LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ dgegs()

 subroutine dgegs ( character JOBVSL, character JOBVSR, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, 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, integer INFO )

DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:
``` This routine is deprecated and has been replaced by routine DGGES.

DGEGS computes the eigenvalues, real Schur form, and, optionally,
left and or/right Schur vectors of a real matrix pair (A,B).
Given two square matrices A and B, the generalized real Schur
factorization has the form

A = Q*S*Z**T,  B = Q*T*Z**T

where Q and Z are orthogonal matrices, T is upper triangular, and S
is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal
blocks, the 2-by-2 blocks corresponding to complex conjugate pairs
of eigenvalues of (A,B).  The columns of Q are the left Schur vectors
and the columns of Z are the right Schur vectors.

If only the eigenvalues of (A,B) are needed, the driver routine
DGEGV should be used instead.  See DGEGV for a description of the
eigenvalues of the generalized nonsymmetric eigenvalue problem
(GNEP).```
Parameters
 [in] JOBVSL ``` JOBVSL is CHARACTER*1 = 'N': do not compute the left Schur vectors; = 'V': compute the left Schur vectors (returned in VSL).``` [in] JOBVSR ``` JOBVSR is CHARACTER*1 = 'N': do not compute the right Schur vectors; = 'V': compute the right Schur vectors (returned in VSR).``` [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 matrix A. On exit, the upper quasi-triangular matrix S from the generalized real Schur factorization.``` [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 matrix B. On exit, the upper triangular matrix T from the generalized real Schur factorization.``` [in] LDB ``` LDB is INTEGER The leading dimension of B. LDB >= max(1,N).``` [out] ALPHAR ``` ALPHAR is DOUBLE PRECISION array, dimension (N) The real parts of each scalar alpha defining an eigenvalue of GNEP.``` [out] ALPHAI ``` ALPHAI is DOUBLE PRECISION array, dimension (N) The imaginary parts of each scalar alpha defining an eigenvalue of GNEP. 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) = -ALPHAI(j).``` [out] BETA ``` BETA is DOUBLE PRECISION array, dimension (N) The scalars beta that define the eigenvalues of GNEP. Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and beta = BETA(j) represent the j-th eigenvalue of the matrix pair (A,B), in one of the forms lambda = alpha/beta or mu = beta/alpha. Since either lambda or mu may overflow, they should not, in general, be computed.``` [out] VSL ``` VSL is DOUBLE PRECISION array, dimension (LDVSL,N) If JOBVSL = 'V', the matrix of left Schur vectors Q. 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', the matrix of right Schur vectors Z. 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. LWORK >= max(1,4*N). For good performance, LWORK must generally be larger. To compute the optimal value of LWORK, call ILAENV to get blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute: NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR The optimal LWORK is 2*N + N*(NB+1). 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] 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: errors that usually indicate LAPACK problems: =N+1: error return from DGGBAL =N+2: error return from DGEQRF =N+3: error return from DORMQR =N+4: error return from DORGQR =N+5: error return from DGGHRD =N+6: error return from DHGEQZ (other than failed iteration) =N+7: error return from DGGBAK (computing VSL) =N+8: error return from DGGBAK (computing VSR) =N+9: error return from DLASCL (various places)```
Date
December 2016

Definition at line 229 of file dgegs.f.

229 *
230 * -- LAPACK driver routine (version 3.7.0) --
231 * -- LAPACK is a software package provided by Univ. of Tennessee, --
232 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
233 * December 2016
234 *
235 * .. Scalar Arguments ..
236  CHARACTER jobvsl, jobvsr
237  INTEGER info, lda, ldb, ldvsl, ldvsr, lwork, n
238 * ..
239 * .. Array Arguments ..
240  DOUBLE PRECISION a( lda, * ), alphai( * ), alphar( * ),
241  \$ b( ldb, * ), beta( * ), vsl( ldvsl, * ),
242  \$ vsr( ldvsr, * ), work( * )
243 * ..
244 *
245 * =====================================================================
246 *
247 * .. Parameters ..
248  DOUBLE PRECISION zero, one
249  parameter( zero = 0.0d0, one = 1.0d0 )
250 * ..
251 * .. Local Scalars ..
252  LOGICAL ilascl, ilbscl, ilvsl, ilvsr, lquery
253  INTEGER icols, ihi, iinfo, ijobvl, ijobvr, ileft, ilo,
254  \$ iright, irows, itau, iwork, lopt, lwkmin,
255  \$ lwkopt, nb, nb1, nb2, nb3
256  DOUBLE PRECISION anrm, anrmto, bignum, bnrm, bnrmto, eps,
257  \$ safmin, smlnum
258 * ..
259 * .. External Subroutines ..
260  EXTERNAL dgeqrf, dggbak, dggbal, dgghrd, dhgeqz, dlacpy,
262 * ..
263 * .. External Functions ..
264  LOGICAL lsame
265  INTEGER ilaenv
266  DOUBLE PRECISION dlamch, dlange
267  EXTERNAL lsame, ilaenv, dlamch, dlange
268 * ..
269 * .. Intrinsic Functions ..
270  INTRINSIC int, max
271 * ..
272 * .. Executable Statements ..
273 *
274 * Decode the input arguments
275 *
276  IF( lsame( jobvsl, 'N' ) ) THEN
277  ijobvl = 1
278  ilvsl = .false.
279  ELSE IF( lsame( jobvsl, 'V' ) ) THEN
280  ijobvl = 2
281  ilvsl = .true.
282  ELSE
283  ijobvl = -1
284  ilvsl = .false.
285  END IF
286 *
287  IF( lsame( jobvsr, 'N' ) ) THEN
288  ijobvr = 1
289  ilvsr = .false.
290  ELSE IF( lsame( jobvsr, 'V' ) ) THEN
291  ijobvr = 2
292  ilvsr = .true.
293  ELSE
294  ijobvr = -1
295  ilvsr = .false.
296  END IF
297 *
298 * Test the input arguments
299 *
300  lwkmin = max( 4*n, 1 )
301  lwkopt = lwkmin
302  work( 1 ) = lwkopt
303  lquery = ( lwork.EQ.-1 )
304  info = 0
305  IF( ijobvl.LE.0 ) THEN
306  info = -1
307  ELSE IF( ijobvr.LE.0 ) THEN
308  info = -2
309  ELSE IF( n.LT.0 ) THEN
310  info = -3
311  ELSE IF( lda.LT.max( 1, n ) ) THEN
312  info = -5
313  ELSE IF( ldb.LT.max( 1, n ) ) THEN
314  info = -7
315  ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
316  info = -12
317  ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
318  info = -14
319  ELSE IF( lwork.LT.lwkmin .AND. .NOT.lquery ) THEN
320  info = -16
321  END IF
322 *
323  IF( info.EQ.0 ) THEN
324  nb1 = ilaenv( 1, 'DGEQRF', ' ', n, n, -1, -1 )
325  nb2 = ilaenv( 1, 'DORMQR', ' ', n, n, n, -1 )
326  nb3 = ilaenv( 1, 'DORGQR', ' ', n, n, n, -1 )
327  nb = max( nb1, nb2, nb3 )
328  lopt = 2*n + n*( nb+1 )
329  work( 1 ) = lopt
330  END IF
331 *
332  IF( info.NE.0 ) THEN
333  CALL xerbla( 'DGEGS ', -info )
334  RETURN
335  ELSE IF( lquery ) THEN
336  RETURN
337  END IF
338 *
339 * Quick return if possible
340 *
341  IF( n.EQ.0 )
342  \$ RETURN
343 *
344 * Get machine constants
345 *
346  eps = dlamch( 'E' )*dlamch( 'B' )
347  safmin = dlamch( 'S' )
348  smlnum = n*safmin / eps
349  bignum = one / smlnum
350 *
351 * Scale A if max element outside range [SMLNUM,BIGNUM]
352 *
353  anrm = dlange( 'M', n, n, a, lda, work )
354  ilascl = .false.
355  IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
356  anrmto = smlnum
357  ilascl = .true.
358  ELSE IF( anrm.GT.bignum ) THEN
359  anrmto = bignum
360  ilascl = .true.
361  END IF
362 *
363  IF( ilascl ) THEN
364  CALL dlascl( 'G', -1, -1, anrm, anrmto, n, n, a, lda, iinfo )
365  IF( iinfo.NE.0 ) THEN
366  info = n + 9
367  RETURN
368  END IF
369  END IF
370 *
371 * Scale B if max element outside range [SMLNUM,BIGNUM]
372 *
373  bnrm = dlange( 'M', n, n, b, ldb, work )
374  ilbscl = .false.
375  IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
376  bnrmto = smlnum
377  ilbscl = .true.
378  ELSE IF( bnrm.GT.bignum ) THEN
379  bnrmto = bignum
380  ilbscl = .true.
381  END IF
382 *
383  IF( ilbscl ) THEN
384  CALL dlascl( 'G', -1, -1, bnrm, bnrmto, n, n, b, ldb, iinfo )
385  IF( iinfo.NE.0 ) THEN
386  info = n + 9
387  RETURN
388  END IF
389  END IF
390 *
391 * Permute the matrix to make it more nearly triangular
392 * Workspace layout: (2*N words -- "work..." not actually used)
393 * left_permutation, right_permutation, work...
394 *
395  ileft = 1
396  iright = n + 1
397  iwork = iright + n
398  CALL dggbal( 'P', n, a, lda, b, ldb, ilo, ihi, work( ileft ),
399  \$ work( iright ), work( iwork ), iinfo )
400  IF( iinfo.NE.0 ) THEN
401  info = n + 1
402  GO TO 10
403  END IF
404 *
405 * Reduce B to triangular form, and initialize VSL and/or VSR
406 * Workspace layout: ("work..." must have at least N words)
407 * left_permutation, right_permutation, tau, work...
408 *
409  irows = ihi + 1 - ilo
410  icols = n + 1 - ilo
411  itau = iwork
412  iwork = itau + irows
413  CALL dgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
414  \$ work( iwork ), lwork+1-iwork, iinfo )
415  IF( iinfo.GE.0 )
416  \$ lwkopt = max( lwkopt, int( work( iwork ) )+iwork-1 )
417  IF( iinfo.NE.0 ) THEN
418  info = n + 2
419  GO TO 10
420  END IF
421 *
422  CALL dormqr( 'L', 'T', irows, icols, irows, b( ilo, ilo ), ldb,
423  \$ work( itau ), a( ilo, ilo ), lda, work( iwork ),
424  \$ lwork+1-iwork, iinfo )
425  IF( iinfo.GE.0 )
426  \$ lwkopt = max( lwkopt, int( work( iwork ) )+iwork-1 )
427  IF( iinfo.NE.0 ) THEN
428  info = n + 3
429  GO TO 10
430  END IF
431 *
432  IF( ilvsl ) THEN
433  CALL dlaset( 'Full', n, n, zero, one, vsl, ldvsl )
434  CALL dlacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
435  \$ vsl( ilo+1, ilo ), ldvsl )
436  CALL dorgqr( irows, irows, irows, vsl( ilo, ilo ), ldvsl,
437  \$ work( itau ), work( iwork ), lwork+1-iwork,
438  \$ iinfo )
439  IF( iinfo.GE.0 )
440  \$ lwkopt = max( lwkopt, int( work( iwork ) )+iwork-1 )
441  IF( iinfo.NE.0 ) THEN
442  info = n + 4
443  GO TO 10
444  END IF
445  END IF
446 *
447  IF( ilvsr )
448  \$ CALL dlaset( 'Full', n, n, zero, one, vsr, ldvsr )
449 *
450 * Reduce to generalized Hessenberg form
451 *
452  CALL dgghrd( jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb, vsl,
453  \$ ldvsl, vsr, ldvsr, iinfo )
454  IF( iinfo.NE.0 ) THEN
455  info = n + 5
456  GO TO 10
457  END IF
458 *
459 * Perform QZ algorithm, computing Schur vectors if desired
460 * Workspace layout: ("work..." must have at least 1 word)
461 * left_permutation, right_permutation, work...
462 *
463  iwork = itau
464  CALL dhgeqz( 'S', jobvsl, jobvsr, n, ilo, ihi, a, lda, b, ldb,
465  \$ alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
466  \$ work( iwork ), lwork+1-iwork, iinfo )
467  IF( iinfo.GE.0 )
468  \$ lwkopt = max( lwkopt, int( work( iwork ) )+iwork-1 )
469  IF( iinfo.NE.0 ) THEN
470  IF( iinfo.GT.0 .AND. iinfo.LE.n ) THEN
471  info = iinfo
472  ELSE IF( iinfo.GT.n .AND. iinfo.LE.2*n ) THEN
473  info = iinfo - n
474  ELSE
475  info = n + 6
476  END IF
477  GO TO 10
478  END IF
479 *
480 * Apply permutation to VSL and VSR
481 *
482  IF( ilvsl ) THEN
483  CALL dggbak( 'P', 'L', n, ilo, ihi, work( ileft ),
484  \$ work( iright ), n, vsl, ldvsl, iinfo )
485  IF( iinfo.NE.0 ) THEN
486  info = n + 7
487  GO TO 10
488  END IF
489  END IF
490  IF( ilvsr ) THEN
491  CALL dggbak( 'P', 'R', n, ilo, ihi, work( ileft ),
492  \$ work( iright ), n, vsr, ldvsr, iinfo )
493  IF( iinfo.NE.0 ) THEN
494  info = n + 8
495  GO TO 10
496  END IF
497  END IF
498 *
499 * Undo scaling
500 *
501  IF( ilascl ) THEN
502  CALL dlascl( 'H', -1, -1, anrmto, anrm, n, n, a, lda, iinfo )
503  IF( iinfo.NE.0 ) THEN
504  info = n + 9
505  RETURN
506  END IF
507  CALL dlascl( 'G', -1, -1, anrmto, anrm, n, 1, alphar, n,
508  \$ iinfo )
509  IF( iinfo.NE.0 ) THEN
510  info = n + 9
511  RETURN
512  END IF
513  CALL dlascl( 'G', -1, -1, anrmto, anrm, n, 1, alphai, n,
514  \$ iinfo )
515  IF( iinfo.NE.0 ) THEN
516  info = n + 9
517  RETURN
518  END IF
519  END IF
520 *
521  IF( ilbscl ) THEN
522  CALL dlascl( 'U', -1, -1, bnrmto, bnrm, n, n, b, ldb, iinfo )
523  IF( iinfo.NE.0 ) THEN
524  info = n + 9
525  RETURN
526  END IF
527  CALL dlascl( 'G', -1, -1, bnrmto, bnrm, n, 1, beta, n, iinfo )
528  IF( iinfo.NE.0 ) THEN
529  info = n + 9
530  RETURN
531  END IF
532  END IF
533 *
534  10 CONTINUE
535  work( 1 ) = lwkopt
536 *
537  RETURN
538 *
539 * End of DGEGS
540 *
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 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 dgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
DGEQRF
Definition: dgeqrf.f:138
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