*> \brief \b SDRGES * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * * Definition: * =========== * * SUBROUTINE SDRGES( NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, * NOUNIT, A, LDA, B, S, T, Q, LDQ, Z, ALPHAR, * ALPHAI, BETA, WORK, LWORK, RESULT, BWORK, * INFO ) * * .. Scalar Arguments .. * INTEGER INFO, LDA, LDQ, LWORK, NOUNIT, NSIZES, NTYPES * REAL THRESH * .. * .. Array Arguments .. * LOGICAL BWORK( * ), DOTYPE( * ) * INTEGER ISEED( 4 ), NN( * ) * REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), * $ B( LDA, * ), BETA( * ), Q( LDQ, * ), * $ RESULT( 13 ), S( LDA, * ), T( LDA, * ), * $ WORK( * ), Z( LDQ, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SDRGES checks the nonsymmetric generalized eigenvalue (Schur form) *> problem driver SGGES. *> *> SGGES 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 SDRGES 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 threshold 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. quasi-triangular 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 j-th and j+1-th 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. quasi-triangular 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 j-th and j+1-th 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=(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) 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, ..., N-3, 0 ) and diag(T2) = *> ( 0, N-3, N-4,..., 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, 1-d, ..., 1-(N-5)*d=s, 0 ) *> diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 ) *> *> N-5 *> (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(N-4), 0 ) *> diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) *> where r1,..., r(N-4) are random. *> *> (22) Q ( big*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) *> diag(T2) = ( 0, 1, ..., 1, 0, 0 ) *> *> (23) Q ( small*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) *> diag(T2) = ( 0, 1, ..., 1, 0, 0 ) *> *> (24) Q ( small*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) *> diag(T2) = ( 0, 1, ..., 1, 0, 0 ) *> *> (25) Q ( big*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) *> diag(T2) = ( 0, 1, ..., 1, 0, 0 ) *> *> (26) Q ( T1, T2 ) Z where T1 and T2 are random upper-triangular *> matrices. *> *> \endverbatim * * Arguments: * ========== * *> \param[in] NSIZES *> \verbatim *> NSIZES is INTEGER *> The number of sizes of matrices to use. If it is zero, *> SDRGES does nothing. NSIZES >= 0. *> \endverbatim *> *> \param[in] NN *> \verbatim *> NN is INTEGER array, dimension (NSIZES) *> An array containing the sizes to be used for the matrices. *> Zero values will be skipped. NN >= 0. *> \endverbatim *> *> \param[in] NTYPES *> \verbatim *> NTYPES is INTEGER *> The number of elements in DOTYPE. If it is zero, SDRGES *> 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. . *> \endverbatim *> *> \param[in] DOTYPE *> \verbatim *> 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. *> \endverbatim *> *> \param[in,out] ISEED *> \verbatim *> 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 SDRGES to continue the same random number *> sequence. *> \endverbatim *> *> \param[in] THRESH *> \verbatim *> THRESH is REAL *> 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. *> \endverbatim *> *> \param[in] NOUNIT *> \verbatim *> NOUNIT is INTEGER *> The FORTRAN unit number for printing out error messages *> (e.g., if a routine returns IINFO not equal to 0.) *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is REAL 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. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of A, B, S, and T. *> It must be at least 1 and at least max( NN ). *> \endverbatim *> *> \param[in,out] B *> \verbatim *> B is REAL 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. *> \endverbatim *> *> \param[out] S *> \verbatim *> S is REAL array, dimension (LDA, max(NN)) *> The Schur form matrix computed from A by SGGES. On exit, S *> contains the Schur form matrix corresponding to the matrix *> in A. *> \endverbatim *> *> \param[out] T *> \verbatim *> T is REAL array, dimension (LDA, max(NN)) *> The upper triangular matrix computed from B by SGGES. *> \endverbatim *> *> \param[out] Q *> \verbatim *> Q is REAL array, dimension (LDQ, max(NN)) *> The (left) orthogonal matrix computed by SGGES. *> \endverbatim *> *> \param[in] LDQ *> \verbatim *> LDQ is INTEGER *> The leading dimension of Q and Z. It must *> be at least 1 and at least max( NN ). *> \endverbatim *> *> \param[out] Z *> \verbatim *> Z is REAL array, dimension( LDQ, max(NN) ) *> The (right) orthogonal matrix computed by SGGES. *> \endverbatim *> *> \param[out] ALPHAR *> \verbatim *> ALPHAR is REAL array, dimension (max(NN)) *> \endverbatim *> *> \param[out] ALPHAI *> \verbatim *> ALPHAI is REAL array, dimension (max(NN)) *> \endverbatim *> *> \param[out] BETA *> \verbatim *> BETA is REAL array, dimension (max(NN)) *> *> The generalized eigenvalues of (A,B) computed by SGGES. *> ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th *> generalized eigenvalue of A and B. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is REAL array, dimension (LWORK) *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> The dimension of the array WORK. *> LWORK >= MAX( 10*(N+1), 3*N*N ), where N is the largest *> matrix dimension. *> \endverbatim *> *> \param[out] RESULT *> \verbatim *> RESULT is REAL array, dimension (15) *> The values computed by the tests described above. *> The values are currently limited to 1/ulp, to avoid overflow. *> \endverbatim *> *> \param[out] BWORK *> \verbatim *> BWORK is LOGICAL array, dimension (N) *> \endverbatim *> *> \param[out] INFO *> \verbatim *> 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. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date June 2016 * *> \ingroup single_eig * * ===================================================================== SUBROUTINE SDRGES( NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, $ NOUNIT, A, LDA, B, S, T, Q, LDQ, Z, ALPHAR, $ ALPHAI, BETA, WORK, LWORK, RESULT, BWORK, $ INFO ) * * -- LAPACK test routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * June 2016 * * .. Scalar Arguments .. INTEGER INFO, LDA, LDQ, LWORK, NOUNIT, NSIZES, NTYPES REAL THRESH * .. * .. Array Arguments .. LOGICAL BWORK( * ), DOTYPE( * ) INTEGER ISEED( 4 ), NN( * ) REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), $ B( LDA, * ), BETA( * ), Q( LDQ, * ), $ RESULT( 13 ), S( LDA, * ), T( LDA, * ), $ WORK( * ), Z( LDQ, * ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) INTEGER MAXTYP PARAMETER ( MAXTYP = 26 ) * .. * .. Local Scalars .. LOGICAL BADNN, ILABAD CHARACTER SORT INTEGER I, I1, IADD, IERR, IINFO, IN, ISORT, J, JC, JR, $ JSIZE, JTYPE, KNTEIG, MAXWRK, MINWRK, MTYPES, $ N, N1, NB, NERRS, NMATS, NMAX, NTEST, NTESTT, $ RSUB, SDIM REAL SAFMAX, SAFMIN, TEMP1, TEMP2, ULP, ULPINV * .. * .. Local Arrays .. INTEGER IASIGN( MAXTYP ), IBSIGN( MAXTYP ), $ IOLDSD( 4 ), KADD( 6 ), KAMAGN( MAXTYP ), $ KATYPE( MAXTYP ), KAZERO( MAXTYP ), $ KBMAGN( MAXTYP ), KBTYPE( MAXTYP ), $ KBZERO( MAXTYP ), KCLASS( MAXTYP ), $ KTRIAN( MAXTYP ), KZ1( 6 ), KZ2( 6 ) REAL RMAGN( 0: 3 ) * .. * .. External Functions .. LOGICAL SLCTES INTEGER ILAENV REAL SLAMCH, SLARND EXTERNAL SLCTES, ILAENV, SLAMCH, SLARND * .. * .. External Subroutines .. EXTERNAL ALASVM, SGET51, SGET53, SGET54, SGGES, SLABAD, $ SLACPY, SLARFG, SLASET, SLATM4, SORM2R, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, REAL, SIGN * .. * .. Data statements .. DATA KCLASS / 15*1, 10*2, 1*3 / DATA KZ1 / 0, 1, 2, 1, 3, 3 / DATA KZ2 / 0, 0, 1, 2, 1, 1 / DATA KADD / 0, 0, 0, 0, 3, 2 / DATA KATYPE / 0, 1, 0, 1, 2, 3, 4, 1, 4, 4, 1, 1, 4, $ 4, 4, 2, 4, 5, 8, 7, 9, 4*4, 0 / DATA KBTYPE / 0, 0, 1, 1, 2, -3, 1, 4, 1, 1, 4, 4, $ 1, 1, -4, 2, -4, 8*8, 0 / DATA KAZERO / 6*1, 2, 1, 2*2, 2*1, 2*2, 3, 1, 3, $ 4*5, 4*3, 1 / DATA KBZERO / 6*1, 1, 2, 2*1, 2*2, 2*1, 4, 1, 4, $ 4*6, 4*4, 1 / DATA KAMAGN / 8*1, 2, 3, 2, 3, 2, 3, 7*1, 2, 3, 3, $ 2, 1 / DATA KBMAGN / 8*1, 3, 2, 3, 2, 2, 3, 7*1, 3, 2, 3, $ 2, 1 / DATA KTRIAN / 16*0, 10*1 / DATA IASIGN / 6*0, 2, 0, 2*2, 2*0, 3*2, 0, 2, 3*0, $ 5*2, 0 / DATA IBSIGN / 7*0, 2, 2*0, 2*2, 2*0, 2, 0, 2, 9*0 / * .. * .. Executable Statements .. * * Check for errors * INFO = 0 * BADNN = .FALSE. NMAX = 1 DO 10 J = 1, NSIZES NMAX = MAX( NMAX, NN( J ) ) IF( NN( J ).LT.0 ) $ BADNN = .TRUE. 10 CONTINUE * IF( NSIZES.LT.0 ) THEN INFO = -1 ELSE IF( BADNN ) THEN INFO = -2 ELSE IF( NTYPES.LT.0 ) THEN INFO = -3 ELSE IF( THRESH.LT.ZERO ) THEN INFO = -6 ELSE IF( LDA.LE.1 .OR. LDA.LT.NMAX ) THEN INFO = -9 ELSE IF( LDQ.LE.1 .OR. LDQ.LT.NMAX ) THEN INFO = -14 END IF * * Compute workspace * (Note: Comments in the code beginning "Workspace:" describe the * minimal amount of workspace needed at that point in the code, * as well as the preferred amount for good performance. * NB refers to the optimal block size for the immediately * following subroutine, as returned by ILAENV. * MINWRK = 1 IF( INFO.EQ.0 .AND. LWORK.GE.1 ) THEN MINWRK = MAX( 10*( NMAX+1 ), 3*NMAX*NMAX ) NB = MAX( 1, ILAENV( 1, 'SGEQRF', ' ', NMAX, NMAX, -1, -1 ), $ ILAENV( 1, 'SORMQR', 'LT', NMAX, NMAX, NMAX, -1 ), $ ILAENV( 1, 'SORGQR', ' ', NMAX, NMAX, NMAX, -1 ) ) MAXWRK = MAX( 10*( NMAX+1 ), 2*NMAX+NMAX*NB, 3*NMAX*NMAX ) WORK( 1 ) = MAXWRK END IF * IF( LWORK.LT.MINWRK ) $ INFO = -20 * IF( INFO.NE.0 ) THEN CALL XERBLA( 'SDRGES', -INFO ) RETURN END IF * * Quick return if possible * IF( NSIZES.EQ.0 .OR. NTYPES.EQ.0 ) $ RETURN * SAFMIN = SLAMCH( 'Safe minimum' ) ULP = SLAMCH( 'Epsilon' )*SLAMCH( 'Base' ) SAFMIN = SAFMIN / ULP SAFMAX = ONE / SAFMIN CALL SLABAD( SAFMIN, SAFMAX ) ULPINV = ONE / ULP * * The values RMAGN(2:3) depend on N, see below. * RMAGN( 0 ) = ZERO RMAGN( 1 ) = ONE * * Loop over matrix sizes * NTESTT = 0 NERRS = 0 NMATS = 0 * DO 190 JSIZE = 1, NSIZES N = NN( JSIZE ) N1 = MAX( 1, N ) RMAGN( 2 ) = SAFMAX*ULP / REAL( N1 ) RMAGN( 3 ) = SAFMIN*ULPINV*REAL( N1 ) * IF( NSIZES.NE.1 ) THEN MTYPES = MIN( MAXTYP, NTYPES ) ELSE MTYPES = MIN( MAXTYP+1, NTYPES ) END IF * * Loop over matrix types * DO 180 JTYPE = 1, MTYPES IF( .NOT.DOTYPE( JTYPE ) ) $ GO TO 180 NMATS = NMATS + 1 NTEST = 0 * * Save ISEED in case of an error. * DO 20 J = 1, 4 IOLDSD( J ) = ISEED( J ) 20 CONTINUE * * Initialize RESULT * DO 30 J = 1, 13 RESULT( J ) = ZERO 30 CONTINUE * * Generate test matrices A and B * * Description of control parameters: * * KCLASS: =1 means w/o rotation, =2 means w/ rotation, * =3 means random. * KATYPE: the "type" to be passed to SLATM4 for computing A. * KAZERO: the pattern of zeros on the diagonal for A: * =1: ( xxx ), =2: (0, xxx ) =3: ( 0, 0, xxx, 0 ), * =4: ( 0, xxx, 0, 0 ), =5: ( 0, 0, 1, xxx, 0 ), * =6: ( 0, 1, 0, xxx, 0 ). (xxx means a string of * non-zero entries.) * KAMAGN: the magnitude of the matrix: =0: zero, =1: O(1), * =2: large, =3: small. * IASIGN: 1 if the diagonal elements of A are to be * multiplied by a random magnitude 1 number, =2 if * randomly chosen diagonal blocks are to be rotated * to form 2x2 blocks. * KBTYPE, KBZERO, KBMAGN, IBSIGN: the same, but for B. * KTRIAN: =0: don't fill in the upper triangle, =1: do. * KZ1, KZ2, KADD: used to implement KAZERO and KBZERO. * RMAGN: used to implement KAMAGN and KBMAGN. * IF( MTYPES.GT.MAXTYP ) $ GO TO 110 IINFO = 0 IF( KCLASS( JTYPE ).LT.3 ) THEN * * Generate A (w/o rotation) * IF( ABS( KATYPE( JTYPE ) ).EQ.3 ) THEN IN = 2*( ( N-1 ) / 2 ) + 1 IF( IN.NE.N ) $ CALL SLASET( 'Full', N, N, ZERO, ZERO, A, LDA ) ELSE IN = N END IF CALL SLATM4( KATYPE( JTYPE ), IN, KZ1( KAZERO( JTYPE ) ), $ KZ2( KAZERO( JTYPE ) ), IASIGN( JTYPE ), $ RMAGN( KAMAGN( JTYPE ) ), ULP, $ RMAGN( KTRIAN( JTYPE )*KAMAGN( JTYPE ) ), 2, $ ISEED, A, LDA ) IADD = KADD( KAZERO( JTYPE ) ) IF( IADD.GT.0 .AND. IADD.LE.N ) $ A( IADD, IADD ) = ONE * * Generate B (w/o rotation) * IF( ABS( KBTYPE( JTYPE ) ).EQ.3 ) THEN IN = 2*( ( N-1 ) / 2 ) + 1 IF( IN.NE.N ) $ CALL SLASET( 'Full', N, N, ZERO, ZERO, B, LDA ) ELSE IN = N END IF CALL SLATM4( KBTYPE( JTYPE ), IN, KZ1( KBZERO( JTYPE ) ), $ KZ2( KBZERO( JTYPE ) ), IBSIGN( JTYPE ), $ RMAGN( KBMAGN( JTYPE ) ), ONE, $ RMAGN( KTRIAN( JTYPE )*KBMAGN( JTYPE ) ), 2, $ ISEED, B, LDA ) IADD = KADD( KBZERO( JTYPE ) ) IF( IADD.NE.0 .AND. IADD.LE.N ) $ B( IADD, IADD ) = ONE * IF( KCLASS( JTYPE ).EQ.2 .AND. N.GT.0 ) THEN * * Include rotations * * Generate Q, Z as Householder transformations times * a diagonal matrix. * DO 50 JC = 1, N - 1 DO 40 JR = JC, N Q( JR, JC ) = SLARND( 3, ISEED ) Z( JR, JC ) = SLARND( 3, ISEED ) 40 CONTINUE CALL SLARFG( N+1-JC, Q( JC, JC ), Q( JC+1, JC ), 1, $ WORK( JC ) ) WORK( 2*N+JC ) = SIGN( ONE, Q( JC, JC ) ) Q( JC, JC ) = ONE CALL SLARFG( N+1-JC, Z( JC, JC ), Z( JC+1, JC ), 1, $ WORK( N+JC ) ) WORK( 3*N+JC ) = SIGN( ONE, Z( JC, JC ) ) Z( JC, JC ) = ONE 50 CONTINUE Q( N, N ) = ONE WORK( N ) = ZERO WORK( 3*N ) = SIGN( ONE, SLARND( 2, ISEED ) ) Z( N, N ) = ONE WORK( 2*N ) = ZERO WORK( 4*N ) = SIGN( ONE, SLARND( 2, ISEED ) ) * * Apply the diagonal matrices * DO 70 JC = 1, N DO 60 JR = 1, N A( JR, JC ) = WORK( 2*N+JR )*WORK( 3*N+JC )* $ A( JR, JC ) B( JR, JC ) = WORK( 2*N+JR )*WORK( 3*N+JC )* $ B( JR, JC ) 60 CONTINUE 70 CONTINUE CALL SORM2R( 'L', 'N', N, N, N-1, Q, LDQ, WORK, A, $ LDA, WORK( 2*N+1 ), IINFO ) IF( IINFO.NE.0 ) $ GO TO 100 CALL SORM2R( 'R', 'T', N, N, N-1, Z, LDQ, WORK( N+1 ), $ A, LDA, WORK( 2*N+1 ), IINFO ) IF( IINFO.NE.0 ) $ GO TO 100 CALL SORM2R( 'L', 'N', N, N, N-1, Q, LDQ, WORK, B, $ LDA, WORK( 2*N+1 ), IINFO ) IF( IINFO.NE.0 ) $ GO TO 100 CALL SORM2R( 'R', 'T', N, N, N-1, Z, LDQ, WORK( N+1 ), $ B, LDA, WORK( 2*N+1 ), IINFO ) IF( IINFO.NE.0 ) $ GO TO 100 END IF ELSE * * Random matrices * DO 90 JC = 1, N DO 80 JR = 1, N A( JR, JC ) = RMAGN( KAMAGN( JTYPE ) )* $ SLARND( 2, ISEED ) B( JR, JC ) = RMAGN( KBMAGN( JTYPE ) )* $ SLARND( 2, ISEED ) 80 CONTINUE 90 CONTINUE END IF * 100 CONTINUE * IF( IINFO.NE.0 ) THEN WRITE( NOUNIT, FMT = 9999 )'Generator', IINFO, N, JTYPE, $ IOLDSD INFO = ABS( IINFO ) RETURN END IF * 110 CONTINUE * DO 120 I = 1, 13 RESULT( I ) = -ONE 120 CONTINUE * * Test with and without sorting of eigenvalues * DO 150 ISORT = 0, 1 IF( ISORT.EQ.0 ) THEN SORT = 'N' RSUB = 0 ELSE SORT = 'S' RSUB = 5 END IF * * Call SGGES to compute H, T, Q, Z, alpha, and beta. * CALL SLACPY( 'Full', N, N, A, LDA, S, LDA ) CALL SLACPY( 'Full', N, N, B, LDA, T, LDA ) NTEST = 1 + RSUB + ISORT RESULT( 1+RSUB+ISORT ) = ULPINV CALL SGGES( 'V', 'V', SORT, SLCTES, N, S, LDA, T, LDA, $ SDIM, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDQ, $ WORK, LWORK, BWORK, IINFO ) IF( IINFO.NE.0 .AND. IINFO.NE.N+2 ) THEN RESULT( 1+RSUB+ISORT ) = ULPINV WRITE( NOUNIT, FMT = 9999 )'SGGES', IINFO, N, JTYPE, $ IOLDSD INFO = ABS( IINFO ) GO TO 160 END IF * NTEST = 4 + RSUB * * Do tests 1--4 (or tests 7--9 when reordering ) * IF( ISORT.EQ.0 ) THEN CALL SGET51( 1, N, A, LDA, S, LDA, Q, LDQ, Z, LDQ, $ WORK, RESULT( 1 ) ) CALL SGET51( 1, N, B, LDA, T, LDA, Q, LDQ, Z, LDQ, $ WORK, RESULT( 2 ) ) ELSE CALL SGET54( N, A, LDA, B, LDA, S, LDA, T, LDA, Q, $ LDQ, Z, LDQ, WORK, RESULT( 7 ) ) END IF CALL SGET51( 3, N, A, LDA, T, LDA, Q, LDQ, Q, LDQ, WORK, $ RESULT( 3+RSUB ) ) CALL SGET51( 3, N, B, LDA, T, LDA, Z, LDQ, Z, LDQ, WORK, $ RESULT( 4+RSUB ) ) * * Do test 5 and 6 (or Tests 10 and 11 when reordering): * check Schur form of A and compare eigenvalues with * diagonals. * NTEST = 6 + RSUB TEMP1 = ZERO * DO 130 J = 1, N ILABAD = .FALSE. IF( ALPHAI( J ).EQ.ZERO ) THEN TEMP2 = ( ABS( ALPHAR( J )-S( J, J ) ) / $ MAX( SAFMIN, ABS( ALPHAR( J ) ), ABS( S( J, $ J ) ) )+ABS( BETA( J )-T( J, J ) ) / $ MAX( SAFMIN, ABS( BETA( J ) ), ABS( T( J, $ J ) ) ) ) / ULP * IF( J.LT.N ) THEN IF( S( J+1, J ).NE.ZERO ) THEN ILABAD = .TRUE. RESULT( 5+RSUB ) = ULPINV END IF END IF IF( J.GT.1 ) THEN IF( S( J, J-1 ).NE.ZERO ) THEN ILABAD = .TRUE. RESULT( 5+RSUB ) = ULPINV END IF END IF * ELSE IF( ALPHAI( J ).GT.ZERO ) THEN I1 = J ELSE I1 = J - 1 END IF IF( I1.LE.0 .OR. I1.GE.N ) THEN ILABAD = .TRUE. ELSE IF( I1.LT.N-1 ) THEN IF( S( I1+2, I1+1 ).NE.ZERO ) THEN ILABAD = .TRUE. RESULT( 5+RSUB ) = ULPINV END IF ELSE IF( I1.GT.1 ) THEN IF( S( I1, I1-1 ).NE.ZERO ) THEN ILABAD = .TRUE. RESULT( 5+RSUB ) = ULPINV END IF END IF IF( .NOT.ILABAD ) THEN CALL SGET53( S( I1, I1 ), LDA, T( I1, I1 ), LDA, $ BETA( J ), ALPHAR( J ), $ ALPHAI( J ), TEMP2, IERR ) IF( IERR.GE.3 ) THEN WRITE( NOUNIT, FMT = 9998 )IERR, J, N, $ JTYPE, IOLDSD INFO = ABS( IERR ) END IF ELSE TEMP2 = ULPINV END IF * END IF TEMP1 = MAX( TEMP1, TEMP2 ) IF( ILABAD ) THEN WRITE( NOUNIT, FMT = 9997 )J, N, JTYPE, IOLDSD END IF 130 CONTINUE RESULT( 6+RSUB ) = TEMP1 * IF( ISORT.GE.1 ) THEN * * Do test 12 * NTEST = 12 RESULT( 12 ) = ZERO KNTEIG = 0 DO 140 I = 1, N IF( SLCTES( ALPHAR( I ), ALPHAI( I ), $ BETA( I ) ) .OR. SLCTES( ALPHAR( I ), $ -ALPHAI( I ), BETA( I ) ) ) THEN KNTEIG = KNTEIG + 1 END IF IF( I.LT.N ) THEN IF( ( SLCTES( ALPHAR( I+1 ), ALPHAI( I+1 ), $ BETA( I+1 ) ) .OR. SLCTES( ALPHAR( I+1 ), $ -ALPHAI( I+1 ), BETA( I+1 ) ) ) .AND. $ ( .NOT.( SLCTES( ALPHAR( I ), ALPHAI( I ), $ BETA( I ) ) .OR. SLCTES( ALPHAR( I ), $ -ALPHAI( I ), BETA( I ) ) ) ) .AND. $ IINFO.NE.N+2 ) THEN RESULT( 12 ) = ULPINV END IF END IF 140 CONTINUE IF( SDIM.NE.KNTEIG ) THEN RESULT( 12 ) = ULPINV END IF END IF * 150 CONTINUE * * End of Loop -- Check for RESULT(j) > THRESH * 160 CONTINUE * NTESTT = NTESTT + NTEST * * Print out tests which fail. * DO 170 JR = 1, NTEST IF( RESULT( JR ).GE.THRESH ) THEN * * If this is the first test to fail, * print a header to the data file. * IF( NERRS.EQ.0 ) THEN WRITE( NOUNIT, FMT = 9996 )'SGS' * * Matrix types * WRITE( NOUNIT, FMT = 9995 ) WRITE( NOUNIT, FMT = 9994 ) WRITE( NOUNIT, FMT = 9993 )'Orthogonal' * * Tests performed * WRITE( NOUNIT, FMT = 9992 )'orthogonal', '''', $ 'transpose', ( '''', J = 1, 8 ) * END IF NERRS = NERRS + 1 IF( RESULT( JR ).LT.10000.0 ) THEN WRITE( NOUNIT, FMT = 9991 )N, JTYPE, IOLDSD, JR, $ RESULT( JR ) ELSE WRITE( NOUNIT, FMT = 9990 )N, JTYPE, IOLDSD, JR, $ RESULT( JR ) END IF END IF 170 CONTINUE * 180 CONTINUE 190 CONTINUE * * Summary * CALL ALASVM( 'SGS', NOUNIT, NERRS, NTESTT, 0 ) * WORK( 1 ) = MAXWRK * RETURN * 9999 FORMAT( ' SDRGES: ', A, ' returned INFO=', I6, '.', / 9X, 'N=', $ I6, ', JTYPE=', I6, ', ISEED=(', 4( I4, ',' ), I5, ')' ) * 9998 FORMAT( ' SDRGES: SGET53 returned INFO=', I1, ' for eigenvalue ', $ I6, '.', / 9X, 'N=', I6, ', JTYPE=', I6, ', ISEED=(', $ 4( I4, ',' ), I5, ')' ) * 9997 FORMAT( ' SDRGES: S not in Schur form at eigenvalue ', I6, '.', $ / 9X, 'N=', I6, ', JTYPE=', I6, ', ISEED=(', 3( I5, ',' ), $ I5, ')' ) * 9996 FORMAT( / 1X, A3, ' -- Real Generalized Schur form driver' ) * 9995 FORMAT( ' Matrix types (see SDRGES for details): ' ) * 9994 FORMAT( ' Special Matrices:', 23X, $ '(J''=transposed Jordan block)', $ / ' 1=(0,0) 2=(I,0) 3=(0,I) 4=(I,I) 5=(J'',J'') ', $ '6=(diag(J'',I), diag(I,J''))', / ' Diagonal Matrices: ( ', $ 'D=diag(0,1,2,...) )', / ' 7=(D,I) 9=(large*D, small*I', $ ') 11=(large*I, small*D) 13=(large*D, large*I)', / $ ' 8=(I,D) 10=(small*D, large*I) 12=(small*I, large*D) ', $ ' 14=(small*D, small*I)', / ' 15=(D, reversed D)' ) 9993 FORMAT( ' Matrices Rotated by Random ', A, ' Matrices U, V:', $ / ' 16=Transposed Jordan Blocks 19=geometric ', $ 'alpha, beta=0,1', / ' 17=arithm. alpha&beta ', $ ' 20=arithmetic alpha, beta=0,1', / ' 18=clustered ', $ 'alpha, beta=0,1 21=random alpha, beta=0,1', $ / ' Large & Small Matrices:', / ' 22=(large, small) ', $ '23=(small,large) 24=(small,small) 25=(large,large)', $ / ' 26=random O(1) matrices.' ) * 9992 FORMAT( / ' Tests performed: (S is Schur, T is triangular, ', $ 'Q and Z are ', A, ',', / 19X, $ 'l and r are the appropriate left and right', / 19X, $ 'eigenvectors, resp., a is alpha, b is beta, and', / 19X, A, $ ' means ', A, '.)', / ' Without ordering: ', $ / ' 1 = | A - Q S Z', A, $ ' | / ( |A| n ulp ) 2 = | B - Q T Z', A, $ ' | / ( |B| n ulp )', / ' 3 = | I - QQ', A, $ ' | / ( n ulp ) 4 = | I - ZZ', A, $ ' | / ( n ulp )', / ' 5 = A is in Schur form S', $ / ' 6 = difference between (alpha,beta)', $ ' and diagonals of (S,T)', / ' With ordering: ', $ / ' 7 = | (A,B) - Q (S,T) Z', A, $ ' | / ( |(A,B)| n ulp ) ', / ' 8 = | I - QQ', A, $ ' | / ( n ulp ) 9 = | I - ZZ', A, $ ' | / ( n ulp )', / ' 10 = A is in Schur form S', $ / ' 11 = difference between (alpha,beta) and diagonals', $ ' of (S,T)', / ' 12 = SDIM is the correct number of ', $ 'selected eigenvalues', / ) 9991 FORMAT( ' Matrix order=', I5, ', type=', I2, ', seed=', $ 4( I4, ',' ), ' result ', I2, ' is', 0P, F8.2 ) 9990 FORMAT( ' Matrix order=', I5, ', type=', I2, ', seed=', $ 4( I4, ',' ), ' result ', I2, ' is', 1P, E10.3 ) * * End of SDRGES * END