1: *> \brief <b> DGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)</b>
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DGGES3 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgges3.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgges3.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgges3.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DGGES3( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,
22: * SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR,
23: * LDVSR, WORK, LWORK, BWORK, INFO )
24: *
25: * .. Scalar Arguments ..
26: * CHARACTER JOBVSL, JOBVSR, SORT
27: * INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
28: * ..
29: * .. Array Arguments ..
30: * LOGICAL BWORK( * )
31: * DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
32: * $ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
33: * $ VSR( LDVSR, * ), WORK( * )
34: * ..
35: * .. Function Arguments ..
36: * LOGICAL SELCTG
37: * EXTERNAL SELCTG
38: * ..
39: *
40: *
41: *> \par Purpose:
42: * =============
43: *>
44: *> \verbatim
45: *>
46: *> DGGES3 computes for a pair of N-by-N real nonsymmetric matrices (A,B),
47: *> the generalized eigenvalues, the generalized real Schur form (S,T),
48: *> optionally, the left and/or right matrices of Schur vectors (VSL and
49: *> VSR). This gives the generalized Schur factorization
50: *>
51: *> (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )
52: *>
53: *> Optionally, it also orders the eigenvalues so that a selected cluster
54: *> of eigenvalues appears in the leading diagonal blocks of the upper
55: *> quasi-triangular matrix S and the upper triangular matrix T.The
56: *> leading columns of VSL and VSR then form an orthonormal basis for the
57: *> corresponding left and right eigenspaces (deflating subspaces).
58: *>
59: *> (If only the generalized eigenvalues are needed, use the driver
60: *> DGGEV instead, which is faster.)
61: *>
62: *> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
63: *> or a ratio alpha/beta = w, such that A - w*B is singular. It is
64: *> usually represented as the pair (alpha,beta), as there is a
65: *> reasonable interpretation for beta=0 or both being zero.
66: *>
67: *> A pair of matrices (S,T) is in generalized real Schur form if T is
68: *> upper triangular with non-negative diagonal and S is block upper
69: *> triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond
70: *> to real generalized eigenvalues, while 2-by-2 blocks of S will be
71: *> "standardized" by making the corresponding elements of T have the
72: *> form:
73: *> [ a 0 ]
74: *> [ 0 b ]
75: *>
76: *> and the pair of corresponding 2-by-2 blocks in S and T will have a
77: *> complex conjugate pair of generalized eigenvalues.
78: *>
79: *> \endverbatim
80: *
81: * Arguments:
82: * ==========
83: *
84: *> \param[in] JOBVSL
85: *> \verbatim
86: *> JOBVSL is CHARACTER*1
87: *> = 'N': do not compute the left Schur vectors;
88: *> = 'V': compute the left Schur vectors.
89: *> \endverbatim
90: *>
91: *> \param[in] JOBVSR
92: *> \verbatim
93: *> JOBVSR is CHARACTER*1
94: *> = 'N': do not compute the right Schur vectors;
95: *> = 'V': compute the right Schur vectors.
96: *> \endverbatim
97: *>
98: *> \param[in] SORT
99: *> \verbatim
100: *> SORT is CHARACTER*1
101: *> Specifies whether or not to order the eigenvalues on the
102: *> diagonal of the generalized Schur form.
103: *> = 'N': Eigenvalues are not ordered;
104: *> = 'S': Eigenvalues are ordered (see SELCTG);
105: *> \endverbatim
106: *>
107: *> \param[in] SELCTG
108: *> \verbatim
109: *> SELCTG is a LOGICAL FUNCTION of three DOUBLE PRECISION arguments
110: *> SELCTG must be declared EXTERNAL in the calling subroutine.
111: *> If SORT = 'N', SELCTG is not referenced.
112: *> If SORT = 'S', SELCTG is used to select eigenvalues to sort
113: *> to the top left of the Schur form.
114: *> An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
115: *> SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
116: *> one of a complex conjugate pair of eigenvalues is selected,
117: *> then both complex eigenvalues are selected.
118: *>
119: *> Note that in the ill-conditioned case, a selected complex
120: *> eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
121: *> BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
122: *> in this case.
123: *> \endverbatim
124: *>
125: *> \param[in] N
126: *> \verbatim
127: *> N is INTEGER
128: *> The order of the matrices A, B, VSL, and VSR. N >= 0.
129: *> \endverbatim
130: *>
131: *> \param[in,out] A
132: *> \verbatim
133: *> A is DOUBLE PRECISION array, dimension (LDA, N)
134: *> On entry, the first of the pair of matrices.
135: *> On exit, A has been overwritten by its generalized Schur
136: *> form S.
137: *> \endverbatim
138: *>
139: *> \param[in] LDA
140: *> \verbatim
141: *> LDA is INTEGER
142: *> The leading dimension of A. LDA >= max(1,N).
143: *> \endverbatim
144: *>
145: *> \param[in,out] B
146: *> \verbatim
147: *> B is DOUBLE PRECISION array, dimension (LDB, N)
148: *> On entry, the second of the pair of matrices.
149: *> On exit, B has been overwritten by its generalized Schur
150: *> form T.
151: *> \endverbatim
152: *>
153: *> \param[in] LDB
154: *> \verbatim
155: *> LDB is INTEGER
156: *> The leading dimension of B. LDB >= max(1,N).
157: *> \endverbatim
158: *>
159: *> \param[out] SDIM
160: *> \verbatim
161: *> SDIM is INTEGER
162: *> If SORT = 'N', SDIM = 0.
163: *> If SORT = 'S', SDIM = number of eigenvalues (after sorting)
164: *> for which SELCTG is true. (Complex conjugate pairs for which
165: *> SELCTG is true for either eigenvalue count as 2.)
166: *> \endverbatim
167: *>
168: *> \param[out] ALPHAR
169: *> \verbatim
170: *> ALPHAR is DOUBLE PRECISION array, dimension (N)
171: *> \endverbatim
172: *>
173: *> \param[out] ALPHAI
174: *> \verbatim
175: *> ALPHAI is DOUBLE PRECISION array, dimension (N)
176: *> \endverbatim
177: *>
178: *> \param[out] BETA
179: *> \verbatim
180: *> BETA is DOUBLE PRECISION array, dimension (N)
181: *> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
182: *> be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i,
183: *> and BETA(j),j=1,...,N are the diagonals of the complex Schur
184: *> form (S,T) that would result if the 2-by-2 diagonal blocks of
185: *> the real Schur form of (A,B) were further reduced to
186: *> triangular form using 2-by-2 complex unitary transformations.
187: *> If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
188: *> positive, then the j-th and (j+1)-st eigenvalues are a
189: *> complex conjugate pair, with ALPHAI(j+1) negative.
190: *>
191: *> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
192: *> may easily over- or underflow, and BETA(j) may even be zero.
193: *> Thus, the user should avoid naively computing the ratio.
194: *> However, ALPHAR and ALPHAI will be always less than and
195: *> usually comparable with norm(A) in magnitude, and BETA always
196: *> less than and usually comparable with norm(B).
197: *> \endverbatim
198: *>
199: *> \param[out] VSL
200: *> \verbatim
201: *> VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
202: *> If JOBVSL = 'V', VSL will contain the left Schur vectors.
203: *> Not referenced if JOBVSL = 'N'.
204: *> \endverbatim
205: *>
206: *> \param[in] LDVSL
207: *> \verbatim
208: *> LDVSL is INTEGER
209: *> The leading dimension of the matrix VSL. LDVSL >=1, and
210: *> if JOBVSL = 'V', LDVSL >= N.
211: *> \endverbatim
212: *>
213: *> \param[out] VSR
214: *> \verbatim
215: *> VSR is DOUBLE PRECISION array, dimension (LDVSR,N)
216: *> If JOBVSR = 'V', VSR will contain the right Schur vectors.
217: *> Not referenced if JOBVSR = 'N'.
218: *> \endverbatim
219: *>
220: *> \param[in] LDVSR
221: *> \verbatim
222: *> LDVSR is INTEGER
223: *> The leading dimension of the matrix VSR. LDVSR >= 1, and
224: *> if JOBVSR = 'V', LDVSR >= N.
225: *> \endverbatim
226: *>
227: *> \param[out] WORK
228: *> \verbatim
229: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
230: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
231: *> \endverbatim
232: *>
233: *> \param[in] LWORK
234: *> \verbatim
235: *> LWORK is INTEGER
236: *> The dimension of the array WORK.
237: *>
238: *> If LWORK = -1, then a workspace query is assumed; the routine
239: *> only calculates the optimal size of the WORK array, returns
240: *> this value as the first entry of the WORK array, and no error
241: *> message related to LWORK is issued by XERBLA.
242: *> \endverbatim
243: *>
244: *> \param[out] BWORK
245: *> \verbatim
246: *> BWORK is LOGICAL array, dimension (N)
247: *> Not referenced if SORT = 'N'.
248: *> \endverbatim
249: *>
250: *> \param[out] INFO
251: *> \verbatim
252: *> INFO is INTEGER
253: *> = 0: successful exit
254: *> < 0: if INFO = -i, the i-th argument had an illegal value.
255: *> = 1,...,N:
256: *> The QZ iteration failed. (A,B) are not in Schur
257: *> form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
258: *> be correct for j=INFO+1,...,N.
259: *> > N: =N+1: other than QZ iteration failed in DLAQZ0.
260: *> =N+2: after reordering, roundoff changed values of
261: *> some complex eigenvalues so that leading
262: *> eigenvalues in the Generalized Schur form no
263: *> longer satisfy SELCTG=.TRUE. This could also
264: *> be caused due to scaling.
265: *> =N+3: reordering failed in DTGSEN.
266: *> \endverbatim
267: *
268: * Authors:
269: * ========
270: *
271: *> \author Univ. of Tennessee
272: *> \author Univ. of California Berkeley
273: *> \author Univ. of Colorado Denver
274: *> \author NAG Ltd.
275: *
276: *> \ingroup doubleGEeigen
277: *
278: * =====================================================================
279: SUBROUTINE DGGES3( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B,
280: $ LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL,
281: $ VSR, LDVSR, WORK, LWORK, BWORK, INFO )
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,
322: $ DLACPY, DLASCL, DLASET, DORGQR, DORMQR, DTGSEN,
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: *
671: END
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