 LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine schkgg ( integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical, dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, real THRESH, logical TSTDIF, real THRSHN, integer NOUNIT, real, dimension( lda, * ) A, integer LDA, real, dimension( lda, * ) B, real, dimension( lda, * ) H, real, dimension( lda, * ) T, real, dimension( lda, * ) S1, real, dimension( lda, * ) S2, real, dimension( lda, * ) P1, real, dimension( lda, * ) P2, real, dimension( ldu, * ) U, integer LDU, real, dimension( ldu, * ) V, real, dimension( ldu, * ) Q, real, dimension( ldu, * ) Z, real, dimension( * ) ALPHR1, real, dimension( * ) ALPHI1, real, dimension( * ) BETA1, real, dimension( * ) ALPHR3, real, dimension( * ) ALPHI3, real, dimension( * ) BETA3, real, dimension( ldu, * ) EVECTL, real, dimension( ldu, * ) EVECTR, real, dimension( * ) WORK, integer LWORK, logical, dimension( * ) LLWORK, real, dimension( 15 ) RESULT, integer INFO )

SCHKGG

Purpose:
``` SCHKGG  checks the nonsymmetric generalized eigenvalue problem
routines.
T          T        T
SGGHRD 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
SHGEQZ 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

STGEVC 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 SCHKGG 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
STGEVC 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
STGEVC 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, SCHKGG 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, SCHKGG 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 SCHKGG to continue the same random number sequence.``` [in] THRESH ``` 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. 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 REAL 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 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.``` [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 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.``` [out] H ``` H is REAL array, dimension (LDA, max(NN)) The upper Hessenberg matrix computed from A by SGGHRD.``` [out] T ``` T is REAL array, dimension (LDA, max(NN)) The upper triangular matrix computed from B by SGGHRD.``` [out] S1 ``` S1 is REAL array, dimension (LDA, max(NN)) The Schur (block upper triangular) matrix computed from H by SHGEQZ when Q and Z are also computed.``` [out] S2 ``` S2 is REAL array, dimension (LDA, max(NN)) The Schur (block upper triangular) matrix computed from H by SHGEQZ when Q and Z are not computed.``` [out] P1 ``` P1 is REAL array, dimension (LDA, max(NN)) The upper triangular matrix computed from T by SHGEQZ when Q and Z are also computed.``` [out] P2 ``` P2 is REAL array, dimension (LDA, max(NN)) The upper triangular matrix computed from T by SHGEQZ when Q and Z are not computed.``` [out] U ``` U is REAL array, dimension (LDU, max(NN)) The (left) orthogonal matrix computed by SGGHRD.``` [in] LDU ``` LDU is INTEGER The leading dimension of U, V, Q, Z, EVECTL, and EVECTR. It must be at least 1 and at least max( NN ).``` [out] V ``` V is REAL array, dimension (LDU, max(NN)) The (right) orthogonal matrix computed by SGGHRD.``` [out] Q ``` Q is REAL array, dimension (LDU, max(NN)) The (left) orthogonal matrix computed by SHGEQZ.``` [out] Z ``` Z is REAL array, dimension (LDU, max(NN)) The (left) orthogonal matrix computed by SHGEQZ.``` [out] ALPHR1 ` ALPHR1 is REAL array, dimension (max(NN))` [out] ALPHI1 ` ALPHI1 is REAL array, dimension (max(NN))` [out] BETA1 ``` BETA1 is REAL array, dimension (max(NN)) The generalized eigenvalues of (A,B) computed by SHGEQZ 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 REAL array, dimension (max(NN))` [out] ALPHI3 ` ALPHI3 is REAL array, dimension (max(NN))` [out] BETA3 ` BETA3 is REAL array, dimension (max(NN))` [out] EVECTL ``` EVECTL is REAL array, dimension (LDU, max(NN)) The (block lower triangular) left eigenvector matrix for the matrices in S1 and P1. (See STGEVC for the format.)``` [out] EVECTR ``` EVECTR is REAL array, dimension (LDU, max(NN)) The (block upper triangular) right eigenvector matrix for the matrices in S1 and P1. (See STGEVC for the format.)``` [out] WORK ` WORK is REAL 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 REAL 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.```
Date
June 2016

Definition at line 513 of file schkgg.f.

513 *
514 * -- LAPACK test routine (version 3.6.1) --
515 * -- LAPACK is a software package provided by Univ. of Tennessee, --
516 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
517 * June 2016
518 *
519 * .. Scalar Arguments ..
520  LOGICAL tstdif
521  INTEGER info, lda, ldu, lwork, nounit, nsizes, ntypes
522  REAL thresh, thrshn
523 * ..
524 * .. Array Arguments ..
525  LOGICAL dotype( * ), llwork( * )
526  INTEGER iseed( 4 ), nn( * )
527  REAL a( lda, * ), alphi1( * ), alphi3( * ),
528  \$ alphr1( * ), alphr3( * ), b( lda, * ),
529  \$ beta1( * ), beta3( * ), evectl( ldu, * ),
530  \$ evectr( ldu, * ), h( lda, * ), p1( lda, * ),
531  \$ p2( lda, * ), q( ldu, * ), result( 15 ),
532  \$ s1( lda, * ), s2( lda, * ), t( lda, * ),
533  \$ u( ldu, * ), v( ldu, * ), work( * ),
534  \$ z( ldu, * )
535 * ..
536 *
537 * =====================================================================
538 *
539 * .. Parameters ..
540  REAL zero, one
541  parameter ( zero = 0.0, one = 1.0 )
542  INTEGER maxtyp
543  parameter ( maxtyp = 26 )
544 * ..
545 * .. Local Scalars ..
547  INTEGER i1, iadd, iinfo, in, j, jc, jr, jsize, jtype,
548  \$ lwkopt, mtypes, n, n1, nerrs, nmats, nmax,
549  \$ ntest, ntestt
550  REAL anorm, bnorm, safmax, safmin, temp1, temp2,
551  \$ ulp, ulpinv
552 * ..
553 * .. Local Arrays ..
554  INTEGER iasign( maxtyp ), ibsign( maxtyp ),
555  \$ ioldsd( 4 ), kadd( 6 ), kamagn( maxtyp ),
556  \$ katype( maxtyp ), kazero( maxtyp ),
557  \$ kbmagn( maxtyp ), kbtype( maxtyp ),
558  \$ kbzero( maxtyp ), kclass( maxtyp ),
559  \$ ktrian( maxtyp ), kz1( 6 ), kz2( 6 )
560  REAL dumma( 4 ), rmagn( 0: 3 )
561 * ..
562 * .. External Functions ..
563  REAL slamch, slange, slarnd
564  EXTERNAL slamch, slange, slarnd
565 * ..
566 * .. External Subroutines ..
567  EXTERNAL sgeqr2, sget51, sget52, sgghrd, shgeqz, slabad,
569  \$ stgevc, xerbla
570 * ..
571 * .. Intrinsic Functions ..
572  INTRINSIC abs, max, min, REAL, sign
573 * ..
574 * .. Data statements ..
575  DATA kclass / 15*1, 10*2, 1*3 /
576  DATA kz1 / 0, 1, 2, 1, 3, 3 /
577  DATA kz2 / 0, 0, 1, 2, 1, 1 /
578  DATA kadd / 0, 0, 0, 0, 3, 2 /
579  DATA katype / 0, 1, 0, 1, 2, 3, 4, 1, 4, 4, 1, 1, 4,
580  \$ 4, 4, 2, 4, 5, 8, 7, 9, 4*4, 0 /
581  DATA kbtype / 0, 0, 1, 1, 2, -3, 1, 4, 1, 1, 4, 4,
582  \$ 1, 1, -4, 2, -4, 8*8, 0 /
583  DATA kazero / 6*1, 2, 1, 2*2, 2*1, 2*2, 3, 1, 3,
584  \$ 4*5, 4*3, 1 /
585  DATA kbzero / 6*1, 1, 2, 2*1, 2*2, 2*1, 4, 1, 4,
586  \$ 4*6, 4*4, 1 /
587  DATA kamagn / 8*1, 2, 3, 2, 3, 2, 3, 7*1, 2, 3, 3,
588  \$ 2, 1 /
589  DATA kbmagn / 8*1, 3, 2, 3, 2, 2, 3, 7*1, 3, 2, 3,
590  \$ 2, 1 /
591  DATA ktrian / 16*0, 10*1 /
592  DATA iasign / 6*0, 2, 0, 2*2, 2*0, 3*2, 0, 2, 3*0,
593  \$ 5*2, 0 /
594  DATA ibsign / 7*0, 2, 2*0, 2*2, 2*0, 2, 0, 2, 9*0 /
595 * ..
596 * .. Executable Statements ..
597 *
598 * Check for errors
599 *
600  info = 0
601 *
603  nmax = 1
604  DO 10 j = 1, nsizes
605  nmax = max( nmax, nn( j ) )
606  IF( nn( j ).LT.0 )
608  10 CONTINUE
609 *
610 * Maximum blocksize and shift -- we assume that blocksize and number
611 * of shifts are monotone increasing functions of N.
612 *
613  lwkopt = max( 6*nmax, 2*nmax*nmax, 1 )
614 *
615 * Check for errors
616 *
617  IF( nsizes.LT.0 ) THEN
618  info = -1
619  ELSE IF( badnn ) THEN
620  info = -2
621  ELSE IF( ntypes.LT.0 ) THEN
622  info = -3
623  ELSE IF( thresh.LT.zero ) THEN
624  info = -6
625  ELSE IF( lda.LE.1 .OR. lda.LT.nmax ) THEN
626  info = -10
627  ELSE IF( ldu.LE.1 .OR. ldu.LT.nmax ) THEN
628  info = -19
629  ELSE IF( lwkopt.GT.lwork ) THEN
630  info = -30
631  END IF
632 *
633  IF( info.NE.0 ) THEN
634  CALL xerbla( 'SCHKGG', -info )
635  RETURN
636  END IF
637 *
638 * Quick return if possible
639 *
640  IF( nsizes.EQ.0 .OR. ntypes.EQ.0 )
641  \$ RETURN
642 *
643  safmin = slamch( 'Safe minimum' )
644  ulp = slamch( 'Epsilon' )*slamch( 'Base' )
645  safmin = safmin / ulp
646  safmax = one / safmin
647  CALL slabad( safmin, safmax )
648  ulpinv = one / ulp
649 *
650 * The values RMAGN(2:3) depend on N, see below.
651 *
652  rmagn( 0 ) = zero
653  rmagn( 1 ) = one
654 *
655 * Loop over sizes, types
656 *
657  ntestt = 0
658  nerrs = 0
659  nmats = 0
660 *
661  DO 240 jsize = 1, nsizes
662  n = nn( jsize )
663  n1 = max( 1, n )
664  rmagn( 2 ) = safmax*ulp / REAL( n1 )
665  rmagn( 3 ) = safmin*ulpinv*n1
666 *
667  IF( nsizes.NE.1 ) THEN
668  mtypes = min( maxtyp, ntypes )
669  ELSE
670  mtypes = min( maxtyp+1, ntypes )
671  END IF
672 *
673  DO 230 jtype = 1, mtypes
674  IF( .NOT.dotype( jtype ) )
675  \$ GO TO 230
676  nmats = nmats + 1
677  ntest = 0
678 *
679 * Save ISEED in case of an error.
680 *
681  DO 20 j = 1, 4
682  ioldsd( j ) = iseed( j )
683  20 CONTINUE
684 *
685 * Initialize RESULT
686 *
687  DO 30 j = 1, 15
688  result( j ) = zero
689  30 CONTINUE
690 *
691 * Compute A and B
692 *
693 * Description of control parameters:
694 *
695 * KCLASS: =1 means w/o rotation, =2 means w/ rotation,
696 * =3 means random.
697 * KATYPE: the "type" to be passed to SLATM4 for computing A.
698 * KAZERO: the pattern of zeros on the diagonal for A:
699 * =1: ( xxx ), =2: (0, xxx ) =3: ( 0, 0, xxx, 0 ),
700 * =4: ( 0, xxx, 0, 0 ), =5: ( 0, 0, 1, xxx, 0 ),
701 * =6: ( 0, 1, 0, xxx, 0 ). (xxx means a string of
702 * non-zero entries.)
703 * KAMAGN: the magnitude of the matrix: =0: zero, =1: O(1),
704 * =2: large, =3: small.
705 * IASIGN: 1 if the diagonal elements of A are to be
706 * multiplied by a random magnitude 1 number, =2 if
707 * randomly chosen diagonal blocks are to be rotated
708 * to form 2x2 blocks.
709 * KBTYPE, KBZERO, KBMAGN, IBSIGN: the same, but for B.
710 * KTRIAN: =0: don't fill in the upper triangle, =1: do.
711 * KZ1, KZ2, KADD: used to implement KAZERO and KBZERO.
712 * RMAGN: used to implement KAMAGN and KBMAGN.
713 *
714  IF( mtypes.GT.maxtyp )
715  \$ GO TO 110
716  iinfo = 0
717  IF( kclass( jtype ).LT.3 ) THEN
718 *
719 * Generate A (w/o rotation)
720 *
721  IF( abs( katype( jtype ) ).EQ.3 ) THEN
722  in = 2*( ( n-1 ) / 2 ) + 1
723  IF( in.NE.n )
724  \$ CALL slaset( 'Full', n, n, zero, zero, a, lda )
725  ELSE
726  in = n
727  END IF
728  CALL slatm4( katype( jtype ), in, kz1( kazero( jtype ) ),
729  \$ kz2( kazero( jtype ) ), iasign( jtype ),
730  \$ rmagn( kamagn( jtype ) ), ulp,
731  \$ rmagn( ktrian( jtype )*kamagn( jtype ) ), 2,
732  \$ iseed, a, lda )
735  \$ a( iadd, iadd ) = rmagn( kamagn( jtype ) )
736 *
737 * Generate B (w/o rotation)
738 *
739  IF( abs( kbtype( jtype ) ).EQ.3 ) THEN
740  in = 2*( ( n-1 ) / 2 ) + 1
741  IF( in.NE.n )
742  \$ CALL slaset( 'Full', n, n, zero, zero, b, lda )
743  ELSE
744  in = n
745  END IF
746  CALL slatm4( kbtype( jtype ), in, kz1( kbzero( jtype ) ),
747  \$ kz2( kbzero( jtype ) ), ibsign( jtype ),
748  \$ rmagn( kbmagn( jtype ) ), one,
749  \$ rmagn( ktrian( jtype )*kbmagn( jtype ) ), 2,
750  \$ iseed, b, lda )
753  \$ b( iadd, iadd ) = rmagn( kbmagn( jtype ) )
754 *
755  IF( kclass( jtype ).EQ.2 .AND. n.GT.0 ) THEN
756 *
757 * Include rotations
758 *
759 * Generate U, V as Householder transformations times
760 * a diagonal matrix.
761 *
762  DO 50 jc = 1, n - 1
763  DO 40 jr = jc, n
764  u( jr, jc ) = slarnd( 3, iseed )
765  v( jr, jc ) = slarnd( 3, iseed )
766  40 CONTINUE
767  CALL slarfg( n+1-jc, u( jc, jc ), u( jc+1, jc ), 1,
768  \$ work( jc ) )
769  work( 2*n+jc ) = sign( one, u( jc, jc ) )
770  u( jc, jc ) = one
771  CALL slarfg( n+1-jc, v( jc, jc ), v( jc+1, jc ), 1,
772  \$ work( n+jc ) )
773  work( 3*n+jc ) = sign( one, v( jc, jc ) )
774  v( jc, jc ) = one
775  50 CONTINUE
776  u( n, n ) = one
777  work( n ) = zero
778  work( 3*n ) = sign( one, slarnd( 2, iseed ) )
779  v( n, n ) = one
780  work( 2*n ) = zero
781  work( 4*n ) = sign( one, slarnd( 2, iseed ) )
782 *
783 * Apply the diagonal matrices
784 *
785  DO 70 jc = 1, n
786  DO 60 jr = 1, n
787  a( jr, jc ) = work( 2*n+jr )*work( 3*n+jc )*
788  \$ a( jr, jc )
789  b( jr, jc ) = work( 2*n+jr )*work( 3*n+jc )*
790  \$ b( jr, jc )
791  60 CONTINUE
792  70 CONTINUE
793  CALL sorm2r( 'L', 'N', n, n, n-1, u, ldu, work, a,
794  \$ lda, work( 2*n+1 ), iinfo )
795  IF( iinfo.NE.0 )
796  \$ GO TO 100
797  CALL sorm2r( 'R', 'T', n, n, n-1, v, ldu, work( n+1 ),
798  \$ a, lda, work( 2*n+1 ), iinfo )
799  IF( iinfo.NE.0 )
800  \$ GO TO 100
801  CALL sorm2r( 'L', 'N', n, n, n-1, u, ldu, work, b,
802  \$ lda, work( 2*n+1 ), iinfo )
803  IF( iinfo.NE.0 )
804  \$ GO TO 100
805  CALL sorm2r( 'R', 'T', n, n, n-1, v, ldu, work( n+1 ),
806  \$ b, lda, work( 2*n+1 ), iinfo )
807  IF( iinfo.NE.0 )
808  \$ GO TO 100
809  END IF
810  ELSE
811 *
812 * Random matrices
813 *
814  DO 90 jc = 1, n
815  DO 80 jr = 1, n
816  a( jr, jc ) = rmagn( kamagn( jtype ) )*
817  \$ slarnd( 2, iseed )
818  b( jr, jc ) = rmagn( kbmagn( jtype ) )*
819  \$ slarnd( 2, iseed )
820  80 CONTINUE
821  90 CONTINUE
822  END IF
823 *
824  anorm = slange( '1', n, n, a, lda, work )
825  bnorm = slange( '1', n, n, b, lda, work )
826 *
827  100 CONTINUE
828 *
829  IF( iinfo.NE.0 ) THEN
830  WRITE( nounit, fmt = 9999 )'Generator', iinfo, n, jtype,
831  \$ ioldsd
832  info = abs( iinfo )
833  RETURN
834  END IF
835 *
836  110 CONTINUE
837 *
838 * Call SGEQR2, SORM2R, and SGGHRD to compute H, T, U, and V
839 *
840  CALL slacpy( ' ', n, n, a, lda, h, lda )
841  CALL slacpy( ' ', n, n, b, lda, t, lda )
842  ntest = 1
843  result( 1 ) = ulpinv
844 *
845  CALL sgeqr2( n, n, t, lda, work, work( n+1 ), iinfo )
846  IF( iinfo.NE.0 ) THEN
847  WRITE( nounit, fmt = 9999 )'SGEQR2', iinfo, n, jtype,
848  \$ ioldsd
849  info = abs( iinfo )
850  GO TO 210
851  END IF
852 *
853  CALL sorm2r( 'L', 'T', n, n, n, t, lda, work, h, lda,
854  \$ work( n+1 ), iinfo )
855  IF( iinfo.NE.0 ) THEN
856  WRITE( nounit, fmt = 9999 )'SORM2R', iinfo, n, jtype,
857  \$ ioldsd
858  info = abs( iinfo )
859  GO TO 210
860  END IF
861 *
862  CALL slaset( 'Full', n, n, zero, one, u, ldu )
863  CALL sorm2r( 'R', 'N', n, n, n, t, lda, work, u, ldu,
864  \$ work( n+1 ), iinfo )
865  IF( iinfo.NE.0 ) THEN
866  WRITE( nounit, fmt = 9999 )'SORM2R', iinfo, n, jtype,
867  \$ ioldsd
868  info = abs( iinfo )
869  GO TO 210
870  END IF
871 *
872  CALL sgghrd( 'V', 'I', n, 1, n, h, lda, t, lda, u, ldu, v,
873  \$ ldu, iinfo )
874  IF( iinfo.NE.0 ) THEN
875  WRITE( nounit, fmt = 9999 )'SGGHRD', iinfo, n, jtype,
876  \$ ioldsd
877  info = abs( iinfo )
878  GO TO 210
879  END IF
880  ntest = 4
881 *
882 * Do tests 1--4
883 *
884  CALL sget51( 1, n, a, lda, h, lda, u, ldu, v, ldu, work,
885  \$ result( 1 ) )
886  CALL sget51( 1, n, b, lda, t, lda, u, ldu, v, ldu, work,
887  \$ result( 2 ) )
888  CALL sget51( 3, n, b, lda, t, lda, u, ldu, u, ldu, work,
889  \$ result( 3 ) )
890  CALL sget51( 3, n, b, lda, t, lda, v, ldu, v, ldu, work,
891  \$ result( 4 ) )
892 *
893 * Call SHGEQZ to compute S1, P1, S2, P2, Q, and Z, do tests.
894 *
895 * Compute T1 and UZ
896 *
897 * Eigenvalues only
898 *
899  CALL slacpy( ' ', n, n, h, lda, s2, lda )
900  CALL slacpy( ' ', n, n, t, lda, p2, lda )
901  ntest = 5
902  result( 5 ) = ulpinv
903 *
904  CALL shgeqz( 'E', 'N', 'N', n, 1, n, s2, lda, p2, lda,
905  \$ alphr3, alphi3, beta3, q, ldu, z, ldu, work,
906  \$ lwork, iinfo )
907  IF( iinfo.NE.0 ) THEN
908  WRITE( nounit, fmt = 9999 )'SHGEQZ(E)', iinfo, n, jtype,
909  \$ ioldsd
910  info = abs( iinfo )
911  GO TO 210
912  END IF
913 *
914 * Eigenvalues and Full Schur Form
915 *
916  CALL slacpy( ' ', n, n, h, lda, s2, lda )
917  CALL slacpy( ' ', n, n, t, lda, p2, lda )
918 *
919  CALL shgeqz( 'S', 'N', 'N', n, 1, n, s2, lda, p2, lda,
920  \$ alphr1, alphi1, beta1, q, ldu, z, ldu, work,
921  \$ lwork, iinfo )
922  IF( iinfo.NE.0 ) THEN
923  WRITE( nounit, fmt = 9999 )'SHGEQZ(S)', iinfo, n, jtype,
924  \$ ioldsd
925  info = abs( iinfo )
926  GO TO 210
927  END IF
928 *
929 * Eigenvalues, Schur Form, and Schur Vectors
930 *
931  CALL slacpy( ' ', n, n, h, lda, s1, lda )
932  CALL slacpy( ' ', n, n, t, lda, p1, lda )
933 *
934  CALL shgeqz( 'S', 'I', 'I', n, 1, n, s1, lda, p1, lda,
935  \$ alphr1, alphi1, beta1, q, ldu, z, ldu, work,
936  \$ lwork, iinfo )
937  IF( iinfo.NE.0 ) THEN
938  WRITE( nounit, fmt = 9999 )'SHGEQZ(V)', iinfo, n, jtype,
939  \$ ioldsd
940  info = abs( iinfo )
941  GO TO 210
942  END IF
943 *
944  ntest = 8
945 *
946 * Do Tests 5--8
947 *
948  CALL sget51( 1, n, h, lda, s1, lda, q, ldu, z, ldu, work,
949  \$ result( 5 ) )
950  CALL sget51( 1, n, t, lda, p1, lda, q, ldu, z, ldu, work,
951  \$ result( 6 ) )
952  CALL sget51( 3, n, t, lda, p1, lda, q, ldu, q, ldu, work,
953  \$ result( 7 ) )
954  CALL sget51( 3, n, t, lda, p1, lda, z, ldu, z, ldu, work,
955  \$ result( 8 ) )
956 *
957 * Compute the Left and Right Eigenvectors of (S1,P1)
958 *
959 * 9: Compute the left eigenvector Matrix without
960 * back transforming:
961 *
962  ntest = 9
963  result( 9 ) = ulpinv
964 *
965 * To test "SELECT" option, compute half of the eigenvectors
966 * in one call, and half in another
967 *
968  i1 = n / 2
969  DO 120 j = 1, i1
970  llwork( j ) = .true.
971  120 CONTINUE
972  DO 130 j = i1 + 1, n
973  llwork( j ) = .false.
974  130 CONTINUE
975 *
976  CALL stgevc( 'L', 'S', llwork, n, s1, lda, p1, lda, evectl,
977  \$ ldu, dumma, ldu, n, in, work, iinfo )
978  IF( iinfo.NE.0 ) THEN
979  WRITE( nounit, fmt = 9999 )'STGEVC(L,S1)', iinfo, n,
980  \$ jtype, ioldsd
981  info = abs( iinfo )
982  GO TO 210
983  END IF
984 *
985  i1 = in
986  DO 140 j = 1, i1
987  llwork( j ) = .false.
988  140 CONTINUE
989  DO 150 j = i1 + 1, n
990  llwork( j ) = .true.
991  150 CONTINUE
992 *
993  CALL stgevc( 'L', 'S', llwork, n, s1, lda, p1, lda,
994  \$ evectl( 1, i1+1 ), ldu, dumma, ldu, n, in,
995  \$ work, iinfo )
996  IF( iinfo.NE.0 ) THEN
997  WRITE( nounit, fmt = 9999 )'STGEVC(L,S2)', iinfo, n,
998  \$ jtype, ioldsd
999  info = abs( iinfo )
1000  GO TO 210
1001  END IF
1002 *
1003  CALL sget52( .true., n, s1, lda, p1, lda, evectl, ldu,
1004  \$ alphr1, alphi1, beta1, work, dumma( 1 ) )
1005  result( 9 ) = dumma( 1 )
1006  IF( dumma( 2 ).GT.thrshn ) THEN
1007  WRITE( nounit, fmt = 9998 )'Left', 'STGEVC(HOWMNY=S)',
1008  \$ dumma( 2 ), n, jtype, ioldsd
1009  END IF
1010 *
1011 * 10: Compute the left eigenvector Matrix with
1012 * back transforming:
1013 *
1014  ntest = 10
1015  result( 10 ) = ulpinv
1016  CALL slacpy( 'F', n, n, q, ldu, evectl, ldu )
1017  CALL stgevc( 'L', 'B', llwork, n, s1, lda, p1, lda, evectl,
1018  \$ ldu, dumma, ldu, n, in, work, iinfo )
1019  IF( iinfo.NE.0 ) THEN
1020  WRITE( nounit, fmt = 9999 )'STGEVC(L,B)', iinfo, n,
1021  \$ jtype, ioldsd
1022  info = abs( iinfo )
1023  GO TO 210
1024  END IF
1025 *
1026  CALL sget52( .true., n, h, lda, t, lda, evectl, ldu, alphr1,
1027  \$ alphi1, beta1, work, dumma( 1 ) )
1028  result( 10 ) = dumma( 1 )
1029  IF( dumma( 2 ).GT.thrshn ) THEN
1030  WRITE( nounit, fmt = 9998 )'Left', 'STGEVC(HOWMNY=B)',
1031  \$ dumma( 2 ), n, jtype, ioldsd
1032  END IF
1033 *
1034 * 11: Compute the right eigenvector Matrix without
1035 * back transforming:
1036 *
1037  ntest = 11
1038  result( 11 ) = ulpinv
1039 *
1040 * To test "SELECT" option, compute half of the eigenvectors
1041 * in one call, and half in another
1042 *
1043  i1 = n / 2
1044  DO 160 j = 1, i1
1045  llwork( j ) = .true.
1046  160 CONTINUE
1047  DO 170 j = i1 + 1, n
1048  llwork( j ) = .false.
1049  170 CONTINUE
1050 *
1051  CALL stgevc( 'R', 'S', llwork, n, s1, lda, p1, lda, dumma,
1052  \$ ldu, evectr, ldu, n, in, work, iinfo )
1053  IF( iinfo.NE.0 ) THEN
1054  WRITE( nounit, fmt = 9999 )'STGEVC(R,S1)', iinfo, n,
1055  \$ jtype, ioldsd
1056  info = abs( iinfo )
1057  GO TO 210
1058  END IF
1059 *
1060  i1 = in
1061  DO 180 j = 1, i1
1062  llwork( j ) = .false.
1063  180 CONTINUE
1064  DO 190 j = i1 + 1, n
1065  llwork( j ) = .true.
1066  190 CONTINUE
1067 *
1068  CALL stgevc( 'R', 'S', llwork, n, s1, lda, p1, lda, dumma,
1069  \$ ldu, evectr( 1, i1+1 ), ldu, n, in, work,
1070  \$ iinfo )
1071  IF( iinfo.NE.0 ) THEN
1072  WRITE( nounit, fmt = 9999 )'STGEVC(R,S2)', iinfo, n,
1073  \$ jtype, ioldsd
1074  info = abs( iinfo )
1075  GO TO 210
1076  END IF
1077 *
1078  CALL sget52( .false., n, s1, lda, p1, lda, evectr, ldu,
1079  \$ alphr1, alphi1, beta1, work, dumma( 1 ) )
1080  result( 11 ) = dumma( 1 )
1081  IF( dumma( 2 ).GT.thresh ) THEN
1082  WRITE( nounit, fmt = 9998 )'Right', 'STGEVC(HOWMNY=S)',
1083  \$ dumma( 2 ), n, jtype, ioldsd
1084  END IF
1085 *
1086 * 12: Compute the right eigenvector Matrix with
1087 * back transforming:
1088 *
1089  ntest = 12
1090  result( 12 ) = ulpinv
1091  CALL slacpy( 'F', n, n, z, ldu, evectr, ldu )
1092  CALL stgevc( 'R', 'B', llwork, n, s1, lda, p1, lda, dumma,
1093  \$ ldu, evectr, ldu, n, in, work, iinfo )
1094  IF( iinfo.NE.0 ) THEN
1095  WRITE( nounit, fmt = 9999 )'STGEVC(R,B)', iinfo, n,
1096  \$ jtype, ioldsd
1097  info = abs( iinfo )
1098  GO TO 210
1099  END IF
1100 *
1101  CALL sget52( .false., n, h, lda, t, lda, evectr, ldu,
1102  \$ alphr1, alphi1, beta1, work, dumma( 1 ) )
1103  result( 12 ) = dumma( 1 )
1104  IF( dumma( 2 ).GT.thresh ) THEN
1105  WRITE( nounit, fmt = 9998 )'Right', 'STGEVC(HOWMNY=B)',
1106  \$ dumma( 2 ), n, jtype, ioldsd
1107  END IF
1108 *
1109 * Tests 13--15 are done only on request
1110 *
1111  IF( tstdif ) THEN
1112 *
1113 * Do Tests 13--14
1114 *
1115  CALL sget51( 2, n, s1, lda, s2, lda, q, ldu, z, ldu,
1116  \$ work, result( 13 ) )
1117  CALL sget51( 2, n, p1, lda, p2, lda, q, ldu, z, ldu,
1118  \$ work, result( 14 ) )
1119 *
1120 * Do Test 15
1121 *
1122  temp1 = zero
1123  temp2 = zero
1124  DO 200 j = 1, n
1125  temp1 = max( temp1, abs( alphr1( j )-alphr3( j ) )+
1126  \$ abs( alphi1( j )-alphi3( j ) ) )
1127  temp2 = max( temp2, abs( beta1( j )-beta3( j ) ) )
1128  200 CONTINUE
1129 *
1130  temp1 = temp1 / max( safmin, ulp*max( temp1, anorm ) )
1131  temp2 = temp2 / max( safmin, ulp*max( temp2, bnorm ) )
1132  result( 15 ) = max( temp1, temp2 )
1133  ntest = 15
1134  ELSE
1135  result( 13 ) = zero
1136  result( 14 ) = zero
1137  result( 15 ) = zero
1138  ntest = 12
1139  END IF
1140 *
1141 * End of Loop -- Check for RESULT(j) > THRESH
1142 *
1143  210 CONTINUE
1144 *
1145  ntestt = ntestt + ntest
1146 *
1147 * Print out tests which fail.
1148 *
1149  DO 220 jr = 1, ntest
1150  IF( result( jr ).GE.thresh ) THEN
1151 *
1152 * If this is the first test to fail,
1153 * print a header to the data file.
1154 *
1155  IF( nerrs.EQ.0 ) THEN
1156  WRITE( nounit, fmt = 9997 )'SGG'
1157 *
1158 * Matrix types
1159 *
1160  WRITE( nounit, fmt = 9996 )
1161  WRITE( nounit, fmt = 9995 )
1162  WRITE( nounit, fmt = 9994 )'Orthogonal'
1163 *
1164 * Tests performed
1165 *
1166  WRITE( nounit, fmt = 9993 )'orthogonal', '''',
1167  \$ 'transpose', ( '''', j = 1, 10 )
1168 *
1169  END IF
1170  nerrs = nerrs + 1
1171  IF( result( jr ).LT.10000.0 ) THEN
1172  WRITE( nounit, fmt = 9992 )n, jtype, ioldsd, jr,
1173  \$ result( jr )
1174  ELSE
1175  WRITE( nounit, fmt = 9991 )n, jtype, ioldsd, jr,
1176  \$ result( jr )
1177  END IF
1178  END IF
1179  220 CONTINUE
1180 *
1181  230 CONTINUE
1182  240 CONTINUE
1183 *
1184 * Summary
1185 *
1186  CALL slasum( 'SGG', nounit, nerrs, ntestt )
1187  RETURN
1188 *
1189  9999 FORMAT( ' SCHKGG: ', a, ' returned INFO=', i6, '.', / 9x, 'N=',
1190  \$ i6, ', JTYPE=', i6, ', ISEED=(', 3( i5, ',' ), i5, ')' )
1191 *
1192  9998 FORMAT( ' SCHKGG: ', a, ' Eigenvectors from ', a, ' incorrectly ',
1193  \$ 'normalized.', / ' Bits of error=', 0p, g10.3, ',', 9x,
1194  \$ 'N=', i6, ', JTYPE=', i6, ', ISEED=(', 3( i5, ',' ), i5,
1195  \$ ')' )
1196 *
1197  9997 FORMAT( / 1x, a3, ' -- Real Generalized eigenvalue problem' )
1198 *
1199  9996 FORMAT( ' Matrix types (see SCHKGG for details): ' )
1200 *
1201  9995 FORMAT( ' Special Matrices:', 23x,
1202  \$ '(J''=transposed Jordan block)',
1203  \$ / ' 1=(0,0) 2=(I,0) 3=(0,I) 4=(I,I) 5=(J'',J'') ',
1204  \$ '6=(diag(J'',I), diag(I,J''))', / ' Diagonal Matrices: ( ',
1205  \$ 'D=diag(0,1,2,...) )', / ' 7=(D,I) 9=(large*D, small*I',
1206  \$ ') 11=(large*I, small*D) 13=(large*D, large*I)', /
1207  \$ ' 8=(I,D) 10=(small*D, large*I) 12=(small*I, large*D) ',
1208  \$ ' 14=(small*D, small*I)', / ' 15=(D, reversed D)' )
1209  9994 FORMAT( ' Matrices Rotated by Random ', a, ' Matrices U, V:',
1210  \$ / ' 16=Transposed Jordan Blocks 19=geometric ',
1211  \$ 'alpha, beta=0,1', / ' 17=arithm. alpha&beta ',
1212  \$ ' 20=arithmetic alpha, beta=0,1', / ' 18=clustered ',
1213  \$ 'alpha, beta=0,1 21=random alpha, beta=0,1',
1214  \$ / ' Large & Small Matrices:', / ' 22=(large, small) ',
1215  \$ '23=(small,large) 24=(small,small) 25=(large,large)',
1216  \$ / ' 26=random O(1) matrices.' )
1217 *
1218  9993 FORMAT( / ' Tests performed: (H is Hessenberg, S is Schur, B, ',
1219  \$ 'T, P are triangular,', / 20x, 'U, V, Q, and Z are ', a,
1220  \$ ', l and r are the', / 20x,
1221  \$ 'appropriate left and right eigenvectors, resp., a is',
1222  \$ / 20x, 'alpha, b is beta, and ', a, ' means ', a, '.)',
1223  \$ / ' 1 = | A - U H V', a,
1224  \$ ' | / ( |A| n ulp ) 2 = | B - U T V', a,
1225  \$ ' | / ( |B| n ulp )', / ' 3 = | I - UU', a,
1226  \$ ' | / ( n ulp ) 4 = | I - VV', a,
1227  \$ ' | / ( n ulp )', / ' 5 = | H - Q S Z', a,
1228  \$ ' | / ( |H| n ulp )', 6x, '6 = | T - Q P Z', a,
1229  \$ ' | / ( |T| n ulp )', / ' 7 = | I - QQ', a,
1230  \$ ' | / ( n ulp ) 8 = | I - ZZ', a,
1231  \$ ' | / ( n ulp )', / ' 9 = max | ( b S - a P )', a,
1232  \$ ' l | / const. 10 = max | ( b H - a T )', a,
1233  \$ ' l | / const.', /
1234  \$ ' 11= max | ( b S - a P ) r | / const. 12 = max | ( b H',
1235  \$ ' - a T ) r | / const.', / 1x )
1236 *
1237  9992 FORMAT( ' Matrix order=', i5, ', type=', i2, ', seed=',
1238  \$ 4( i4, ',' ), ' result ', i2, ' is', 0p, f8.2 )
1239  9991 FORMAT( ' Matrix order=', i5, ', type=', i2, ', seed=',
1240  \$ 4( i4, ',' ), ' result ', i2, ' is', 1p, e10.3 )
1241 *
1242 * End of SCHKGG
1243 *
subroutine slatm4(ITYPE, N, NZ1, NZ2, ISIGN, AMAGN, RCOND, TRIANG, IDIST, ISEED, A, LDA)
SLATM4
Definition: slatm4.f:177
subroutine slarfg(N, ALPHA, X, INCX, TAU)
SLARFG generates an elementary reflector (Householder matrix).
Definition: slarfg.f:108
subroutine stgevc(SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL, LDVL, VR, LDVR, MM, M, WORK, INFO)
STGEVC
Definition: stgevc.f:297
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:105
subroutine sgeqr2(M, N, A, LDA, TAU, WORK, INFO)
SGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm...
Definition: sgeqr2.f:123
subroutine shgeqz(JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
SHGEQZ
Definition: shgeqz.f:306
subroutine slaset(UPLO, M, N, ALPHA, BETA, A, LDA)
SLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values...
Definition: slaset.f:112
subroutine sget51(ITYPE, N, A, LDA, B, LDB, U, LDU, V, LDV, WORK, RESULT)
SGET51
Definition: sget51.f:151
real function slange(NORM, M, N, A, LDA, WORK)
SLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: slange.f:116
real function slarnd(IDIST, ISEED)
SLARND
Definition: slarnd.f:75
subroutine sorm2r(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
SORM2R multiplies a general matrix by the orthogonal matrix from a QR factorization determined by sge...
Definition: sorm2r.f:161
subroutine sgghrd(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
SGGHRD
Definition: sgghrd.f:209
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
subroutine slasum(TYPE, IOUNIT, IE, NRUN)
SLASUM
Definition: slasum.f:42
subroutine sget52(LEFT, N, A, LDA, B, LDB, E, LDE, ALPHAR, ALPHAI, BETA, WORK, RESULT)
SGET52
Definition: sget52.f:201

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