda, q, ldu, z, ldu, work,
946  \$ result( 5 ) )
947  CALL dget51( 1, n, t, lda, p1, lda, q, ldu, z, ldu, work,
948  \$ result( 6 ) )
949  CALL dget51( 3, n, t, lda, p1, lda, q, ldu, q, ldu, work,
950  \$ result( 7 ) )
951  CALL dget51( 3, n, t, lda, p1, lda, z, ldu, z, ldu, work,
952  \$ result( 8 ) )
953 *
954 * Compute the Left and Right Eigenvectors of (S1,P1)
955 *
956 * 9: Compute the left eigenvector Matrix without
957 * back transforming:
958 *
959  ntest = 9
960  result( 9 ) = ulpinv
961 *
962 * To test "SELECT" option, compute half of the eigenvectors
963 * in one call, and half in another
964 *
965  i1 = n / 2
966  DO 120 j = 1, i1
967  llwork( j ) = .true.
968  120 CONTINUE
969  DO 130 j = i1 + 1, n
970  llwork( j ) = .false.
971  130 CONTINUE
972 *
973  CALL dtgevc( 'L', 'S', llwork, n, s1, lda, p1, lda, evectl,
974  \$ ldu, dumma, ldu, n, in, work, iinfo )
975  IF( iinfo.NE.0 ) THEN
976  WRITE( nounit, fmt = 9999 )'DTGEVC(L,S1)', iinfo, n,
977  \$ jtype, ioldsd
978  info = abs( iinfo )
979  GO TO 210
980  END IF
981 *
982  i1 = in
983  DO 140 j = 1, i1
984  llwork( j ) = .false.
985  140 CONTINUE
986  DO 150 j = i1 + 1, n
987  llwork( j ) = .true.
988  150 CONTINUE
989 *
990  CALL dtgevc( 'L', 'S', llwork, n, s1, lda, p1, lda,
991  \$ evectl( 1, i1+1 ), ldu, dumma, ldu, n, in,
992  \$ work, iinfo )
993  IF( iinfo.NE.0 ) THEN
994  WRITE( nounit, fmt = 9999 )'DTGEVC(L,S2)', iinfo, n,
995  \$ jtype, ioldsd
996  info = abs( iinfo )
997  GO TO 210
998  END IF
999 *
1000  CALL dget52( .true., n, s1, lda, p1, lda, evectl, ldu,
1001  \$ alphr1, alphi1, beta1, work, dumma( 1 ) )
1002  result( 9 ) = dumma( 1 )
1003  IF( dumma( 2 ).GT.thrshn ) THEN
1004  WRITE( nounit, fmt = 9998 )'Left', 'DTGEVC(HOWMNY=S)',
1005  \$ dumma( 2 ), n, jtype, ioldsd
1006  END IF
1007 *
1008 * 10: Compute the left eigenvector Matrix with
1009 * back transforming:
1010 *
1011  ntest = 10
1012  result( 10 ) = ulpinv
1013  CALL dlacpy( 'F', n, n, q, ldu, evectl, ldu )
1014  CALL dtgevc( 'L', 'B', llwork, n, s1, lda, p1, lda, evectl,
1015  \$ ldu, dumma, ldu, n, in, work, iinfo )
1016  IF( iinfo.NE.0 ) THEN
1017  WRITE( nounit, fmt = 9999 )'DTGEVC(L,B)', iinfo, n,
1018  \$ jtype, ioldsd
1019  info = abs( iinfo )
1020  GO TO 210
1021  END IF
1022 *
1023  CALL dget52( .true., n, h, lda, t, lda, evectl, ldu, alphr1,
1024  \$ alphi1, beta1, work, dumma( 1 ) )
1025  result( 10 ) = dumma( 1 )
1026  IF( dumma( 2 ).GT.thrshn ) THEN
1027  WRITE( nounit, fmt = 9998 )'Left', 'DTGEVC(HOWMNY=B)',
1028  \$ dumma( 2 ), n, jtype, ioldsd
1029  END IF
1030 *
1031 * 11: Compute the right eigenvector Matrix without
1032 * back transforming:
1033 *
1034  ntest = 11
1035  result( 11 ) = ulpinv
1036 *
1037 * To test "SELECT" option, compute half of the eigenvectors
1038 * in one call, and half in another
1039 *
1040  i1 = n / 2
1041  DO 160 j = 1, i1
1042  llwork( j ) = .true.
1043  160 CONTINUE
1044  DO 170 j = i1 + 1, n
1045  llwork( j ) = .false.
1046  170 CONTINUE
1047 *
1048  CALL dtgevc( 'R', 'S', llwork, n, s1, lda, p1, lda, dumma,
1049  \$ ldu, evectr, ldu, n, in, work, iinfo )
1050  IF( iinfo.NE.0 ) THEN
1051  WRITE( nounit, fmt = 9999 )'DTGEVC(R,S1)', iinfo, n,
1052  \$ jtype, ioldsd
1053  info = abs( iinfo )
1054  GO TO 210
1055  END IF
1056 *
1057  i1 = in
1058  DO 180 j = 1, i1
1059  llwork( j ) = .false.
1060  180 CONTINUE
1061  DO 190 j = i1 + 1, n
1062  llwork( j ) = .true.
1063  190 CONTINUE
1064 *
1065  CALL dtgevc( 'R', 'S', llwork, n, s1, lda, p1, lda, dumma,
1066  \$ ldu, evectr( 1, i1+1 ), ldu, n, in, work,
1067  \$ iinfo )
1068  IF( iinfo.NE.0 ) THEN
1069  WRITE( nounit, fmt = 9999 )'DTGEVC(R,S2)', iinfo, n,
1070  \$ jtype, ioldsd
1071  info = abs( iinfo )
1072  GO TO 210
1073  END IF
1074 *
1075  CALL dget52( .false., n, s1, lda, p1, lda, evectr, ldu,
1076  \$ alphr1, alphi1, beta1, work, dumma( 1 ) )
1077  result( 11 ) = dumma( 1 )
1078  IF( dumma( 2 ).GT.thresh ) THEN
1079  WRITE( nounit, fmt = 9998 )'Right', 'DTGEVC(HOWMNY=S)',
1080  \$ dumma( 2 ), n, jtype, ioldsd
1081  END IF
1082 *
1083 * 12: Compute the right eigenvector Matrix with
1084 * back transforming:
1085 *
1086  ntest = 12
1087  result( 12 ) = ulpinv
1088  CALL dlacpy( 'F', n, n, z, ldu, evectr, ldu )
1089  CALL dtgevc( 'R', 'B', llwork, n, s1, lda, p1, lda, dumma,
1090  \$ ldu, evectr, ldu, n, in, work, iinfo )
1091  IF( iinfo.NE.0 ) THEN
1092  WRITE( nounit, fmt = 9999 )'DTGEVC(R,B)', iinfo, n,
1093  \$ jtype, ioldsd
1094  info = abs( iinfo )
1095  GO TO 210
1096  END IF
1097 *
1098  CALL dget52( .false., n, h, lda, t, lda, evectr, ldu,
1099  \$ alphr1, alphi1, beta1, work, dumma( 1 ) )
1100  result( 12 ) = dumma( 1 )
1101  IF( dumma( 2 ).GT.thresh ) THEN
1102  WRITE( nounit, fmt = 9998 )'Right', 'DTGEVC(HOWMNY=B)',
1103  \$ dumma( 2 ), n, jtype, ioldsd
1104  END IF
1105 *
1106 * Tests 13--15 are done only on request
1107 *
1108  IF( tstdif ) THEN
1109 *
1110 * Do Tests 13--14
1111 *
1112  CALL dget51( 2, n, s1, lda, s2, lda, q, ldu, z, ldu,
1113  \$ work, result( 13 ) )
1114  CALL dget51( 2, n, p1, lda, p2, lda, q, ldu, z, ldu,
1115  \$ work, result( 14 ) )
1116 *
1117 * Do Test 15
1118 *
1119  temp1 = zero
1120  temp2 = zero
1121  DO 200 j = 1, n
1122  temp1 = max( temp1, abs( alphr1( j )-alphr3( j ) )+
1123  \$ abs( alphi1( j )-alphi3( j ) ) )
1124  temp2 = max( temp2, abs( beta1( j )-beta3( j ) ) )
1125  200 CONTINUE
1126 *
1127  temp1 = temp1 / max( safmin, ulp*max( temp1, anorm ) )
1128  temp2 = temp2 / max( safmin, ulp*max( temp2, bnorm ) )
1129  result( 15 ) = max( temp1, temp2 )
1130  ntest = 15
1131  ELSE
1132  result( 13 ) = zero
1133  result( 14 ) = zero
1134  result( 15 ) = zero
1135  ntest = 12
1136  END IF
1137 *
1138 * End of Loop -- Check for RESULT(j) > THRESH
1139 *
1140  210 CONTINUE
1141 *
1142  ntestt = ntestt + ntest
1143 *
1144 * Print out tests which fail.
1145 *
1146  DO 220 jr = 1, ntest
1147  IF( result( jr ).GE.thresh ) THEN
1148 *
1149 * If this is the first test to fail,
1150 * print a header to the data file.
1151 *
1152  IF( nerrs.EQ.0 ) THEN
1153  WRITE( nounit, fmt = 9997 )'DGG'
1154 *
1155 * Matrix types
1156 *
1157  WRITE( nounit, fmt = 9996 )
1158  WRITE( nounit, fmt = 9995 )
1159  WRITE( nounit, fmt = 9994 )'Orthogonal'
1160 *
1161 * Tests performed
1162 *
1163  WRITE( nounit, fmt = 9993 )'orthogonal', '''',
1164  \$ 'transpose', ( '''', j = 1, 10 )
1165 *
1166  END IF
1167  nerrs = nerrs + 1
1168  IF( result( jr ).LT.10000.0d0 ) THEN
1169  WRITE( nounit, fmt = 9992 )n, jtype, ioldsd, jr,
1170  \$ result( jr )
1171  ELSE
1172  WRITE( nounit, fmt = 9991 )n, jtype, ioldsd, jr,
1173  \$ result( jr )
1174  END IF
1175  END IF
1176  220 CONTINUE
1177 *
1178  230 CONTINUE
1179  240 CONTINUE
1180 *
1181 * Summary
1182 *
1183  CALL dlasum( 'DGG', nounit, nerrs, ntestt )
1184  RETURN
1185 *
1186  9999 FORMAT( ' DCHKGG: ', a, ' returned INFO=', i6, '.', / 9x, 'N=',
1187  \$ i6, ', JTYPE=', i6, ', ISEED=(', 3( i5, ',' ), i5, ')' )
1188 *
1189  9998 FORMAT( ' DCHKGG: ', a, ' Eigenvectors from ', a, ' incorrectly ',
1190  \$ 'normalized.', / ' Bits of error=', 0p, g10.3, ',', 9x,
1191  \$ 'N=', i6, ', JTYPE=', i6, ', ISEED=(', 3( i5, ',' ), i5,
1192  \$ ')' )
1193 *
1194  9997 FORMAT( / 1x, a3, ' -- Real Generalized eigenvalue problem' )
1195 *
1196  9996 FORMAT( ' Matrix types (see DCHKGG for details): ' )
1197 *
1198  9995 FORMAT( ' Special Matrices:', 23x,
1199  \$ '(J''=transposed Jordan block)',
1200  \$ / ' 1=(0,0) 2=(I,0) 3=(0,I) 4=(I,I) 5=(J'',J'') ',
1201  \$ '6=(diag(J'',I), diag(I,J''))', / ' Diagonal Matrices: ( ',
1202  \$ 'D=diag(0,1,2,...) )', / ' 7=(D,I) 9=(large*D, small*I',
1203  \$ ') 11=(large*I, small*D) 13=(large*D, large*I)', /
1204  \$ ' 8=(I,D) 10=(small*D, large*I) 12=(small*I, large*D) ',
1205  \$ ' 14=(small*D, small*I)', / ' 15=(D, reversed D)' )
1206  9994 FORMAT( ' Matrices Rotated by Random ', a, ' Matrices U, V:',
1207  \$ / ' 16=Transposed Jordan Blocks 19=geometric ',
1208  \$ 'alpha, beta=0,1', / ' 17=arithm. alpha&beta ',
1209  \$ ' 20=arithmetic alpha, beta=0,1', / ' 18=clustered ',
1210  \$ 'alpha, beta=0,1 21=random alpha, beta=0,1',
1211  \$ / ' Large & Small Matrices:', / ' 22=(large, small) ',
1212  \$ '23=(small,large) 24=(small,small) 25=(large,large)',
1213  \$ / ' 26=random O(1) matrices.' )
1214 *
1215  9993 FORMAT( / ' Tests performed: (H is Hessenberg, S is Schur, B, ',
1216  \$ 'T, P are triangular,', / 20x, 'U, V, Q, and Z are ', a,
1217  \$ ', l and r are the', / 20x,
1218  \$ 'appropriate left and right eigenvectors, resp., a is',
1219  \$ / 20x, 'alpha, b is beta, and ', a, ' means ', a, '.)',
1220  \$ / ' 1 = | A - U H V', a,
1221  \$ ' | / ( |A| n ulp ) 2 = | B - U T V', a,
1222  \$ ' | / ( |B| n ulp )', / ' 3 = | I - UU', a,
1223  \$ ' | / ( n ulp ) 4 = | I - VV', a,
1224  \$ ' | / ( n ulp )', / ' 5 = | H - Q S Z', a,
1225  \$ ' | / ( |H| n ulp )', 6x, '6 = | T - Q P Z', a,
1226  \$ ' | / ( |T| n ulp )', / ' 7 = | I - QQ', a,
1227  \$ ' | / ( n ulp ) 8 = | I - ZZ', a,
1228  \$ ' | / ( n ulp )', / ' 9 = max | ( b S - a P )', a,
1229  \$ ' l | / const. 10 = max | ( b H - a T )', a,
1230  \$ ' l | / const.', /
1231  \$ ' 11= max | ( b S - a P ) r | / const. 12 = max | ( b H',
1232  \$ ' - a T ) r | / const.', / 1x )
1233 *
1234  9992 FORMAT( ' Matrix order=', i5, ', type=', i2, ', seed=',
1235  \$ 4( i4, ',' ), ' result ', i2, ' is', 0p, f8.2 )
1236  9991 FORMAT( ' Matrix order=', i5, ', type=', i2, ', seed=',
1237  \$ 4( i4, ',' ), ' result ', i2, ' is', 1p, d10.3 )
1238 *
1239 * End of DCHKGG
1240 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
subroutine dlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: dlaset.f:110
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dlatm4(ITYPE, N, NZ1, NZ2, ISIGN, AMAGN, RCOND, TRIANG, IDIST, ISEED, A, LDA)
DLATM4
Definition: dlatm4.f:175
subroutine dlasum(TYPE, IOUNIT, IE, NRUN)
DLASUM
Definition: dlasum.f:43
subroutine dget52(LEFT, N, A, LDA, B, LDB, E, LDE, ALPHAR, ALPHAI, BETA, WORK, RESULT)
DGET52
Definition: dget52.f:199
subroutine dget51(ITYPE, N, A, LDA, B, LDB, U, LDU, V, LDV, WORK, RESULT)
DGET51
Definition: dget51.f:149
double precision function dlarnd(IDIST, ISEED)
DLARND
Definition: dlarnd.f:73
double precision function dlange(NORM, M, N, A, LDA, WORK)
DLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: dlange.f:114
subroutine dhgeqz(JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
DHGEQZ
Definition: dhgeqz.f:304
subroutine dtgevc(SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL, LDVL, VR, LDVR, MM, M, WORK, INFO)
DTGEVC
Definition: dtgevc.f:295
subroutine dgeqr2(M, N, A, LDA, TAU, WORK, INFO)
DGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm.
Definition: dgeqr2.f:130
subroutine dlarfg(N, ALPHA, X, INCX, TAU)
DLARFG generates an elementary reflector (Householder matrix).
Definition: dlarfg.f:106
subroutine dgghrd(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
DGGHRD
Definition: dgghrd.f:207
subroutine dorm2r(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
DORM2R multiplies a general matrix by the orthogonal matrix from a QR factorization determined by sge...
Definition: dorm2r.f:159
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