LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ dgges()

subroutine dgges ( character  JOBVSL,
character  JOBVSR,
character  SORT,
external  SELCTG,
integer  N,
double precision, dimension( lda, * )  A,
integer  LDA,
double precision, dimension( ldb, * )  B,
integer  LDB,
integer  SDIM,
double precision, dimension( * )  ALPHAR,
double precision, dimension( * )  ALPHAI,
double precision, dimension( * )  BETA,
double precision, dimension( ldvsl, * )  VSL,
integer  LDVSL,
double precision, dimension( ldvsr, * )  VSR,
integer  LDVSR,
double precision, dimension( * )  WORK,
integer  LWORK,
logical, dimension( * )  BWORK,
integer  INFO 
)

DGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

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

Purpose:
 DGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B),
 the generalized eigenvalues, the generalized real Schur form (S,T),
 optionally, the left and/or right matrices of Schur vectors (VSL and
 VSR). This gives the generalized Schur factorization

          (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )

 Optionally, it also orders the eigenvalues so that a selected cluster
 of eigenvalues appears in the leading diagonal blocks of the upper
 quasi-triangular matrix S and the upper triangular matrix T.The
 leading columns of VSL and VSR then form an orthonormal basis for the
 corresponding left and right eigenspaces (deflating subspaces).

 (If only the generalized eigenvalues are needed, use the driver
 DGGEV instead, which is faster.)

 A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
 or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
 usually represented as the pair (alpha,beta), as there is a
 reasonable interpretation for beta=0 or both being zero.

 A pair of matrices (S,T) is in generalized real Schur form if T is
 upper triangular with non-negative diagonal and S is block upper
 triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
 to real generalized eigenvalues, while 2-by-2 blocks of S will be
 "standardized" by making the corresponding elements of T have the
 form:
         [  a  0  ]
         [  0  b  ]

 and the pair of corresponding 2-by-2 blocks in S and T will have a
 complex conjugate pair of generalized eigenvalues.
Parameters
[in]JOBVSL
          JOBVSL is CHARACTER*1
          = 'N':  do not compute the left Schur vectors;
          = 'V':  compute the left Schur vectors.
[in]JOBVSR
          JOBVSR is CHARACTER*1
          = 'N':  do not compute the right Schur vectors;
          = 'V':  compute the right Schur vectors.
[in]SORT
          SORT is CHARACTER*1
          Specifies whether or not to order the eigenvalues on the
          diagonal of the generalized Schur form.
          = 'N':  Eigenvalues are not ordered;
          = 'S':  Eigenvalues are ordered (see SELCTG);
[in]SELCTG
          SELCTG is a LOGICAL FUNCTION of three DOUBLE PRECISION arguments
          SELCTG must be declared EXTERNAL in the calling subroutine.
          If SORT = 'N', SELCTG is not referenced.
          If SORT = 'S', SELCTG is used to select eigenvalues to sort
          to the top left of the Schur form.
          An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
          SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
          one of a complex conjugate pair of eigenvalues is selected,
          then both complex eigenvalues are selected.

          Note that in the ill-conditioned case, a selected complex
          eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
          BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
          in this case.
[in]N
          N is INTEGER
          The order of the matrices A, B, VSL, and VSR.  N >= 0.
[in,out]A
          A is DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the first of the pair of matrices.
          On exit, A has been overwritten by its generalized Schur
          form S.
[in]LDA
          LDA is INTEGER
          The leading dimension of A.  LDA >= max(1,N).
[in,out]B
          B is DOUBLE PRECISION array, dimension (LDB, N)
          On entry, the second of the pair of matrices.
          On exit, B has been overwritten by its generalized Schur
          form T.
[in]LDB
          LDB is INTEGER
          The leading dimension of B.  LDB >= max(1,N).
[out]SDIM
          SDIM is INTEGER
          If SORT = 'N', SDIM = 0.
          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
          for which SELCTG is true.  (Complex conjugate pairs for which
          SELCTG is true for either eigenvalue count as 2.)
[out]ALPHAR
          ALPHAR is DOUBLE PRECISION array, dimension (N)
[out]ALPHAI
          ALPHAI is DOUBLE PRECISION array, dimension (N)
[out]BETA
          BETA is DOUBLE PRECISION array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i,
          and  BETA(j),j=1,...,N are the diagonals of the complex Schur
          form (S,T) that would result if the 2-by-2 diagonal blocks of
          the real Schur form of (A,B) were further reduced to
          triangular form using 2-by-2 complex unitary transformations.
          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
          positive, then the j-th and (j+1)-st eigenvalues are a
          complex conjugate pair, with ALPHAI(j+1) negative.

          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
          may easily over- or underflow, and BETA(j) may even be zero.
          Thus, the user should avoid naively computing the ratio.
          However, ALPHAR and ALPHAI will be always less than and
          usually comparable with norm(A) in magnitude, and BETA always
          less than and usually comparable with norm(B).
[out]VSL
          VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
          If JOBVSL = 'V', VSL will contain the left Schur vectors.
          Not referenced if JOBVSL = 'N'.
[in]LDVSL
          LDVSL is INTEGER
          The leading dimension of the matrix VSL. LDVSL >=1, and
          if JOBVSL = 'V', LDVSL >= N.
[out]VSR
          VSR is DOUBLE PRECISION array, dimension (LDVSR,N)
          If JOBVSR = 'V', VSR will contain the right Schur vectors.
          Not referenced if JOBVSR = 'N'.
[in]LDVSR
          LDVSR is INTEGER
          The leading dimension of the matrix VSR. LDVSR >= 1, and
          if JOBVSR = 'V', LDVSR >= N.
[out]WORK
          WORK is DOUBLE PRECISION 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.
          If N = 0, LWORK >= 1, else LWORK >= 8*N+16.
          For good performance , LWORK must generally be larger.

          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]BWORK
          BWORK is LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'.
[out]INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  (A,B) are not in Schur
                form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
                be correct for j=INFO+1,...,N.
          > N:  =N+1: other than QZ iteration failed in DHGEQZ.
                =N+2: after reordering, roundoff changed values of
                      some complex eigenvalues so that leading
                      eigenvalues in the Generalized Schur form no
                      longer satisfy SELCTG=.TRUE.  This could also
                      be caused due to scaling.
                =N+3: reordering failed in DTGSEN.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
December 2016

Definition at line 286 of file dgges.f.

286 *
287 * -- LAPACK driver routine (version 3.7.0) --
288 * -- LAPACK is a software package provided by Univ. of Tennessee, --
289 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
290 * December 2016
291 *
292 * .. Scalar Arguments ..
293  CHARACTER jobvsl, jobvsr, sort
294  INTEGER info, lda, ldb, ldvsl, ldvsr, lwork, n, sdim
295 * ..
296 * .. Array Arguments ..
297  LOGICAL bwork( * )
298  DOUBLE PRECISION a( lda, * ), alphai( * ), alphar( * ),
299  $ b( ldb, * ), beta( * ), vsl( ldvsl, * ),
300  $ vsr( ldvsr, * ), work( * )
301 * ..
302 * .. Function Arguments ..
303  LOGICAL selctg
304  EXTERNAL selctg
305 * ..
306 *
307 * =====================================================================
308 *
309 * .. Parameters ..
310  DOUBLE PRECISION zero, one
311  parameter( zero = 0.0d+0, one = 1.0d+0 )
312 * ..
313 * .. Local Scalars ..
314  LOGICAL cursl, ilascl, ilbscl, ilvsl, ilvsr, lastsl,
315  $ lquery, lst2sl, wantst
316  INTEGER i, icols, ierr, ihi, ijobvl, ijobvr, ileft,
317  $ ilo, ip, iright, irows, itau, iwrk, maxwrk,
318  $ minwrk
319  DOUBLE PRECISION anrm, anrmto, bignum, bnrm, bnrmto, eps, pvsl,
320  $ pvsr, safmax, safmin, smlnum
321 * ..
322 * .. Local Arrays ..
323  INTEGER idum( 1 )
324  DOUBLE PRECISION dif( 2 )
325 * ..
326 * .. External Subroutines ..
327  EXTERNAL dgeqrf, dggbak, dggbal, dgghrd, dhgeqz, dlabad,
329  $ xerbla
330 * ..
331 * .. External Functions ..
332  LOGICAL lsame
333  INTEGER ilaenv
334  DOUBLE PRECISION dlamch, dlange
335  EXTERNAL lsame, ilaenv, dlamch, dlange
336 * ..
337 * .. Intrinsic Functions ..
338  INTRINSIC abs, max, sqrt
339 * ..
340 * .. Executable Statements ..
341 *
342 * Decode the input arguments
343 *
344  IF( lsame( jobvsl, 'N' ) ) THEN
345  ijobvl = 1
346  ilvsl = .false.
347  ELSE IF( lsame( jobvsl, 'V' ) ) THEN
348  ijobvl = 2
349  ilvsl = .true.
350  ELSE
351  ijobvl = -1
352  ilvsl = .false.
353  END IF
354 *
355  IF( lsame( jobvsr, 'N' ) ) THEN
356  ijobvr = 1
357  ilvsr = .false.
358  ELSE IF( lsame( jobvsr, 'V' ) ) THEN
359  ijobvr = 2
360  ilvsr = .true.
361  ELSE
362  ijobvr = -1
363  ilvsr = .false.
364  END IF
365 *
366  wantst = lsame( sort, 'S' )
367 *
368 * Test the input arguments
369 *
370  info = 0
371  lquery = ( lwork.EQ.-1 )
372  IF( ijobvl.LE.0 ) THEN
373  info = -1
374  ELSE IF( ijobvr.LE.0 ) THEN
375  info = -2
376  ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
377  info = -3
378  ELSE IF( n.LT.0 ) THEN
379  info = -5
380  ELSE IF( lda.LT.max( 1, n ) ) THEN
381  info = -7
382  ELSE IF( ldb.LT.max( 1, n ) ) THEN
383  info = -9
384  ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
385  info = -15
386  ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
387  info = -17
388  END IF
389 *
390 * Compute workspace
391 * (Note: Comments in the code beginning "Workspace:" describe the
392 * minimal amount of workspace needed at that point in the code,
393 * as well as the preferred amount for good performance.
394 * NB refers to the optimal block size for the immediately
395 * following subroutine, as returned by ILAENV.)
396 *
397  IF( info.EQ.0 ) THEN
398  IF( n.GT.0 )THEN
399  minwrk = max( 8*n, 6*n + 16 )
400  maxwrk = minwrk - n +
401  $ n*ilaenv( 1, 'DGEQRF', ' ', n, 1, n, 0 )
402  maxwrk = max( maxwrk, minwrk - n +
403  $ n*ilaenv( 1, 'DORMQR', ' ', n, 1, n, -1 ) )
404  IF( ilvsl ) THEN
405  maxwrk = max( maxwrk, minwrk - n +
406  $ n*ilaenv( 1, 'DORGQR', ' ', n, 1, n, -1 ) )
407  END IF
408  ELSE
409  minwrk = 1
410  maxwrk = 1
411  END IF
412  work( 1 ) = maxwrk
413 *
414  IF( lwork.LT.minwrk .AND. .NOT.lquery )
415  $ info = -19
416  END IF
417 *
418  IF( info.NE.0 ) THEN
419  CALL xerbla( 'DGGES ', -info )
420  RETURN
421  ELSE IF( lquery ) THEN
422  RETURN
423  END IF
424 *
425 * Quick return if possible
426 *
427  IF( n.EQ.0 ) THEN
428  sdim = 0
429  RETURN
430  END IF
431 *
432 * Get machine constants
433 *
434  eps = dlamch( 'P' )
435  safmin = dlamch( 'S' )
436  safmax = one / safmin
437  CALL dlabad( safmin, safmax )
438  smlnum = sqrt( safmin ) / eps
439  bignum = one / smlnum
440 *
441 * Scale A if max element outside range [SMLNUM,BIGNUM]
442 *
443  anrm = dlange( 'M', n, n, a, lda, work )
444  ilascl = .false.
445  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
446  anrmto = smlnum
447  ilascl = .true.
448  ELSE IF( anrm.GT.bignum ) THEN
449  anrmto = bignum
450  ilascl = .true.
451  END IF
452  IF( ilascl )
453  $ CALL dlascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
454 *
455 * Scale B if max element outside range [SMLNUM,BIGNUM]
456 *
457  bnrm = dlange( 'M', n, n, b, ldb, work )
458  ilbscl = .false.
459  IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
460  bnrmto = smlnum
461  ilbscl = .true.
462  ELSE IF( bnrm.GT.bignum ) THEN
463  bnrmto = bignum
464  ilbscl = .true.
465  END IF
466  IF( ilbscl )
467  $ CALL dlascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
468 *
469 * Permute the matrix to make it more nearly triangular
470 * (Workspace: need 6*N + 2*N space for storing balancing factors)
471 *
472  ileft = 1
473  iright = n + 1
474  iwrk = iright + n
475  CALL dggbal( 'P', n, a, lda, b, ldb, ilo, ihi, work( ileft ),
476  $ work( iright ), work( iwrk ), ierr )
477 *
478 * Reduce B to triangular form (QR decomposition of B)
479 * (Workspace: need N, prefer N*NB)
480 *
481  irows = ihi + 1 - ilo
482  icols = n + 1 - ilo
483  itau = iwrk
484  iwrk = itau + irows
485  CALL dgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
486  $ work( iwrk ), lwork+1-iwrk, ierr )
487 *
488 * Apply the orthogonal transformation to matrix A
489 * (Workspace: need N, prefer N*NB)
490 *
491  CALL dormqr( 'L', 'T', irows, icols, irows, b( ilo, ilo ), ldb,
492  $ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
493  $ lwork+1-iwrk, ierr )
494 *
495 * Initialize VSL
496 * (Workspace: need N, prefer N*NB)
497 *
498  IF( ilvsl ) THEN
499  CALL dlaset( 'Full', n, n, zero, one, vsl, ldvsl )
500  IF( irows.GT.1 ) THEN
501  CALL dlacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
502  $ vsl( ilo+1, ilo ), ldvsl )
503  END IF
504  CALL dorgqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
505  $ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
506  END IF
507 *
508 * Initialize VSR
509 *
510  IF( ilvsr )
511  $ CALL dlaset( 'Full', n, n, zero, one, vsr, ldvsr )
512 *
513 * Reduce to generalized Hessenberg form
514 * (Workspace: none needed)
515 *
516  CALL dgghrd( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
517  $ ldvsl, vsr, ldvsr, ierr )
518 *
519 * Perform QZ algorithm, computing Schur vectors if desired
520 * (Workspace: need N)
521 *
522  iwrk = itau
523  CALL dhgeqz( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
524  $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
525  $ work( iwrk ), lwork+1-iwrk, ierr )
526  IF( ierr.NE.0 ) THEN
527  IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
528  info = ierr
529  ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
530  info = ierr - n
531  ELSE
532  info = n + 1
533  END IF
534  GO TO 50
535  END IF
536 *
537 * Sort eigenvalues ALPHA/BETA if desired
538 * (Workspace: need 4*N+16 )
539 *
540  sdim = 0
541  IF( wantst ) THEN
542 *
543 * Undo scaling on eigenvalues before SELCTGing
544 *
545  IF( ilascl ) THEN
546  CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n,
547  $ ierr )
548  CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n,
549  $ ierr )
550  END IF
551  IF( ilbscl )
552  $ CALL dlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
553 *
554 * Select eigenvalues
555 *
556  DO 10 i = 1, n
557  bwork( i ) = selctg( alphar( i ), alphai( i ), beta( i ) )
558  10 CONTINUE
559 *
560  CALL dtgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb, alphar,
561  $ alphai, beta, vsl, ldvsl, vsr, ldvsr, sdim, pvsl,
562  $ pvsr, dif, work( iwrk ), lwork-iwrk+1, idum, 1,
563  $ ierr )
564  IF( ierr.EQ.1 )
565  $ info = n + 3
566 *
567  END IF
568 *
569 * Apply back-permutation to VSL and VSR
570 * (Workspace: none needed)
571 *
572  IF( ilvsl )
573  $ CALL dggbak( 'P', 'L', n, ilo, ihi, work( ileft ),
574  $ work( iright ), n, vsl, ldvsl, ierr )
575 *
576  IF( ilvsr )
577  $ CALL dggbak( 'P', 'R', n, ilo, ihi, work( ileft ),
578  $ work( iright ), n, vsr, ldvsr, ierr )
579 *
580 * Check if unscaling would cause over/underflow, if so, rescale
581 * (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
582 * B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
583 *
584  IF( ilascl ) THEN
585  DO 20 i = 1, n
586  IF( alphai( i ).NE.zero ) THEN
587  IF( ( alphar( i ) / safmax ).GT.( anrmto / anrm ) .OR.
588  $ ( safmin / alphar( i ) ).GT.( anrm / anrmto ) ) THEN
589  work( 1 ) = abs( a( i, i ) / alphar( i ) )
590  beta( i ) = beta( i )*work( 1 )
591  alphar( i ) = alphar( i )*work( 1 )
592  alphai( i ) = alphai( i )*work( 1 )
593  ELSE IF( ( alphai( i ) / safmax ).GT.
594  $ ( anrmto / anrm ) .OR.
595  $ ( safmin / alphai( i ) ).GT.( anrm / anrmto ) )
596  $ THEN
597  work( 1 ) = abs( a( i, i+1 ) / alphai( i ) )
598  beta( i ) = beta( i )*work( 1 )
599  alphar( i ) = alphar( i )*work( 1 )
600  alphai( i ) = alphai( i )*work( 1 )
601  END IF
602  END IF
603  20 CONTINUE
604  END IF
605 *
606  IF( ilbscl ) THEN
607  DO 30 i = 1, n
608  IF( alphai( i ).NE.zero ) THEN
609  IF( ( beta( i ) / safmax ).GT.( bnrmto / bnrm ) .OR.
610  $ ( safmin / beta( i ) ).GT.( bnrm / bnrmto ) ) THEN
611  work( 1 ) = abs( b( i, i ) / beta( i ) )
612  beta( i ) = beta( i )*work( 1 )
613  alphar( i ) = alphar( i )*work( 1 )
614  alphai( i ) = alphai( i )*work( 1 )
615  END IF
616  END IF
617  30 CONTINUE
618  END IF
619 *
620 * Undo scaling
621 *
622  IF( ilascl ) THEN
623  CALL dlascl( 'H', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
624  CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n, ierr )
625  CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n, ierr )
626  END IF
627 *
628  IF( ilbscl ) THEN
629  CALL dlascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
630  CALL dlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
631  END IF
632 *
633  IF( wantst ) THEN
634 *
635 * Check if reordering is correct
636 *
637  lastsl = .true.
638  lst2sl = .true.
639  sdim = 0
640  ip = 0
641  DO 40 i = 1, n
642  cursl = selctg( alphar( i ), alphai( i ), beta( i ) )
643  IF( alphai( i ).EQ.zero ) THEN
644  IF( cursl )
645  $ sdim = sdim + 1
646  ip = 0
647  IF( cursl .AND. .NOT.lastsl )
648  $ info = n + 2
649  ELSE
650  IF( ip.EQ.1 ) THEN
651 *
652 * Last eigenvalue of conjugate pair
653 *
654  cursl = cursl .OR. lastsl
655  lastsl = cursl
656  IF( cursl )
657  $ sdim = sdim + 2
658  ip = -1
659  IF( cursl .AND. .NOT.lst2sl )
660  $ info = n + 2
661  ELSE
662 *
663 * First eigenvalue of conjugate pair
664 *
665  ip = 1
666  END IF
667  END IF
668  lst2sl = lastsl
669  lastsl = cursl
670  40 CONTINUE
671 *
672  END IF
673 *
674  50 CONTINUE
675 *
676  work( 1 ) = maxwrk
677 *
678  RETURN
679 *
680 * End of DGGES
681 *
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:105
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
subroutine dtgsen(IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO)
DTGSEN
Definition: dtgsen.f:453
subroutine dggbak(JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, LDV, INFO)
DGGBAK
Definition: dggbak.f:149
subroutine dgghrd(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
DGGHRD
Definition: dgghrd.f:209
subroutine dlascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
DLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: dlascl.f:145
subroutine dggbal(JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO)
DGGBAL
Definition: dggbal.f:179
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:112
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: tstiee.f:83
subroutine dormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMQR
Definition: dormqr.f:169
subroutine dorgqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
DORGQR
Definition: dorgqr.f:130
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:116
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:306
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine dlabad(SMALL, LARGE)
DLABAD
Definition: dlabad.f:76
subroutine dgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
DGEQRF
Definition: dgeqrf.f:138
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