LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ 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.

Definition at line 281 of file dgges.f.

284*
285* -- LAPACK driver routine --
286* -- LAPACK is a software package provided by Univ. of Tennessee, --
287* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
288*
289* .. Scalar Arguments ..
290 CHARACTER JOBVSL, JOBVSR, SORT
291 INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
292* ..
293* .. Array Arguments ..
294 LOGICAL BWORK( * )
295 DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
296 $ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
297 $ VSR( LDVSR, * ), WORK( * )
298* ..
299* .. Function Arguments ..
300 LOGICAL SELCTG
301 EXTERNAL selctg
302* ..
303*
304* =====================================================================
305*
306* .. Parameters ..
307 DOUBLE PRECISION ZERO, ONE
308 parameter( zero = 0.0d+0, one = 1.0d+0 )
309* ..
310* .. Local Scalars ..
311 LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
312 $ LQUERY, LST2SL, WANTST
313 INTEGER I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT,
314 $ ILO, IP, IRIGHT, IROWS, ITAU, IWRK, MAXWRK,
315 $ MINWRK
316 DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
317 $ PVSR, SAFMAX, SAFMIN, SMLNUM
318* ..
319* .. Local Arrays ..
320 INTEGER IDUM( 1 )
321 DOUBLE PRECISION DIF( 2 )
322* ..
323* .. External Subroutines ..
324 EXTERNAL dgeqrf, dggbak, dggbal, dgghrd, dhgeqz, dlacpy,
326* ..
327* .. External Functions ..
328 LOGICAL LSAME
329 INTEGER ILAENV
330 DOUBLE PRECISION DLAMCH, DLANGE
331 EXTERNAL lsame, ilaenv, dlamch, dlange
332* ..
333* .. Intrinsic Functions ..
334 INTRINSIC abs, max, sqrt
335* ..
336* .. Executable Statements ..
337*
338* Decode the input arguments
339*
340 IF( lsame( jobvsl, 'N' ) ) THEN
341 ijobvl = 1
342 ilvsl = .false.
343 ELSE IF( lsame( jobvsl, 'V' ) ) THEN
344 ijobvl = 2
345 ilvsl = .true.
346 ELSE
347 ijobvl = -1
348 ilvsl = .false.
349 END IF
350*
351 IF( lsame( jobvsr, 'N' ) ) THEN
352 ijobvr = 1
353 ilvsr = .false.
354 ELSE IF( lsame( jobvsr, 'V' ) ) THEN
355 ijobvr = 2
356 ilvsr = .true.
357 ELSE
358 ijobvr = -1
359 ilvsr = .false.
360 END IF
361*
362 wantst = lsame( sort, 'S' )
363*
364* Test the input arguments
365*
366 info = 0
367 lquery = ( lwork.EQ.-1 )
368 IF( ijobvl.LE.0 ) THEN
369 info = -1
370 ELSE IF( ijobvr.LE.0 ) THEN
371 info = -2
372 ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
373 info = -3
374 ELSE IF( n.LT.0 ) THEN
375 info = -5
376 ELSE IF( lda.LT.max( 1, n ) ) THEN
377 info = -7
378 ELSE IF( ldb.LT.max( 1, n ) ) THEN
379 info = -9
380 ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
381 info = -15
382 ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
383 info = -17
384 END IF
385*
386* Compute workspace
387* (Note: Comments in the code beginning "Workspace:" describe the
388* minimal amount of workspace needed at that point in the code,
389* as well as the preferred amount for good performance.
390* NB refers to the optimal block size for the immediately
391* following subroutine, as returned by ILAENV.)
392*
393 IF( info.EQ.0 ) THEN
394 IF( n.GT.0 )THEN
395 minwrk = max( 8*n, 6*n + 16 )
396 maxwrk = minwrk - n +
397 $ n*ilaenv( 1, 'DGEQRF', ' ', n, 1, n, 0 )
398 maxwrk = max( maxwrk, minwrk - n +
399 $ n*ilaenv( 1, 'DORMQR', ' ', n, 1, n, -1 ) )
400 IF( ilvsl ) THEN
401 maxwrk = max( maxwrk, minwrk - n +
402 $ n*ilaenv( 1, 'DORGQR', ' ', n, 1, n, -1 ) )
403 END IF
404 ELSE
405 minwrk = 1
406 maxwrk = 1
407 END IF
408 work( 1 ) = maxwrk
409*
410 IF( lwork.LT.minwrk .AND. .NOT.lquery )
411 $ info = -19
412 END IF
413*
414 IF( info.NE.0 ) THEN
415 CALL xerbla( 'DGGES ', -info )
416 RETURN
417 ELSE IF( lquery ) THEN
418 RETURN
419 END IF
420*
421* Quick return if possible
422*
423 IF( n.EQ.0 ) THEN
424 sdim = 0
425 RETURN
426 END IF
427*
428* Get machine constants
429*
430 eps = dlamch( 'P' )
431 safmin = dlamch( 'S' )
432 safmax = one / safmin
433 smlnum = sqrt( safmin ) / eps
434 bignum = one / smlnum
435*
436* Scale A if max element outside range [SMLNUM,BIGNUM]
437*
438 anrm = dlange( 'M', n, n, a, lda, work )
439 ilascl = .false.
440 IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
441 anrmto = smlnum
442 ilascl = .true.
443 ELSE IF( anrm.GT.bignum ) THEN
444 anrmto = bignum
445 ilascl = .true.
446 END IF
447 IF( ilascl )
448 $ CALL dlascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
449*
450* Scale B if max element outside range [SMLNUM,BIGNUM]
451*
452 bnrm = dlange( 'M', n, n, b, ldb, work )
453 ilbscl = .false.
454 IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
455 bnrmto = smlnum
456 ilbscl = .true.
457 ELSE IF( bnrm.GT.bignum ) THEN
458 bnrmto = bignum
459 ilbscl = .true.
460 END IF
461 IF( ilbscl )
462 $ CALL dlascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
463*
464* Permute the matrix to make it more nearly triangular
465* (Workspace: need 6*N + 2*N space for storing balancing factors)
466*
467 ileft = 1
468 iright = n + 1
469 iwrk = iright + n
470 CALL dggbal( 'P', n, a, lda, b, ldb, ilo, ihi, work( ileft ),
471 $ work( iright ), work( iwrk ), ierr )
472*
473* Reduce B to triangular form (QR decomposition of B)
474* (Workspace: need N, prefer N*NB)
475*
476 irows = ihi + 1 - ilo
477 icols = n + 1 - ilo
478 itau = iwrk
479 iwrk = itau + irows
480 CALL dgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
481 $ work( iwrk ), lwork+1-iwrk, ierr )
482*
483* Apply the orthogonal transformation to matrix A
484* (Workspace: need N, prefer N*NB)
485*
486 CALL dormqr( 'L', 'T', irows, icols, irows, b( ilo, ilo ), ldb,
487 $ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
488 $ lwork+1-iwrk, ierr )
489*
490* Initialize VSL
491* (Workspace: need N, prefer N*NB)
492*
493 IF( ilvsl ) THEN
494 CALL dlaset( 'Full', n, n, zero, one, vsl, ldvsl )
495 IF( irows.GT.1 ) THEN
496 CALL dlacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
497 $ vsl( ilo+1, ilo ), ldvsl )
498 END IF
499 CALL dorgqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
500 $ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
501 END IF
502*
503* Initialize VSR
504*
505 IF( ilvsr )
506 $ CALL dlaset( 'Full', n, n, zero, one, vsr, ldvsr )
507*
508* Reduce to generalized Hessenberg form
509* (Workspace: none needed)
510*
511 CALL dgghrd( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
512 $ ldvsl, vsr, ldvsr, ierr )
513*
514* Perform QZ algorithm, computing Schur vectors if desired
515* (Workspace: need N)
516*
517 iwrk = itau
518 CALL dhgeqz( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
519 $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
520 $ work( iwrk ), lwork+1-iwrk, ierr )
521 IF( ierr.NE.0 ) THEN
522 IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
523 info = ierr
524 ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
525 info = ierr - n
526 ELSE
527 info = n + 1
528 END IF
529 GO TO 50
530 END IF
531*
532* Sort eigenvalues ALPHA/BETA if desired
533* (Workspace: need 4*N+16 )
534*
535 sdim = 0
536 IF( wantst ) THEN
537*
538* Undo scaling on eigenvalues before SELCTGing
539*
540 IF( ilascl ) THEN
541 CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n,
542 $ ierr )
543 CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n,
544 $ ierr )
545 END IF
546 IF( ilbscl )
547 $ CALL dlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
548*
549* Select eigenvalues
550*
551 DO 10 i = 1, n
552 bwork( i ) = selctg( alphar( i ), alphai( i ), beta( i ) )
553 10 CONTINUE
554*
555 CALL dtgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb, alphar,
556 $ alphai, beta, vsl, ldvsl, vsr, ldvsr, sdim, pvsl,
557 $ pvsr, dif, work( iwrk ), lwork-iwrk+1, idum, 1,
558 $ ierr )
559 IF( ierr.EQ.1 )
560 $ info = n + 3
561*
562 END IF
563*
564* Apply back-permutation to VSL and VSR
565* (Workspace: none needed)
566*
567 IF( ilvsl )
568 $ CALL dggbak( 'P', 'L', n, ilo, ihi, work( ileft ),
569 $ work( iright ), n, vsl, ldvsl, ierr )
570*
571 IF( ilvsr )
572 $ CALL dggbak( 'P', 'R', n, ilo, ihi, work( ileft ),
573 $ work( iright ), n, vsr, ldvsr, ierr )
574*
575* Check if unscaling would cause over/underflow, if so, rescale
576* (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
577* B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
578*
579 IF( ilascl ) THEN
580 DO 20 i = 1, n
581 IF( alphai( i ).NE.zero ) THEN
582 IF( ( alphar( i ) / safmax ).GT.( anrmto / anrm ) .OR.
583 $ ( safmin / alphar( i ) ).GT.( anrm / anrmto ) ) THEN
584 work( 1 ) = abs( a( i, i ) / alphar( i ) )
585 beta( i ) = beta( i )*work( 1 )
586 alphar( i ) = alphar( i )*work( 1 )
587 alphai( i ) = alphai( i )*work( 1 )
588 ELSE IF( ( alphai( i ) / safmax ).GT.
589 $ ( anrmto / anrm ) .OR.
590 $ ( safmin / alphai( i ) ).GT.( anrm / anrmto ) )
591 $ THEN
592 work( 1 ) = abs( a( i, i+1 ) / alphai( i ) )
593 beta( i ) = beta( i )*work( 1 )
594 alphar( i ) = alphar( i )*work( 1 )
595 alphai( i ) = alphai( i )*work( 1 )
596 END IF
597 END IF
598 20 CONTINUE
599 END IF
600*
601 IF( ilbscl ) THEN
602 DO 30 i = 1, n
603 IF( alphai( i ).NE.zero ) THEN
604 IF( ( beta( i ) / safmax ).GT.( bnrmto / bnrm ) .OR.
605 $ ( safmin / beta( i ) ).GT.( bnrm / bnrmto ) ) THEN
606 work( 1 ) = abs( b( i, i ) / beta( i ) )
607 beta( i ) = beta( i )*work( 1 )
608 alphar( i ) = alphar( i )*work( 1 )
609 alphai( i ) = alphai( i )*work( 1 )
610 END IF
611 END IF
612 30 CONTINUE
613 END IF
614*
615* Undo scaling
616*
617 IF( ilascl ) THEN
618 CALL dlascl( 'H', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
619 CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n, ierr )
620 CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n, ierr )
621 END IF
622*
623 IF( ilbscl ) THEN
624 CALL dlascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
625 CALL dlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
626 END IF
627*
628 IF( wantst ) THEN
629*
630* Check if reordering is correct
631*
632 lastsl = .true.
633 lst2sl = .true.
634 sdim = 0
635 ip = 0
636 DO 40 i = 1, n
637 cursl = selctg( alphar( i ), alphai( i ), beta( i ) )
638 IF( alphai( i ).EQ.zero ) THEN
639 IF( cursl )
640 $ sdim = sdim + 1
641 ip = 0
642 IF( cursl .AND. .NOT.lastsl )
643 $ info = n + 2
644 ELSE
645 IF( ip.EQ.1 ) THEN
646*
647* Last eigenvalue of conjugate pair
648*
649 cursl = cursl .OR. lastsl
650 lastsl = cursl
651 IF( cursl )
652 $ sdim = sdim + 2
653 ip = -1
654 IF( cursl .AND. .NOT.lst2sl )
655 $ info = n + 2
656 ELSE
657*
658* First eigenvalue of conjugate pair
659*
660 ip = 1
661 END IF
662 END IF
663 lst2sl = lastsl
664 lastsl = cursl
665 40 CONTINUE
666*
667 END IF
668*
669 50 CONTINUE
670*
671 work( 1 ) = maxwrk
672*
673 RETURN
674*
675* End of DGGES
676*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine dgeqrf(m, n, a, lda, tau, work, lwork, info)
DGEQRF
Definition dgeqrf.f:146
subroutine dggbak(job, side, n, ilo, ihi, lscale, rscale, m, v, ldv, info)
DGGBAK
Definition dggbak.f:147
subroutine dggbal(job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info)
DGGBAL
Definition dggbal.f:177
subroutine dgghrd(compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info)
DGGHRD
Definition dgghrd.f:207
subroutine dhgeqz(job, compq, compz, n, ilo, ihi, h, ldh, t, ldt, alphar, alphai, beta, q, ldq, z, ldz, work, lwork, info)
DHGEQZ
Definition dhgeqz.f:304
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:162
subroutine dlacpy(uplo, m, n, a, lda, b, ldb)
DLACPY copies all or part of one two-dimensional array to another.
Definition dlacpy.f:103
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
double precision function dlange(norm, m, n, a, lda, work)
DLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition dlange.f:114
subroutine 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:143
subroutine dlaset(uplo, m, n, alpha, beta, a, lda)
DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition dlaset.f:110
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
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:451
subroutine dorgqr(m, n, k, a, lda, tau, work, lwork, info)
DORGQR
Definition dorgqr.f:128
subroutine dormqr(side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
DORMQR
Definition dormqr.f:167
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