LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ stgsna()

subroutine stgsna ( character  JOB,
character  HOWMNY,
logical, dimension( * )  SELECT,
integer  N,
real, dimension( lda, * )  A,
integer  LDA,
real, dimension( ldb, * )  B,
integer  LDB,
real, dimension( ldvl, * )  VL,
integer  LDVL,
real, dimension( ldvr, * )  VR,
integer  LDVR,
real, dimension( * )  S,
real, dimension( * )  DIF,
integer  MM,
integer  M,
real, dimension( * )  WORK,
integer  LWORK,
integer, dimension( * )  IWORK,
integer  INFO 
)

STGSNA

Download STGSNA + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 STGSNA estimates reciprocal condition numbers for specified
 eigenvalues and/or eigenvectors of a matrix pair (A, B) in
 generalized real Schur canonical form (or of any matrix pair
 (Q*A*Z**T, Q*B*Z**T) with orthogonal matrices Q and Z, where
 Z**T denotes the transpose of Z.

 (A, B) must be in generalized real Schur form (as returned by SGGES),
 i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal
 blocks. B is upper triangular.
Parameters
[in]JOB
          JOB is CHARACTER*1
          Specifies whether condition numbers are required for
          eigenvalues (S) or eigenvectors (DIF):
          = 'E': for eigenvalues only (S);
          = 'V': for eigenvectors only (DIF);
          = 'B': for both eigenvalues and eigenvectors (S and DIF).
[in]HOWMNY
          HOWMNY is CHARACTER*1
          = 'A': compute condition numbers for all eigenpairs;
          = 'S': compute condition numbers for selected eigenpairs
                 specified by the array SELECT.
[in]SELECT
          SELECT is LOGICAL array, dimension (N)
          If HOWMNY = 'S', SELECT specifies the eigenpairs for which
          condition numbers are required. To select condition numbers
          for the eigenpair corresponding to a real eigenvalue w(j),
          SELECT(j) must be set to .TRUE.. To select condition numbers
          corresponding to a complex conjugate pair of eigenvalues w(j)
          and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
          set to .TRUE..
          If HOWMNY = 'A', SELECT is not referenced.
[in]N
          N is INTEGER
          The order of the square matrix pair (A, B). N >= 0.
[in]A
          A is REAL array, dimension (LDA,N)
          The upper quasi-triangular matrix A in the pair (A,B).
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,N).
[in]B
          B is REAL array, dimension (LDB,N)
          The upper triangular matrix B in the pair (A,B).
[in]LDB
          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,N).
[in]VL
          VL is REAL array, dimension (LDVL,M)
          If JOB = 'E' or 'B', VL must contain left eigenvectors of
          (A, B), corresponding to the eigenpairs specified by HOWMNY
          and SELECT. The eigenvectors must be stored in consecutive
          columns of VL, as returned by STGEVC.
          If JOB = 'V', VL is not referenced.
[in]LDVL
          LDVL is INTEGER
          The leading dimension of the array VL. LDVL >= 1.
          If JOB = 'E' or 'B', LDVL >= N.
[in]VR
          VR is REAL array, dimension (LDVR,M)
          If JOB = 'E' or 'B', VR must contain right eigenvectors of
          (A, B), corresponding to the eigenpairs specified by HOWMNY
          and SELECT. The eigenvectors must be stored in consecutive
          columns ov VR, as returned by STGEVC.
          If JOB = 'V', VR is not referenced.
[in]LDVR
          LDVR is INTEGER
          The leading dimension of the array VR. LDVR >= 1.
          If JOB = 'E' or 'B', LDVR >= N.
[out]S
          S is REAL array, dimension (MM)
          If JOB = 'E' or 'B', the reciprocal condition numbers of the
          selected eigenvalues, stored in consecutive elements of the
          array. For a complex conjugate pair of eigenvalues two
          consecutive elements of S are set to the same value. Thus
          S(j), DIF(j), and the j-th columns of VL and VR all
          correspond to the same eigenpair (but not in general the
          j-th eigenpair, unless all eigenpairs are selected).
          If JOB = 'V', S is not referenced.
[out]DIF
          DIF is REAL array, dimension (MM)
          If JOB = 'V' or 'B', the estimated reciprocal condition
          numbers of the selected eigenvectors, stored in consecutive
          elements of the array. For a complex eigenvector two
          consecutive elements of DIF are set to the same value. If
          the eigenvalues cannot be reordered to compute DIF(j), DIF(j)
          is set to 0; this can only occur when the true value would be
          very small anyway.
          If JOB = 'E', DIF is not referenced.
[in]MM
          MM is INTEGER
          The number of elements in the arrays S and DIF. MM >= M.
[out]M
          M is INTEGER
          The number of elements of the arrays S and DIF used to store
          the specified condition numbers; for each selected real
          eigenvalue one element is used, and for each selected complex
          conjugate pair of eigenvalues, two elements are used.
          If HOWMNY = 'A', M is set to N.
[out]WORK
          WORK is REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
[in]LWORK
          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,N).
          If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.
[out]IWORK
          IWORK is INTEGER array, dimension (N + 6)
          If JOB = 'E', IWORK is not referenced.
[out]INFO
          INFO is INTEGER
          =0: Successful exit
          <0: If INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
  The reciprocal of the condition number of a generalized eigenvalue
  w = (a, b) is defined as

       S(w) = (|u**TAv|**2 + |u**TBv|**2)**(1/2) / (norm(u)*norm(v))

  where u and v are the left and right eigenvectors of (A, B)
  corresponding to w; |z| denotes the absolute value of the complex
  number, and norm(u) denotes the 2-norm of the vector u.
  The pair (a, b) corresponds to an eigenvalue w = a/b (= u**TAv/u**TBv)
  of the matrix pair (A, B). If both a and b equal zero, then (A B) is
  singular and S(I) = -1 is returned.

  An approximate error bound on the chordal distance between the i-th
  computed generalized eigenvalue w and the corresponding exact
  eigenvalue lambda is

       chord(w, lambda) <= EPS * norm(A, B) / S(I)

  where EPS is the machine precision.

  The reciprocal of the condition number DIF(i) of right eigenvector u
  and left eigenvector v corresponding to the generalized eigenvalue w
  is defined as follows:

  a) If the i-th eigenvalue w = (a,b) is real

     Suppose U and V are orthogonal transformations such that

              U**T*(A, B)*V  = (S, T) = ( a   *  ) ( b  *  )  1
                                        ( 0  S22 ),( 0 T22 )  n-1
                                          1  n-1     1 n-1

     Then the reciprocal condition number DIF(i) is

                Difl((a, b), (S22, T22)) = sigma-min( Zl ),

     where sigma-min(Zl) denotes the smallest singular value of the
     2(n-1)-by-2(n-1) matrix

         Zl = [ kron(a, In-1)  -kron(1, S22) ]
              [ kron(b, In-1)  -kron(1, T22) ] .

     Here In-1 is the identity matrix of size n-1. kron(X, Y) is the
     Kronecker product between the matrices X and Y.

     Note that if the default method for computing DIF(i) is wanted
     (see SLATDF), then the parameter DIFDRI (see below) should be
     changed from 3 to 4 (routine SLATDF(IJOB = 2 will be used)).
     See STGSYL for more details.

  b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,

     Suppose U and V are orthogonal transformations such that

              U**T*(A, B)*V = (S, T) = ( S11  *   ) ( T11  *  )  2
                                       ( 0    S22 ),( 0    T22) n-2
                                         2    n-2     2    n-2

     and (S11, T11) corresponds to the complex conjugate eigenvalue
     pair (w, conjg(w)). There exist unitary matrices U1 and V1 such
     that

       U1**T*S11*V1 = ( s11 s12 ) and U1**T*T11*V1 = ( t11 t12 )
                      (  0  s22 )                    (  0  t22 )

     where the generalized eigenvalues w = s11/t11 and
     conjg(w) = s22/t22.

     Then the reciprocal condition number DIF(i) is bounded by

         min( d1, max( 1, |real(s11)/real(s22)| )*d2 )

     where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where
     Z1 is the complex 2-by-2 matrix

              Z1 =  [ s11  -s22 ]
                    [ t11  -t22 ],

     This is done by computing (using real arithmetic) the
     roots of the characteristical polynomial det(Z1**T * Z1 - lambda I),
     where Z1**T denotes the transpose of Z1 and det(X) denotes
     the determinant of X.

     and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an
     upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)

              Z2 = [ kron(S11**T, In-2)  -kron(I2, S22) ]
                   [ kron(T11**T, In-2)  -kron(I2, T22) ]

     Note that if the default method for computing DIF is wanted (see
     SLATDF), then the parameter DIFDRI (see below) should be changed
     from 3 to 4 (routine SLATDF(IJOB = 2 will be used)). See STGSYL
     for more details.

  For each eigenvalue/vector specified by SELECT, DIF stores a
  Frobenius norm-based estimate of Difl.

  An approximate error bound for the i-th computed eigenvector VL(i) or
  VR(i) is given by

             EPS * norm(A, B) / DIF(i).

  See ref. [2-3] for more details and further references.
Contributors:
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.
References:
  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
      M.S. Moonen et al (eds), Linear Algebra for Large Scale and
      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

  [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
      Eigenvalues of a Regular Matrix Pair (A, B) and Condition
      Estimation: Theory, Algorithms and Software,
      Report UMINF - 94.04, Department of Computing Science, Umea
      University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
      Note 87. To appear in Numerical Algorithms, 1996.

  [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
      for Solving the Generalized Sylvester Equation and Estimating the
      Separation between Regular Matrix Pairs, Report UMINF - 93.23,
      Department of Computing Science, Umea University, S-901 87 Umea,
      Sweden, December 1993, Revised April 1994, Also as LAPACK Working
      Note 75.  To appear in ACM Trans. on Math. Software, Vol 22,
      No 1, 1996.

Definition at line 378 of file stgsna.f.

381*
382* -- LAPACK computational routine --
383* -- LAPACK is a software package provided by Univ. of Tennessee, --
384* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
385*
386* .. Scalar Arguments ..
387 CHARACTER HOWMNY, JOB
388 INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N
389* ..
390* .. Array Arguments ..
391 LOGICAL SELECT( * )
392 INTEGER IWORK( * )
393 REAL A( LDA, * ), B( LDB, * ), DIF( * ), S( * ),
394 $ VL( LDVL, * ), VR( LDVR, * ), WORK( * )
395* ..
396*
397* =====================================================================
398*
399* .. Parameters ..
400 INTEGER DIFDRI
401 parameter( difdri = 3 )
402 REAL ZERO, ONE, TWO, FOUR
403 parameter( zero = 0.0e+0, one = 1.0e+0, two = 2.0e+0,
404 $ four = 4.0e+0 )
405* ..
406* .. Local Scalars ..
407 LOGICAL LQUERY, PAIR, SOMCON, WANTBH, WANTDF, WANTS
408 INTEGER I, IERR, IFST, ILST, IZ, K, KS, LWMIN, N1, N2
409 REAL ALPHAI, ALPHAR, ALPRQT, BETA, C1, C2, COND,
410 $ EPS, LNRM, RNRM, ROOT1, ROOT2, SCALE, SMLNUM,
411 $ TMPII, TMPIR, TMPRI, TMPRR, UHAV, UHAVI, UHBV,
412 $ UHBVI
413* ..
414* .. Local Arrays ..
415 REAL DUMMY( 1 ), DUMMY1( 1 )
416* ..
417* .. External Functions ..
418 LOGICAL LSAME
419 REAL SDOT, SLAMCH, SLAPY2, SNRM2
420 EXTERNAL lsame, sdot, slamch, slapy2, snrm2
421* ..
422* .. External Subroutines ..
423 EXTERNAL sgemv, slacpy, slag2, stgexc, stgsyl, xerbla
424* ..
425* .. Intrinsic Functions ..
426 INTRINSIC max, min, sqrt
427* ..
428* .. Executable Statements ..
429*
430* Decode and test the input parameters
431*
432 wantbh = lsame( job, 'B' )
433 wants = lsame( job, 'E' ) .OR. wantbh
434 wantdf = lsame( job, 'V' ) .OR. wantbh
435*
436 somcon = lsame( howmny, 'S' )
437*
438 info = 0
439 lquery = ( lwork.EQ.-1 )
440*
441 IF( .NOT.wants .AND. .NOT.wantdf ) THEN
442 info = -1
443 ELSE IF( .NOT.lsame( howmny, 'A' ) .AND. .NOT.somcon ) THEN
444 info = -2
445 ELSE IF( n.LT.0 ) THEN
446 info = -4
447 ELSE IF( lda.LT.max( 1, n ) ) THEN
448 info = -6
449 ELSE IF( ldb.LT.max( 1, n ) ) THEN
450 info = -8
451 ELSE IF( wants .AND. ldvl.LT.n ) THEN
452 info = -10
453 ELSE IF( wants .AND. ldvr.LT.n ) THEN
454 info = -12
455 ELSE
456*
457* Set M to the number of eigenpairs for which condition numbers
458* are required, and test MM.
459*
460 IF( somcon ) THEN
461 m = 0
462 pair = .false.
463 DO 10 k = 1, n
464 IF( pair ) THEN
465 pair = .false.
466 ELSE
467 IF( k.LT.n ) THEN
468 IF( a( k+1, k ).EQ.zero ) THEN
469 IF( SELECT( k ) )
470 $ m = m + 1
471 ELSE
472 pair = .true.
473 IF( SELECT( k ) .OR. SELECT( k+1 ) )
474 $ m = m + 2
475 END IF
476 ELSE
477 IF( SELECT( n ) )
478 $ m = m + 1
479 END IF
480 END IF
481 10 CONTINUE
482 ELSE
483 m = n
484 END IF
485*
486 IF( n.EQ.0 ) THEN
487 lwmin = 1
488 ELSE IF( lsame( job, 'V' ) .OR. lsame( job, 'B' ) ) THEN
489 lwmin = 2*n*( n + 2 ) + 16
490 ELSE
491 lwmin = n
492 END IF
493 work( 1 ) = lwmin
494*
495 IF( mm.LT.m ) THEN
496 info = -15
497 ELSE IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
498 info = -18
499 END IF
500 END IF
501*
502 IF( info.NE.0 ) THEN
503 CALL xerbla( 'STGSNA', -info )
504 RETURN
505 ELSE IF( lquery ) THEN
506 RETURN
507 END IF
508*
509* Quick return if possible
510*
511 IF( n.EQ.0 )
512 $ RETURN
513*
514* Get machine constants
515*
516 eps = slamch( 'P' )
517 smlnum = slamch( 'S' ) / eps
518 ks = 0
519 pair = .false.
520*
521 DO 20 k = 1, n
522*
523* Determine whether A(k,k) begins a 1-by-1 or 2-by-2 block.
524*
525 IF( pair ) THEN
526 pair = .false.
527 GO TO 20
528 ELSE
529 IF( k.LT.n )
530 $ pair = a( k+1, k ).NE.zero
531 END IF
532*
533* Determine whether condition numbers are required for the k-th
534* eigenpair.
535*
536 IF( somcon ) THEN
537 IF( pair ) THEN
538 IF( .NOT.SELECT( k ) .AND. .NOT.SELECT( k+1 ) )
539 $ GO TO 20
540 ELSE
541 IF( .NOT.SELECT( k ) )
542 $ GO TO 20
543 END IF
544 END IF
545*
546 ks = ks + 1
547*
548 IF( wants ) THEN
549*
550* Compute the reciprocal condition number of the k-th
551* eigenvalue.
552*
553 IF( pair ) THEN
554*
555* Complex eigenvalue pair.
556*
557 rnrm = slapy2( snrm2( n, vr( 1, ks ), 1 ),
558 $ snrm2( n, vr( 1, ks+1 ), 1 ) )
559 lnrm = slapy2( snrm2( n, vl( 1, ks ), 1 ),
560 $ snrm2( n, vl( 1, ks+1 ), 1 ) )
561 CALL sgemv( 'N', n, n, one, a, lda, vr( 1, ks ), 1, zero,
562 $ work, 1 )
563 tmprr = sdot( n, work, 1, vl( 1, ks ), 1 )
564 tmpri = sdot( n, work, 1, vl( 1, ks+1 ), 1 )
565 CALL sgemv( 'N', n, n, one, a, lda, vr( 1, ks+1 ), 1,
566 $ zero, work, 1 )
567 tmpii = sdot( n, work, 1, vl( 1, ks+1 ), 1 )
568 tmpir = sdot( n, work, 1, vl( 1, ks ), 1 )
569 uhav = tmprr + tmpii
570 uhavi = tmpir - tmpri
571 CALL sgemv( 'N', n, n, one, b, ldb, vr( 1, ks ), 1, zero,
572 $ work, 1 )
573 tmprr = sdot( n, work, 1, vl( 1, ks ), 1 )
574 tmpri = sdot( n, work, 1, vl( 1, ks+1 ), 1 )
575 CALL sgemv( 'N', n, n, one, b, ldb, vr( 1, ks+1 ), 1,
576 $ zero, work, 1 )
577 tmpii = sdot( n, work, 1, vl( 1, ks+1 ), 1 )
578 tmpir = sdot( n, work, 1, vl( 1, ks ), 1 )
579 uhbv = tmprr + tmpii
580 uhbvi = tmpir - tmpri
581 uhav = slapy2( uhav, uhavi )
582 uhbv = slapy2( uhbv, uhbvi )
583 cond = slapy2( uhav, uhbv )
584 s( ks ) = cond / ( rnrm*lnrm )
585 s( ks+1 ) = s( ks )
586*
587 ELSE
588*
589* Real eigenvalue.
590*
591 rnrm = snrm2( n, vr( 1, ks ), 1 )
592 lnrm = snrm2( n, vl( 1, ks ), 1 )
593 CALL sgemv( 'N', n, n, one, a, lda, vr( 1, ks ), 1, zero,
594 $ work, 1 )
595 uhav = sdot( n, work, 1, vl( 1, ks ), 1 )
596 CALL sgemv( 'N', n, n, one, b, ldb, vr( 1, ks ), 1, zero,
597 $ work, 1 )
598 uhbv = sdot( n, work, 1, vl( 1, ks ), 1 )
599 cond = slapy2( uhav, uhbv )
600 IF( cond.EQ.zero ) THEN
601 s( ks ) = -one
602 ELSE
603 s( ks ) = cond / ( rnrm*lnrm )
604 END IF
605 END IF
606 END IF
607*
608 IF( wantdf ) THEN
609 IF( n.EQ.1 ) THEN
610 dif( ks ) = slapy2( a( 1, 1 ), b( 1, 1 ) )
611 GO TO 20
612 END IF
613*
614* Estimate the reciprocal condition number of the k-th
615* eigenvectors.
616 IF( pair ) THEN
617*
618* Copy the 2-by 2 pencil beginning at (A(k,k), B(k, k)).
619* Compute the eigenvalue(s) at position K.
620*
621 work( 1 ) = a( k, k )
622 work( 2 ) = a( k+1, k )
623 work( 3 ) = a( k, k+1 )
624 work( 4 ) = a( k+1, k+1 )
625 work( 5 ) = b( k, k )
626 work( 6 ) = b( k+1, k )
627 work( 7 ) = b( k, k+1 )
628 work( 8 ) = b( k+1, k+1 )
629 CALL slag2( work, 2, work( 5 ), 2, smlnum*eps, beta,
630 $ dummy1( 1 ), alphar, dummy( 1 ), alphai )
631 alprqt = one
632 c1 = two*( alphar*alphar+alphai*alphai+beta*beta )
633 c2 = four*beta*beta*alphai*alphai
634 root1 = c1 + sqrt( c1*c1-4.0*c2 )
635 root2 = c2 / root1
636 root1 = root1 / two
637 cond = min( sqrt( root1 ), sqrt( root2 ) )
638 END IF
639*
640* Copy the matrix (A, B) to the array WORK and swap the
641* diagonal block beginning at A(k,k) to the (1,1) position.
642*
643 CALL slacpy( 'Full', n, n, a, lda, work, n )
644 CALL slacpy( 'Full', n, n, b, ldb, work( n*n+1 ), n )
645 ifst = k
646 ilst = 1
647*
648 CALL stgexc( .false., .false., n, work, n, work( n*n+1 ), n,
649 $ dummy, 1, dummy1, 1, ifst, ilst,
650 $ work( n*n*2+1 ), lwork-2*n*n, ierr )
651*
652 IF( ierr.GT.0 ) THEN
653*
654* Ill-conditioned problem - swap rejected.
655*
656 dif( ks ) = zero
657 ELSE
658*
659* Reordering successful, solve generalized Sylvester
660* equation for R and L,
661* A22 * R - L * A11 = A12
662* B22 * R - L * B11 = B12,
663* and compute estimate of Difl((A11,B11), (A22, B22)).
664*
665 n1 = 1
666 IF( work( 2 ).NE.zero )
667 $ n1 = 2
668 n2 = n - n1
669 IF( n2.EQ.0 ) THEN
670 dif( ks ) = cond
671 ELSE
672 i = n*n + 1
673 iz = 2*n*n + 1
674 CALL stgsyl( 'N', difdri, n2, n1, work( n*n1+n1+1 ),
675 $ n, work, n, work( n1+1 ), n,
676 $ work( n*n1+n1+i ), n, work( i ), n,
677 $ work( n1+i ), n, scale, dif( ks ),
678 $ work( iz+1 ), lwork-2*n*n, iwork, ierr )
679*
680 IF( pair )
681 $ dif( ks ) = min( max( one, alprqt )*dif( ks ),
682 $ cond )
683 END IF
684 END IF
685 IF( pair )
686 $ dif( ks+1 ) = dif( ks )
687 END IF
688 IF( pair )
689 $ ks = ks + 1
690*
691 20 CONTINUE
692 work( 1 ) = lwmin
693 RETURN
694*
695* End of STGSNA
696*
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
real function slapy2(X, Y)
SLAPY2 returns sqrt(x2+y2).
Definition: slapy2.f:63
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine stgexc(WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, ILST, WORK, LWORK, INFO)
STGEXC
Definition: stgexc.f:220
subroutine slag2(A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2, WI)
SLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary ...
Definition: slag2.f:156
subroutine stgsyl(TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO)
STGSYL
Definition: stgsyl.f:299
real function sdot(N, SX, INCX, SY, INCY)
SDOT
Definition: sdot.f:82
real(wp) function snrm2(n, x, incx)
SNRM2
Definition: snrm2.f90:89
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:156
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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