LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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dgeevx.f
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1*> \brief <b> DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b>
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeevx.f">
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13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeevx.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE DGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI,
22* VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM,
23* RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )
24*
25* .. Scalar Arguments ..
26* CHARACTER BALANC, JOBVL, JOBVR, SENSE
27* INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
28* DOUBLE PRECISION ABNRM
29* ..
30* .. Array Arguments ..
31* INTEGER IWORK( * )
32* DOUBLE PRECISION A( LDA, * ), RCONDE( * ), RCONDV( * ),
33* \$ SCALE( * ), VL( LDVL, * ), VR( LDVR, * ),
34* \$ WI( * ), WORK( * ), WR( * )
35* ..
36*
37*
38*> \par Purpose:
39* =============
40*>
41*> \verbatim
42*>
43*> DGEEVX computes for an N-by-N real nonsymmetric matrix A, the
44*> eigenvalues and, optionally, the left and/or right eigenvectors.
45*>
46*> Optionally also, it computes a balancing transformation to improve
47*> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
48*> SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
49*> (RCONDE), and reciprocal condition numbers for the right
50*> eigenvectors (RCONDV).
51*>
52*> The right eigenvector v(j) of A satisfies
53*> A * v(j) = lambda(j) * v(j)
54*> where lambda(j) is its eigenvalue.
55*> The left eigenvector u(j) of A satisfies
56*> u(j)**H * A = lambda(j) * u(j)**H
57*> where u(j)**H denotes the conjugate-transpose of u(j).
58*>
59*> The computed eigenvectors are normalized to have Euclidean norm
60*> equal to 1 and largest component real.
61*>
62*> Balancing a matrix means permuting the rows and columns to make it
63*> more nearly upper triangular, and applying a diagonal similarity
64*> transformation D * A * D**(-1), where D is a diagonal matrix, to
65*> make its rows and columns closer in norm and the condition numbers
66*> of its eigenvalues and eigenvectors smaller. The computed
67*> reciprocal condition numbers correspond to the balanced matrix.
68*> Permuting rows and columns will not change the condition numbers
69*> (in exact arithmetic) but diagonal scaling will. For further
70*> explanation of balancing, see section 4.10.2 of the LAPACK
71*> Users' Guide.
72*> \endverbatim
73*
74* Arguments:
75* ==========
76*
77*> \param[in] BALANC
78*> \verbatim
79*> BALANC is CHARACTER*1
80*> Indicates how the input matrix should be diagonally scaled
81*> and/or permuted to improve the conditioning of its
82*> eigenvalues.
83*> = 'N': Do not diagonally scale or permute;
84*> = 'P': Perform permutations to make the matrix more nearly
85*> upper triangular. Do not diagonally scale;
86*> = 'S': Diagonally scale the matrix, i.e. replace A by
87*> D*A*D**(-1), where D is a diagonal matrix chosen
88*> to make the rows and columns of A more equal in
89*> norm. Do not permute;
90*> = 'B': Both diagonally scale and permute A.
91*>
92*> Computed reciprocal condition numbers will be for the matrix
93*> after balancing and/or permuting. Permuting does not change
94*> condition numbers (in exact arithmetic), but balancing does.
95*> \endverbatim
96*>
97*> \param[in] JOBVL
98*> \verbatim
99*> JOBVL is CHARACTER*1
100*> = 'N': left eigenvectors of A are not computed;
101*> = 'V': left eigenvectors of A are computed.
102*> If SENSE = 'E' or 'B', JOBVL must = 'V'.
103*> \endverbatim
104*>
105*> \param[in] JOBVR
106*> \verbatim
107*> JOBVR is CHARACTER*1
108*> = 'N': right eigenvectors of A are not computed;
109*> = 'V': right eigenvectors of A are computed.
110*> If SENSE = 'E' or 'B', JOBVR must = 'V'.
111*> \endverbatim
112*>
113*> \param[in] SENSE
114*> \verbatim
115*> SENSE is CHARACTER*1
116*> Determines which reciprocal condition numbers are computed.
117*> = 'N': None are computed;
118*> = 'E': Computed for eigenvalues only;
119*> = 'V': Computed for right eigenvectors only;
120*> = 'B': Computed for eigenvalues and right eigenvectors.
121*>
122*> If SENSE = 'E' or 'B', both left and right eigenvectors
123*> must also be computed (JOBVL = 'V' and JOBVR = 'V').
124*> \endverbatim
125*>
126*> \param[in] N
127*> \verbatim
128*> N is INTEGER
129*> The order of the matrix A. N >= 0.
130*> \endverbatim
131*>
132*> \param[in,out] A
133*> \verbatim
134*> A is DOUBLE PRECISION array, dimension (LDA,N)
135*> On entry, the N-by-N matrix A.
136*> On exit, A has been overwritten. If JOBVL = 'V' or
137*> JOBVR = 'V', A contains the real Schur form of the balanced
138*> version of the input matrix A.
139*> \endverbatim
140*>
141*> \param[in] LDA
142*> \verbatim
143*> LDA is INTEGER
144*> The leading dimension of the array A. LDA >= max(1,N).
145*> \endverbatim
146*>
147*> \param[out] WR
148*> \verbatim
149*> WR is DOUBLE PRECISION array, dimension (N)
150*> \endverbatim
151*>
152*> \param[out] WI
153*> \verbatim
154*> WI is DOUBLE PRECISION array, dimension (N)
155*> WR and WI contain the real and imaginary parts,
156*> respectively, of the computed eigenvalues. Complex
157*> conjugate pairs of eigenvalues will appear consecutively
158*> with the eigenvalue having the positive imaginary part
159*> first.
160*> \endverbatim
161*>
162*> \param[out] VL
163*> \verbatim
164*> VL is DOUBLE PRECISION array, dimension (LDVL,N)
165*> If JOBVL = 'V', the left eigenvectors u(j) are stored one
166*> after another in the columns of VL, in the same order
167*> as their eigenvalues.
168*> If JOBVL = 'N', VL is not referenced.
169*> If the j-th eigenvalue is real, then u(j) = VL(:,j),
170*> the j-th column of VL.
171*> If the j-th and (j+1)-st eigenvalues form a complex
172*> conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
173*> u(j+1) = VL(:,j) - i*VL(:,j+1).
174*> \endverbatim
175*>
176*> \param[in] LDVL
177*> \verbatim
178*> LDVL is INTEGER
179*> The leading dimension of the array VL. LDVL >= 1; if
180*> JOBVL = 'V', LDVL >= N.
181*> \endverbatim
182*>
183*> \param[out] VR
184*> \verbatim
185*> VR is DOUBLE PRECISION array, dimension (LDVR,N)
186*> If JOBVR = 'V', the right eigenvectors v(j) are stored one
187*> after another in the columns of VR, in the same order
188*> as their eigenvalues.
189*> If JOBVR = 'N', VR is not referenced.
190*> If the j-th eigenvalue is real, then v(j) = VR(:,j),
191*> the j-th column of VR.
192*> If the j-th and (j+1)-st eigenvalues form a complex
193*> conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
194*> v(j+1) = VR(:,j) - i*VR(:,j+1).
195*> \endverbatim
196*>
197*> \param[in] LDVR
198*> \verbatim
199*> LDVR is INTEGER
200*> The leading dimension of the array VR. LDVR >= 1, and if
201*> JOBVR = 'V', LDVR >= N.
202*> \endverbatim
203*>
204*> \param[out] ILO
205*> \verbatim
206*> ILO is INTEGER
207*> \endverbatim
208*>
209*> \param[out] IHI
210*> \verbatim
211*> IHI is INTEGER
212*> ILO and IHI are integer values determined when A was
213*> balanced. The balanced A(i,j) = 0 if I > J and
214*> J = 1,...,ILO-1 or I = IHI+1,...,N.
215*> \endverbatim
216*>
217*> \param[out] SCALE
218*> \verbatim
219*> SCALE is DOUBLE PRECISION array, dimension (N)
220*> Details of the permutations and scaling factors applied
221*> when balancing A. If P(j) is the index of the row and column
222*> interchanged with row and column j, and D(j) is the scaling
223*> factor applied to row and column j, then
224*> SCALE(J) = P(J), for J = 1,...,ILO-1
225*> = D(J), for J = ILO,...,IHI
226*> = P(J) for J = IHI+1,...,N.
227*> The order in which the interchanges are made is N to IHI+1,
228*> then 1 to ILO-1.
229*> \endverbatim
230*>
231*> \param[out] ABNRM
232*> \verbatim
233*> ABNRM is DOUBLE PRECISION
234*> The one-norm of the balanced matrix (the maximum
235*> of the sum of absolute values of elements of any column).
236*> \endverbatim
237*>
238*> \param[out] RCONDE
239*> \verbatim
240*> RCONDE is DOUBLE PRECISION array, dimension (N)
241*> RCONDE(j) is the reciprocal condition number of the j-th
242*> eigenvalue.
243*> \endverbatim
244*>
245*> \param[out] RCONDV
246*> \verbatim
247*> RCONDV is DOUBLE PRECISION array, dimension (N)
248*> RCONDV(j) is the reciprocal condition number of the j-th
249*> right eigenvector.
250*> \endverbatim
251*>
252*> \param[out] WORK
253*> \verbatim
254*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
255*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
256*> \endverbatim
257*>
258*> \param[in] LWORK
259*> \verbatim
260*> LWORK is INTEGER
261*> The dimension of the array WORK. If SENSE = 'N' or 'E',
262*> LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V',
263*> LWORK >= 3*N. If SENSE = 'V' or 'B', LWORK >= N*(N+6).
264*> For good performance, LWORK must generally be larger.
265*>
266*> If LWORK = -1, then a workspace query is assumed; the routine
267*> only calculates the optimal size of the WORK array, returns
268*> this value as the first entry of the WORK array, and no error
269*> message related to LWORK is issued by XERBLA.
270*> \endverbatim
271*>
272*> \param[out] IWORK
273*> \verbatim
274*> IWORK is INTEGER array, dimension (2*N-2)
275*> If SENSE = 'N' or 'E', not referenced.
276*> \endverbatim
277*>
278*> \param[out] INFO
279*> \verbatim
280*> INFO is INTEGER
281*> = 0: successful exit
282*> < 0: if INFO = -i, the i-th argument had an illegal value.
283*> > 0: if INFO = i, the QR algorithm failed to compute all the
284*> eigenvalues, and no eigenvectors or condition numbers
285*> have been computed; elements 1:ILO-1 and i+1:N of WR
286*> and WI contain eigenvalues which have converged.
287*> \endverbatim
288*
289* Authors:
290* ========
291*
292*> \author Univ. of Tennessee
293*> \author Univ. of California Berkeley
294*> \author Univ. of Colorado Denver
295*> \author NAG Ltd.
296*
297*
298* @precisions fortran d -> s
299*
300*> \ingroup doubleGEeigen
301*
302* =====================================================================
303 SUBROUTINE dgeevx( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI,
304 \$ VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM,
305 \$ RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )
306 implicit none
307*
308* -- LAPACK driver routine --
309* -- LAPACK is a software package provided by Univ. of Tennessee, --
310* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
311*
312* .. Scalar Arguments ..
313 CHARACTER BALANC, JOBVL, JOBVR, SENSE
314 INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
315 DOUBLE PRECISION ABNRM
316* ..
317* .. Array Arguments ..
318 INTEGER IWORK( * )
319 DOUBLE PRECISION A( LDA, * ), RCONDE( * ), RCONDV( * ),
320 \$ scale( * ), vl( ldvl, * ), vr( ldvr, * ),
321 \$ wi( * ), work( * ), wr( * )
322* ..
323*
324* =====================================================================
325*
326* .. Parameters ..
327 DOUBLE PRECISION ZERO, ONE
328 PARAMETER ( ZERO = 0.0d0, one = 1.0d0 )
329* ..
330* .. Local Scalars ..
331 LOGICAL LQUERY, SCALEA, WANTVL, WANTVR, WNTSNB, WNTSNE,
332 \$ WNTSNN, WNTSNV
333 CHARACTER JOB, SIDE
334 INTEGER HSWORK, I, ICOND, IERR, ITAU, IWRK, K,
335 \$ lwork_trevc, maxwrk, minwrk, nout
336 DOUBLE PRECISION ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM,
337 \$ SN
338* ..
339* .. Local Arrays ..
340 LOGICAL SELECT( 1 )
341 DOUBLE PRECISION DUM( 1 )
342* ..
343* .. External Subroutines ..
344 EXTERNAL dgebak, dgebal, dgehrd, dhseqr, dlabad, dlacpy,
346 \$ dtrsna, xerbla
347* ..
348* .. External Functions ..
349 LOGICAL LSAME
350 INTEGER IDAMAX, ILAENV
351 DOUBLE PRECISION DLAMCH, DLANGE, DLAPY2, DNRM2
352 EXTERNAL lsame, idamax, ilaenv, dlamch, dlange, dlapy2,
353 \$ dnrm2
354* ..
355* .. Intrinsic Functions ..
356 INTRINSIC max, sqrt
357* ..
358* .. Executable Statements ..
359*
360* Test the input arguments
361*
362 info = 0
363 lquery = ( lwork.EQ.-1 )
364 wantvl = lsame( jobvl, 'V' )
365 wantvr = lsame( jobvr, 'V' )
366 wntsnn = lsame( sense, 'N' )
367 wntsne = lsame( sense, 'E' )
368 wntsnv = lsame( sense, 'V' )
369 wntsnb = lsame( sense, 'B' )
370 IF( .NOT.( lsame( balanc, 'N' ) .OR. lsame( balanc, 'S' )
371 \$ .OR. lsame( balanc, 'P' ) .OR. lsame( balanc, 'B' ) ) )
372 \$ THEN
373 info = -1
374 ELSE IF( ( .NOT.wantvl ) .AND. ( .NOT.lsame( jobvl, 'N' ) ) ) THEN
375 info = -2
376 ELSE IF( ( .NOT.wantvr ) .AND. ( .NOT.lsame( jobvr, 'N' ) ) ) THEN
377 info = -3
378 ELSE IF( .NOT.( wntsnn .OR. wntsne .OR. wntsnb .OR. wntsnv ) .OR.
379 \$ ( ( wntsne .OR. wntsnb ) .AND. .NOT.( wantvl .AND.
380 \$ wantvr ) ) ) THEN
381 info = -4
382 ELSE IF( n.LT.0 ) THEN
383 info = -5
384 ELSE IF( lda.LT.max( 1, n ) ) THEN
385 info = -7
386 ELSE IF( ldvl.LT.1 .OR. ( wantvl .AND. ldvl.LT.n ) ) THEN
387 info = -11
388 ELSE IF( ldvr.LT.1 .OR. ( wantvr .AND. ldvr.LT.n ) ) THEN
389 info = -13
390 END IF
391*
392* Compute workspace
393* (Note: Comments in the code beginning "Workspace:" describe the
394* minimal amount of workspace needed at that point in the code,
395* as well as the preferred amount for good performance.
396* NB refers to the optimal block size for the immediately
397* following subroutine, as returned by ILAENV.
398* HSWORK refers to the workspace preferred by DHSEQR, as
399* calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
400* the worst case.)
401*
402 IF( info.EQ.0 ) THEN
403 IF( n.EQ.0 ) THEN
404 minwrk = 1
405 maxwrk = 1
406 ELSE
407 maxwrk = n + n*ilaenv( 1, 'DGEHRD', ' ', n, 1, n, 0 )
408*
409 IF( wantvl ) THEN
410 CALL dtrevc3( 'L', 'B', SELECT, n, a, lda,
411 \$ vl, ldvl, vr, ldvr,
412 \$ n, nout, work, -1, ierr )
413 lwork_trevc = int( work(1) )
414 maxwrk = max( maxwrk, n + lwork_trevc )
415 CALL dhseqr( 'S', 'V', n, 1, n, a, lda, wr, wi, vl, ldvl,
416 \$ work, -1, info )
417 ELSE IF( wantvr ) THEN
418 CALL dtrevc3( 'R', 'B', SELECT, n, a, lda,
419 \$ vl, ldvl, vr, ldvr,
420 \$ n, nout, work, -1, ierr )
421 lwork_trevc = int( work(1) )
422 maxwrk = max( maxwrk, n + lwork_trevc )
423 CALL dhseqr( 'S', 'V', n, 1, n, a, lda, wr, wi, vr, ldvr,
424 \$ work, -1, info )
425 ELSE
426 IF( wntsnn ) THEN
427 CALL dhseqr( 'E', 'N', n, 1, n, a, lda, wr, wi, vr,
428 \$ ldvr, work, -1, info )
429 ELSE
430 CALL dhseqr( 'S', 'N', n, 1, n, a, lda, wr, wi, vr,
431 \$ ldvr, work, -1, info )
432 END IF
433 END IF
434 hswork = int( work(1) )
435*
436 IF( ( .NOT.wantvl ) .AND. ( .NOT.wantvr ) ) THEN
437 minwrk = 2*n
438 IF( .NOT.wntsnn )
439 \$ minwrk = max( minwrk, n*n+6*n )
440 maxwrk = max( maxwrk, hswork )
441 IF( .NOT.wntsnn )
442 \$ maxwrk = max( maxwrk, n*n + 6*n )
443 ELSE
444 minwrk = 3*n
445 IF( ( .NOT.wntsnn ) .AND. ( .NOT.wntsne ) )
446 \$ minwrk = max( minwrk, n*n + 6*n )
447 maxwrk = max( maxwrk, hswork )
448 maxwrk = max( maxwrk, n + ( n - 1 )*ilaenv( 1, 'DORGHR',
449 \$ ' ', n, 1, n, -1 ) )
450 IF( ( .NOT.wntsnn ) .AND. ( .NOT.wntsne ) )
451 \$ maxwrk = max( maxwrk, n*n + 6*n )
452 maxwrk = max( maxwrk, 3*n )
453 END IF
454 maxwrk = max( maxwrk, minwrk )
455 END IF
456 work( 1 ) = maxwrk
457*
458 IF( lwork.LT.minwrk .AND. .NOT.lquery ) THEN
459 info = -21
460 END IF
461 END IF
462*
463 IF( info.NE.0 ) THEN
464 CALL xerbla( 'DGEEVX', -info )
465 RETURN
466 ELSE IF( lquery ) THEN
467 RETURN
468 END IF
469*
470* Quick return if possible
471*
472 IF( n.EQ.0 )
473 \$ RETURN
474*
475* Get machine constants
476*
477 eps = dlamch( 'P' )
478 smlnum = dlamch( 'S' )
479 bignum = one / smlnum
480 CALL dlabad( smlnum, bignum )
481 smlnum = sqrt( smlnum ) / eps
482 bignum = one / smlnum
483*
484* Scale A if max element outside range [SMLNUM,BIGNUM]
485*
486 icond = 0
487 anrm = dlange( 'M', n, n, a, lda, dum )
488 scalea = .false.
489 IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
490 scalea = .true.
491 cscale = smlnum
492 ELSE IF( anrm.GT.bignum ) THEN
493 scalea = .true.
494 cscale = bignum
495 END IF
496 IF( scalea )
497 \$ CALL dlascl( 'G', 0, 0, anrm, cscale, n, n, a, lda, ierr )
498*
499* Balance the matrix and compute ABNRM
500*
501 CALL dgebal( balanc, n, a, lda, ilo, ihi, scale, ierr )
502 abnrm = dlange( '1', n, n, a, lda, dum )
503 IF( scalea ) THEN
504 dum( 1 ) = abnrm
505 CALL dlascl( 'G', 0, 0, cscale, anrm, 1, 1, dum, 1, ierr )
506 abnrm = dum( 1 )
507 END IF
508*
509* Reduce to upper Hessenberg form
510* (Workspace: need 2*N, prefer N+N*NB)
511*
512 itau = 1
513 iwrk = itau + n
514 CALL dgehrd( n, ilo, ihi, a, lda, work( itau ), work( iwrk ),
515 \$ lwork-iwrk+1, ierr )
516*
517 IF( wantvl ) THEN
518*
519* Want left eigenvectors
520* Copy Householder vectors to VL
521*
522 side = 'L'
523 CALL dlacpy( 'L', n, n, a, lda, vl, ldvl )
524*
525* Generate orthogonal matrix in VL
526* (Workspace: need 2*N-1, prefer N+(N-1)*NB)
527*
528 CALL dorghr( n, ilo, ihi, vl, ldvl, work( itau ), work( iwrk ),
529 \$ lwork-iwrk+1, ierr )
530*
531* Perform QR iteration, accumulating Schur vectors in VL
532* (Workspace: need 1, prefer HSWORK (see comments) )
533*
534 iwrk = itau
535 CALL dhseqr( 'S', 'V', n, ilo, ihi, a, lda, wr, wi, vl, ldvl,
536 \$ work( iwrk ), lwork-iwrk+1, info )
537*
538 IF( wantvr ) THEN
539*
540* Want left and right eigenvectors
541* Copy Schur vectors to VR
542*
543 side = 'B'
544 CALL dlacpy( 'F', n, n, vl, ldvl, vr, ldvr )
545 END IF
546*
547 ELSE IF( wantvr ) THEN
548*
549* Want right eigenvectors
550* Copy Householder vectors to VR
551*
552 side = 'R'
553 CALL dlacpy( 'L', n, n, a, lda, vr, ldvr )
554*
555* Generate orthogonal matrix in VR
556* (Workspace: need 2*N-1, prefer N+(N-1)*NB)
557*
558 CALL dorghr( n, ilo, ihi, vr, ldvr, work( itau ), work( iwrk ),
559 \$ lwork-iwrk+1, ierr )
560*
561* Perform QR iteration, accumulating Schur vectors in VR
562* (Workspace: need 1, prefer HSWORK (see comments) )
563*
564 iwrk = itau
565 CALL dhseqr( 'S', 'V', n, ilo, ihi, a, lda, wr, wi, vr, ldvr,
566 \$ work( iwrk ), lwork-iwrk+1, info )
567*
568 ELSE
569*
570* Compute eigenvalues only
571* If condition numbers desired, compute Schur form
572*
573 IF( wntsnn ) THEN
574 job = 'E'
575 ELSE
576 job = 'S'
577 END IF
578*
579* (Workspace: need 1, prefer HSWORK (see comments) )
580*
581 iwrk = itau
582 CALL dhseqr( job, 'N', n, ilo, ihi, a, lda, wr, wi, vr, ldvr,
583 \$ work( iwrk ), lwork-iwrk+1, info )
584 END IF
585*
586* If INFO .NE. 0 from DHSEQR, then quit
587*
588 IF( info.NE.0 )
589 \$ GO TO 50
590*
591 IF( wantvl .OR. wantvr ) THEN
592*
593* Compute left and/or right eigenvectors
594* (Workspace: need 3*N, prefer N + 2*N*NB)
595*
596 CALL dtrevc3( side, 'B', SELECT, n, a, lda, vl, ldvl, vr, ldvr,
597 \$ n, nout, work( iwrk ), lwork-iwrk+1, ierr )
598 END IF
599*
600* Compute condition numbers if desired
601* (Workspace: need N*N+6*N unless SENSE = 'E')
602*
603 IF( .NOT.wntsnn ) THEN
604 CALL dtrsna( sense, 'A', SELECT, n, a, lda, vl, ldvl, vr, ldvr,
605 \$ rconde, rcondv, n, nout, work( iwrk ), n, iwork,
606 \$ icond )
607 END IF
608*
609 IF( wantvl ) THEN
610*
611* Undo balancing of left eigenvectors
612*
613 CALL dgebak( balanc, 'L', n, ilo, ihi, scale, n, vl, ldvl,
614 \$ ierr )
615*
616* Normalize left eigenvectors and make largest component real
617*
618 DO 20 i = 1, n
619 IF( wi( i ).EQ.zero ) THEN
620 scl = one / dnrm2( n, vl( 1, i ), 1 )
621 CALL dscal( n, scl, vl( 1, i ), 1 )
622 ELSE IF( wi( i ).GT.zero ) THEN
623 scl = one / dlapy2( dnrm2( n, vl( 1, i ), 1 ),
624 \$ dnrm2( n, vl( 1, i+1 ), 1 ) )
625 CALL dscal( n, scl, vl( 1, i ), 1 )
626 CALL dscal( n, scl, vl( 1, i+1 ), 1 )
627 DO 10 k = 1, n
628 work( k ) = vl( k, i )**2 + vl( k, i+1 )**2
629 10 CONTINUE
630 k = idamax( n, work, 1 )
631 CALL dlartg( vl( k, i ), vl( k, i+1 ), cs, sn, r )
632 CALL drot( n, vl( 1, i ), 1, vl( 1, i+1 ), 1, cs, sn )
633 vl( k, i+1 ) = zero
634 END IF
635 20 CONTINUE
636 END IF
637*
638 IF( wantvr ) THEN
639*
640* Undo balancing of right eigenvectors
641*
642 CALL dgebak( balanc, 'R', n, ilo, ihi, scale, n, vr, ldvr,
643 \$ ierr )
644*
645* Normalize right eigenvectors and make largest component real
646*
647 DO 40 i = 1, n
648 IF( wi( i ).EQ.zero ) THEN
649 scl = one / dnrm2( n, vr( 1, i ), 1 )
650 CALL dscal( n, scl, vr( 1, i ), 1 )
651 ELSE IF( wi( i ).GT.zero ) THEN
652 scl = one / dlapy2( dnrm2( n, vr( 1, i ), 1 ),
653 \$ dnrm2( n, vr( 1, i+1 ), 1 ) )
654 CALL dscal( n, scl, vr( 1, i ), 1 )
655 CALL dscal( n, scl, vr( 1, i+1 ), 1 )
656 DO 30 k = 1, n
657 work( k ) = vr( k, i )**2 + vr( k, i+1 )**2
658 30 CONTINUE
659 k = idamax( n, work, 1 )
660 CALL dlartg( vr( k, i ), vr( k, i+1 ), cs, sn, r )
661 CALL drot( n, vr( 1, i ), 1, vr( 1, i+1 ), 1, cs, sn )
662 vr( k, i+1 ) = zero
663 END IF
664 40 CONTINUE
665 END IF
666*
667* Undo scaling if necessary
668*
669 50 CONTINUE
670 IF( scalea ) THEN
671 CALL dlascl( 'G', 0, 0, cscale, anrm, n-info, 1, wr( info+1 ),
672 \$ max( n-info, 1 ), ierr )
673 CALL dlascl( 'G', 0, 0, cscale, anrm, n-info, 1, wi( info+1 ),
674 \$ max( n-info, 1 ), ierr )
675 IF( info.EQ.0 ) THEN
676 IF( ( wntsnv .OR. wntsnb ) .AND. icond.EQ.0 )
677 \$ CALL dlascl( 'G', 0, 0, cscale, anrm, n, 1, rcondv, n,
678 \$ ierr )
679 ELSE
680 CALL dlascl( 'G', 0, 0, cscale, anrm, ilo-1, 1, wr, n,
681 \$ ierr )
682 CALL dlascl( 'G', 0, 0, cscale, anrm, ilo-1, 1, wi, n,
683 \$ ierr )
684 END IF
685 END IF
686*
687 work( 1 ) = maxwrk
688 RETURN
689*
690* End of DGEEVX
691*
692 END
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 dlartg(f, g, c, s, r)
DLARTG generates a plane rotation with real cosine and real sine.
Definition: dlartg.f90:111
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine drot(N, DX, INCX, DY, INCY, C, S)
DROT
Definition: drot.f:92
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
subroutine dgehrd(N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
DGEHRD
Definition: dgehrd.f:167
subroutine dgebal(JOB, N, A, LDA, ILO, IHI, SCALE, INFO)
DGEBAL
Definition: dgebal.f:160
subroutine dgebak(JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV, INFO)
DGEBAK
Definition: dgebak.f:130
subroutine dgeevx(BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, RCONDV, WORK, LWORK, IWORK, INFO)
DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
Definition: dgeevx.f:306
subroutine dtrsna(JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M, WORK, LDWORK, IWORK, INFO)
DTRSNA
Definition: dtrsna.f:265
subroutine dhseqr(JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, LDZ, WORK, LWORK, INFO)
DHSEQR
Definition: dhseqr.f:316
subroutine dorghr(N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
DORGHR
Definition: dorghr.f:126
subroutine dtrevc3(SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, MM, M, WORK, LWORK, INFO)
DTREVC3
Definition: dtrevc3.f:237