LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ dgges3()

subroutine dgges3 ( 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 
)

DGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

Download DGGES3 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 DGGES3 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 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 DLAQZ0.
                =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 279 of file dgges3.f.

282*
283* -- LAPACK driver routine --
284* -- LAPACK is a software package provided by Univ. of Tennessee, --
285* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
286*
287* .. Scalar Arguments ..
288 CHARACTER JOBVSL, JOBVSR, SORT
289 INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
290* ..
291* .. Array Arguments ..
292 LOGICAL BWORK( * )
293 DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
294 $ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
295 $ VSR( LDVSR, * ), WORK( * )
296* ..
297* .. Function Arguments ..
298 LOGICAL SELCTG
299 EXTERNAL selctg
300* ..
301*
302* =====================================================================
303*
304* .. Parameters ..
305 DOUBLE PRECISION ZERO, ONE
306 parameter( zero = 0.0d+0, one = 1.0d+0 )
307* ..
308* .. Local Scalars ..
309 LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
310 $ LQUERY, LST2SL, WANTST
311 INTEGER I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT,
312 $ ILO, IP, IRIGHT, IROWS, ITAU, IWRK, LWKOPT
313 DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL,
314 $ PVSR, SAFMAX, SAFMIN, SMLNUM
315* ..
316* .. Local Arrays ..
317 INTEGER IDUM( 1 )
318 DOUBLE PRECISION DIF( 2 )
319* ..
320* .. External Subroutines ..
321 EXTERNAL dgeqrf, dggbak, dggbal, dgghd3, dlaqz0, dlabad,
323 $ xerbla
324* ..
325* .. External Functions ..
326 LOGICAL LSAME
327 DOUBLE PRECISION DLAMCH, DLANGE
328 EXTERNAL lsame, dlamch, dlange
329* ..
330* .. Intrinsic Functions ..
331 INTRINSIC abs, max, sqrt
332* ..
333* .. Executable Statements ..
334*
335* Decode the input arguments
336*
337 IF( lsame( jobvsl, 'N' ) ) THEN
338 ijobvl = 1
339 ilvsl = .false.
340 ELSE IF( lsame( jobvsl, 'V' ) ) THEN
341 ijobvl = 2
342 ilvsl = .true.
343 ELSE
344 ijobvl = -1
345 ilvsl = .false.
346 END IF
347*
348 IF( lsame( jobvsr, 'N' ) ) THEN
349 ijobvr = 1
350 ilvsr = .false.
351 ELSE IF( lsame( jobvsr, 'V' ) ) THEN
352 ijobvr = 2
353 ilvsr = .true.
354 ELSE
355 ijobvr = -1
356 ilvsr = .false.
357 END IF
358*
359 wantst = lsame( sort, 'S' )
360*
361* Test the input arguments
362*
363 info = 0
364 lquery = ( lwork.EQ.-1 )
365 IF( ijobvl.LE.0 ) THEN
366 info = -1
367 ELSE IF( ijobvr.LE.0 ) THEN
368 info = -2
369 ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
370 info = -3
371 ELSE IF( n.LT.0 ) THEN
372 info = -5
373 ELSE IF( lda.LT.max( 1, n ) ) THEN
374 info = -7
375 ELSE IF( ldb.LT.max( 1, n ) ) THEN
376 info = -9
377 ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
378 info = -15
379 ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
380 info = -17
381 ELSE IF( lwork.LT.6*n+16 .AND. .NOT.lquery ) THEN
382 info = -19
383 END IF
384*
385* Compute workspace
386*
387 IF( info.EQ.0 ) THEN
388 CALL dgeqrf( n, n, b, ldb, work, work, -1, ierr )
389 lwkopt = max( 6*n+16, 3*n+int( work( 1 ) ) )
390 CALL dormqr( 'L', 'T', n, n, n, b, ldb, work, a, lda, work,
391 $ -1, ierr )
392 lwkopt = max( lwkopt, 3*n+int( work( 1 ) ) )
393 IF( ilvsl ) THEN
394 CALL dorgqr( n, n, n, vsl, ldvsl, work, work, -1, ierr )
395 lwkopt = max( lwkopt, 3*n+int( work( 1 ) ) )
396 END IF
397 CALL dgghd3( jobvsl, jobvsr, n, 1, n, a, lda, b, ldb, vsl,
398 $ ldvsl, vsr, ldvsr, work, -1, ierr )
399 lwkopt = max( lwkopt, 3*n+int( work( 1 ) ) )
400 CALL dlaqz0( 'S', jobvsl, jobvsr, n, 1, n, a, lda, b, ldb,
401 $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
402 $ work, -1, 0, ierr )
403 lwkopt = max( lwkopt, 2*n+int( work( 1 ) ) )
404 IF( wantst ) THEN
405 CALL dtgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb,
406 $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
407 $ sdim, pvsl, pvsr, dif, work, -1, idum, 1,
408 $ ierr )
409 lwkopt = max( lwkopt, 2*n+int( work( 1 ) ) )
410 END IF
411 work( 1 ) = lwkopt
412 END IF
413*
414 IF( info.NE.0 ) THEN
415 CALL xerbla( 'DGGES3 ', -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 CALL dlabad( safmin, safmax )
434 smlnum = sqrt( safmin ) / eps
435 bignum = one / smlnum
436*
437* Scale A if max element outside range [SMLNUM,BIGNUM]
438*
439 anrm = dlange( 'M', n, n, a, lda, work )
440 ilascl = .false.
441 IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
442 anrmto = smlnum
443 ilascl = .true.
444 ELSE IF( anrm.GT.bignum ) THEN
445 anrmto = bignum
446 ilascl = .true.
447 END IF
448 IF( ilascl )
449 $ CALL dlascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
450*
451* Scale B if max element outside range [SMLNUM,BIGNUM]
452*
453 bnrm = dlange( 'M', n, n, b, ldb, work )
454 ilbscl = .false.
455 IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
456 bnrmto = smlnum
457 ilbscl = .true.
458 ELSE IF( bnrm.GT.bignum ) THEN
459 bnrmto = bignum
460 ilbscl = .true.
461 END IF
462 IF( ilbscl )
463 $ CALL dlascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
464*
465* Permute the matrix to make it more nearly triangular
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*
475 irows = ihi + 1 - ilo
476 icols = n + 1 - ilo
477 itau = iwrk
478 iwrk = itau + irows
479 CALL dgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
480 $ work( iwrk ), lwork+1-iwrk, ierr )
481*
482* Apply the orthogonal transformation to matrix A
483*
484 CALL dormqr( 'L', 'T', irows, icols, irows, b( ilo, ilo ), ldb,
485 $ work( itau ), a( ilo, ilo ), lda, work( iwrk ),
486 $ lwork+1-iwrk, ierr )
487*
488* Initialize VSL
489*
490 IF( ilvsl ) THEN
491 CALL dlaset( 'Full', n, n, zero, one, vsl, ldvsl )
492 IF( irows.GT.1 ) THEN
493 CALL dlacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
494 $ vsl( ilo+1, ilo ), ldvsl )
495 END IF
496 CALL dorgqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
497 $ work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
498 END IF
499*
500* Initialize VSR
501*
502 IF( ilvsr )
503 $ CALL dlaset( 'Full', n, n, zero, one, vsr, ldvsr )
504*
505* Reduce to generalized Hessenberg form
506*
507 CALL dgghd3( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
508 $ ldvsl, vsr, ldvsr, work( iwrk ), lwork+1-iwrk,
509 $ ierr )
510*
511* Perform QZ algorithm, computing Schur vectors if desired
512*
513 iwrk = itau
514 CALL dlaqz0( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
515 $ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
516 $ work( iwrk ), lwork+1-iwrk, 0, ierr )
517 IF( ierr.NE.0 ) THEN
518 IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
519 info = ierr
520 ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
521 info = ierr - n
522 ELSE
523 info = n + 1
524 END IF
525 GO TO 50
526 END IF
527*
528* Sort eigenvalues ALPHA/BETA if desired
529*
530 sdim = 0
531 IF( wantst ) THEN
532*
533* Undo scaling on eigenvalues before SELCTGing
534*
535 IF( ilascl ) THEN
536 CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n,
537 $ ierr )
538 CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n,
539 $ ierr )
540 END IF
541 IF( ilbscl )
542 $ CALL dlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
543*
544* Select eigenvalues
545*
546 DO 10 i = 1, n
547 bwork( i ) = selctg( alphar( i ), alphai( i ), beta( i ) )
548 10 CONTINUE
549*
550 CALL dtgsen( 0, ilvsl, ilvsr, bwork, n, a, lda, b, ldb, alphar,
551 $ alphai, beta, vsl, ldvsl, vsr, ldvsr, sdim, pvsl,
552 $ pvsr, dif, work( iwrk ), lwork-iwrk+1, idum, 1,
553 $ ierr )
554 IF( ierr.EQ.1 )
555 $ info = n + 3
556*
557 END IF
558*
559* Apply back-permutation to VSL and VSR
560*
561 IF( ilvsl )
562 $ CALL dggbak( 'P', 'L', n, ilo, ihi, work( ileft ),
563 $ work( iright ), n, vsl, ldvsl, ierr )
564*
565 IF( ilvsr )
566 $ CALL dggbak( 'P', 'R', n, ilo, ihi, work( ileft ),
567 $ work( iright ), n, vsr, ldvsr, ierr )
568*
569* Check if unscaling would cause over/underflow, if so, rescale
570* (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
571* B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
572*
573 IF( ilascl ) THEN
574 DO 20 i = 1, n
575 IF( alphai( i ).NE.zero ) THEN
576 IF( ( alphar( i ) / safmax ).GT.( anrmto / anrm ) .OR.
577 $ ( safmin / alphar( i ) ).GT.( anrm / anrmto ) ) THEN
578 work( 1 ) = abs( a( i, i ) / alphar( i ) )
579 beta( i ) = beta( i )*work( 1 )
580 alphar( i ) = alphar( i )*work( 1 )
581 alphai( i ) = alphai( i )*work( 1 )
582 ELSE IF( ( alphai( i ) / safmax ).GT.
583 $ ( anrmto / anrm ) .OR.
584 $ ( safmin / alphai( i ) ).GT.( anrm / anrmto ) )
585 $ THEN
586 work( 1 ) = abs( a( i, i+1 ) / alphai( i ) )
587 beta( i ) = beta( i )*work( 1 )
588 alphar( i ) = alphar( i )*work( 1 )
589 alphai( i ) = alphai( i )*work( 1 )
590 END IF
591 END IF
592 20 CONTINUE
593 END IF
594*
595 IF( ilbscl ) THEN
596 DO 30 i = 1, n
597 IF( alphai( i ).NE.zero ) THEN
598 IF( ( beta( i ) / safmax ).GT.( bnrmto / bnrm ) .OR.
599 $ ( safmin / beta( i ) ).GT.( bnrm / bnrmto ) ) THEN
600 work( 1 ) = abs( b( i, i ) / beta( i ) )
601 beta( i ) = beta( i )*work( 1 )
602 alphar( i ) = alphar( i )*work( 1 )
603 alphai( i ) = alphai( i )*work( 1 )
604 END IF
605 END IF
606 30 CONTINUE
607 END IF
608*
609* Undo scaling
610*
611 IF( ilascl ) THEN
612 CALL dlascl( 'H', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
613 CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n, ierr )
614 CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n, ierr )
615 END IF
616*
617 IF( ilbscl ) THEN
618 CALL dlascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
619 CALL dlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
620 END IF
621*
622 IF( wantst ) THEN
623*
624* Check if reordering is correct
625*
626 lastsl = .true.
627 lst2sl = .true.
628 sdim = 0
629 ip = 0
630 DO 40 i = 1, n
631 cursl = selctg( alphar( i ), alphai( i ), beta( i ) )
632 IF( alphai( i ).EQ.zero ) THEN
633 IF( cursl )
634 $ sdim = sdim + 1
635 ip = 0
636 IF( cursl .AND. .NOT.lastsl )
637 $ info = n + 2
638 ELSE
639 IF( ip.EQ.1 ) THEN
640*
641* Last eigenvalue of conjugate pair
642*
643 cursl = cursl .OR. lastsl
644 lastsl = cursl
645 IF( cursl )
646 $ sdim = sdim + 2
647 ip = -1
648 IF( cursl .AND. .NOT.lst2sl )
649 $ info = n + 2
650 ELSE
651*
652* First eigenvalue of conjugate pair
653*
654 ip = 1
655 END IF
656 END IF
657 lst2sl = lastsl
658 lastsl = cursl
659 40 CONTINUE
660*
661 END IF
662*
663 50 CONTINUE
664*
665 work( 1 ) = lwkopt
666*
667 RETURN
668*
669* End of DGGES3
670*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine dlabad(SMALL, LARGE)
DLABAD
Definition: dlabad.f:74
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 dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
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
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
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
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 dgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
DGEQRF
Definition: dgeqrf.f:146
recursive subroutine dlaqz0(WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, REC, INFO)
DLAQZ0
Definition: dlaqz0.f:306
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
subroutine dgghd3(COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
DGGHD3
Definition: dgghd3.f:230
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
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