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
9: *> Download DGEGV + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgegv.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgegv.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgegv.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
22: * BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER JOBVL, JOBVR
26: * INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
30: * $ B( LDB, * ), BETA( * ), VL( LDVL, * ),
31: * $ VR( LDVR, * ), WORK( * )
32: * ..
33: *
34: *
35: *> \par Purpose:
36: * =============
37: *>
38: *> \verbatim
39: *>
40: *> This routine is deprecated and has been replaced by routine DGGEV.
41: *>
42: *> DGEGV computes the eigenvalues and, optionally, the left and/or right
43: *> eigenvectors of a real matrix pair (A,B).
44: *> Given two square matrices A and B,
45: *> the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
46: *> eigenvalues lambda and corresponding (non-zero) eigenvectors x such
47: *> that
48: *>
49: *> A*x = lambda*B*x.
50: *>
51: *> An alternate form is to find the eigenvalues mu and corresponding
52: *> eigenvectors y such that
53: *>
54: *> mu*A*y = B*y.
55: *>
56: *> These two forms are equivalent with mu = 1/lambda and x = y if
57: *> neither lambda nor mu is zero. In order to deal with the case that
58: *> lambda or mu is zero or small, two values alpha and beta are returned
59: *> for each eigenvalue, such that lambda = alpha/beta and
60: *> mu = beta/alpha.
61: *>
62: *> The vectors x and y in the above equations are right eigenvectors of
63: *> the matrix pair (A,B). Vectors u and v satisfying
64: *>
65: *> u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B
66: *>
67: *> are left eigenvectors of (A,B).
68: *>
69: *> Note: this routine performs "full balancing" on A and B
70: *> \endverbatim
71: *
72: * Arguments:
73: * ==========
74: *
75: *> \param[in] JOBVL
76: *> \verbatim
77: *> JOBVL is CHARACTER*1
78: *> = 'N': do not compute the left generalized eigenvectors;
79: *> = 'V': compute the left generalized eigenvectors (returned
80: *> in VL).
81: *> \endverbatim
82: *>
83: *> \param[in] JOBVR
84: *> \verbatim
85: *> JOBVR is CHARACTER*1
86: *> = 'N': do not compute the right generalized eigenvectors;
87: *> = 'V': compute the right generalized eigenvectors (returned
88: *> in VR).
89: *> \endverbatim
90: *>
91: *> \param[in] N
92: *> \verbatim
93: *> N is INTEGER
94: *> The order of the matrices A, B, VL, and VR. N >= 0.
95: *> \endverbatim
96: *>
97: *> \param[in,out] A
98: *> \verbatim
99: *> A is DOUBLE PRECISION array, dimension (LDA, N)
100: *> On entry, the matrix A.
101: *> If JOBVL = 'V' or JOBVR = 'V', then on exit A
102: *> contains the real Schur form of A from the generalized Schur
103: *> factorization of the pair (A,B) after balancing.
104: *> If no eigenvectors were computed, then only the diagonal
105: *> blocks from the Schur form will be correct. See DGGHRD and
106: *> DHGEQZ for details.
107: *> \endverbatim
108: *>
109: *> \param[in] LDA
110: *> \verbatim
111: *> LDA is INTEGER
112: *> The leading dimension of A. LDA >= max(1,N).
113: *> \endverbatim
114: *>
115: *> \param[in,out] B
116: *> \verbatim
117: *> B is DOUBLE PRECISION array, dimension (LDB, N)
118: *> On entry, the matrix B.
119: *> If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
120: *> upper triangular matrix obtained from B in the generalized
121: *> Schur factorization of the pair (A,B) after balancing.
122: *> If no eigenvectors were computed, then only those elements of
123: *> B corresponding to the diagonal blocks from the Schur form of
124: *> A will be correct. See DGGHRD and DHGEQZ for details.
125: *> \endverbatim
126: *>
127: *> \param[in] LDB
128: *> \verbatim
129: *> LDB is INTEGER
130: *> The leading dimension of B. LDB >= max(1,N).
131: *> \endverbatim
132: *>
133: *> \param[out] ALPHAR
134: *> \verbatim
135: *> ALPHAR is DOUBLE PRECISION array, dimension (N)
136: *> The real parts of each scalar alpha defining an eigenvalue of
137: *> GNEP.
138: *> \endverbatim
139: *>
140: *> \param[out] ALPHAI
141: *> \verbatim
142: *> ALPHAI is DOUBLE PRECISION array, dimension (N)
143: *> The imaginary parts of each scalar alpha defining an
144: *> eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th
145: *> eigenvalue is real; if positive, then the j-th and
146: *> (j+1)-st eigenvalues are a complex conjugate pair, with
147: *> ALPHAI(j+1) = -ALPHAI(j).
148: *> \endverbatim
149: *>
150: *> \param[out] BETA
151: *> \verbatim
152: *> BETA is DOUBLE PRECISION array, dimension (N)
153: *> The scalars beta that define the eigenvalues of GNEP.
154: *>
155: *> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
156: *> beta = BETA(j) represent the j-th eigenvalue of the matrix
157: *> pair (A,B), in one of the forms lambda = alpha/beta or
158: *> mu = beta/alpha. Since either lambda or mu may overflow,
159: *> they should not, in general, be computed.
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
166: *> in the columns of VL, in the same order as their eigenvalues.
167: *> If the j-th eigenvalue is real, then u(j) = VL(:,j).
168: *> If the j-th and (j+1)-st eigenvalues form a complex conjugate
169: *> pair, then
170: *> u(j) = VL(:,j) + i*VL(:,j+1)
171: *> and
172: *> u(j+1) = VL(:,j) - i*VL(:,j+1).
173: *>
174: *> Each eigenvector is scaled so that its largest component has
175: *> abs(real part) + abs(imag. part) = 1, except for eigenvectors
176: *> corresponding to an eigenvalue with alpha = beta = 0, which
177: *> are set to zero.
178: *> Not referenced if JOBVL = 'N'.
179: *> \endverbatim
180: *>
181: *> \param[in] LDVL
182: *> \verbatim
183: *> LDVL is INTEGER
184: *> The leading dimension of the matrix VL. LDVL >= 1, and
185: *> if JOBVL = 'V', LDVL >= N.
186: *> \endverbatim
187: *>
188: *> \param[out] VR
189: *> \verbatim
190: *> VR is DOUBLE PRECISION array, dimension (LDVR,N)
191: *> If JOBVR = 'V', the right eigenvectors x(j) are stored
192: *> in the columns of VR, in the same order as their eigenvalues.
193: *> If the j-th eigenvalue is real, then x(j) = VR(:,j).
194: *> If the j-th and (j+1)-st eigenvalues form a complex conjugate
195: *> pair, then
196: *> x(j) = VR(:,j) + i*VR(:,j+1)
197: *> and
198: *> x(j+1) = VR(:,j) - i*VR(:,j+1).
199: *>
200: *> Each eigenvector is scaled so that its largest component has
201: *> abs(real part) + abs(imag. part) = 1, except for eigenvalues
202: *> corresponding to an eigenvalue with alpha = beta = 0, which
203: *> are set to zero.
204: *> Not referenced if JOBVR = 'N'.
205: *> \endverbatim
206: *>
207: *> \param[in] LDVR
208: *> \verbatim
209: *> LDVR is INTEGER
210: *> The leading dimension of the matrix VR. LDVR >= 1, and
211: *> if JOBVR = 'V', LDVR >= N.
212: *> \endverbatim
213: *>
214: *> \param[out] WORK
215: *> \verbatim
216: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
217: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
218: *> \endverbatim
219: *>
220: *> \param[in] LWORK
221: *> \verbatim
222: *> LWORK is INTEGER
223: *> The dimension of the array WORK. LWORK >= max(1,8*N).
224: *> For good performance, LWORK must generally be larger.
225: *> To compute the optimal value of LWORK, call ILAENV to get
226: *> blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute:
227: *> NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR;
228: *> The optimal LWORK is:
229: *> 2*N + MAX( 6*N, N*(NB+1) ).
230: *>
231: *> If LWORK = -1, then a workspace query is assumed; the routine
232: *> only calculates the optimal size of the WORK array, returns
233: *> this value as the first entry of the WORK array, and no error
234: *> message related to LWORK is issued by XERBLA.
235: *> \endverbatim
236: *>
237: *> \param[out] INFO
238: *> \verbatim
239: *> INFO is INTEGER
240: *> = 0: successful exit
241: *> < 0: if INFO = -i, the i-th argument had an illegal value.
242: *> = 1,...,N:
243: *> The QZ iteration failed. No eigenvectors have been
244: *> calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
245: *> should be correct for j=INFO+1,...,N.
246: *> > N: errors that usually indicate LAPACK problems:
247: *> =N+1: error return from DGGBAL
248: *> =N+2: error return from DGEQRF
249: *> =N+3: error return from DORMQR
250: *> =N+4: error return from DORGQR
251: *> =N+5: error return from DGGHRD
252: *> =N+6: error return from DHGEQZ (other than failed
253: *> iteration)
254: *> =N+7: error return from DTGEVC
255: *> =N+8: error return from DGGBAK (computing VL)
256: *> =N+9: error return from DGGBAK (computing VR)
257: *> =N+10: error return from DLASCL (various calls)
258: *> \endverbatim
259: *
260: * Authors:
261: * ========
262: *
263: *> \author Univ. of Tennessee
264: *> \author Univ. of California Berkeley
265: *> \author Univ. of Colorado Denver
266: *> \author NAG Ltd.
267: *
268: *> \date December 2016
269: *
270: *> \ingroup doubleGEeigen
271: *
272: *> \par Further Details:
273: * =====================
274: *>
275: *> \verbatim
276: *>
277: *> Balancing
278: *> ---------
279: *>
280: *> This driver calls DGGBAL to both permute and scale rows and columns
281: *> of A and B. The permutations PL and PR are chosen so that PL*A*PR
282: *> and PL*B*R will be upper triangular except for the diagonal blocks
283: *> A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
284: *> possible. The diagonal scaling matrices DL and DR are chosen so
285: *> that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
286: *> one (except for the elements that start out zero.)
287: *>
288: *> After the eigenvalues and eigenvectors of the balanced matrices
289: *> have been computed, DGGBAK transforms the eigenvectors back to what
290: *> they would have been (in perfect arithmetic) if they had not been
291: *> balanced.
292: *>
293: *> Contents of A and B on Exit
294: *> -------- -- - --- - -- ----
295: *>
296: *> If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
297: *> both), then on exit the arrays A and B will contain the real Schur
298: *> form[*] of the "balanced" versions of A and B. If no eigenvectors
299: *> are computed, then only the diagonal blocks will be correct.
300: *>
301: *> [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations",
302: *> by Golub & van Loan, pub. by Johns Hopkins U. Press.
303: *> \endverbatim
304: *>
305: * =====================================================================
306: SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
307: $ BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
308: *
309: * -- LAPACK driver routine (version 3.7.0) --
310: * -- LAPACK is a software package provided by Univ. of Tennessee, --
311: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
312: * December 2016
313: *
314: * .. Scalar Arguments ..
315: CHARACTER JOBVL, JOBVR
316: INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
317: * ..
318: * .. Array Arguments ..
319: DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
320: $ B( LDB, * ), BETA( * ), VL( LDVL, * ),
321: $ VR( LDVR, * ), WORK( * )
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 ILIMIT, ILV, ILVL, ILVR, LQUERY
332: CHARACTER CHTEMP
333: INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
334: $ IN, IRIGHT, IROWS, ITAU, IWORK, JC, JR, LOPT,
335: $ LWKMIN, LWKOPT, NB, NB1, NB2, NB3
336: DOUBLE PRECISION ABSAI, ABSAR, ABSB, ANRM, ANRM1, ANRM2, BNRM,
337: $ BNRM1, BNRM2, EPS, ONEPLS, SAFMAX, SAFMIN,
338: $ SALFAI, SALFAR, SBETA, SCALE, TEMP
339: * ..
340: * .. Local Arrays ..
341: LOGICAL LDUMMA( 1 )
342: * ..
343: * .. External Subroutines ..
344: EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLACPY,
345: $ DLASCL, DLASET, DORGQR, DORMQR, DTGEVC, XERBLA
346: * ..
347: * .. External Functions ..
348: LOGICAL LSAME
349: INTEGER ILAENV
350: DOUBLE PRECISION DLAMCH, DLANGE
351: EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
352: * ..
353: * .. Intrinsic Functions ..
354: INTRINSIC ABS, INT, MAX
355: * ..
356: * .. Executable Statements ..
357: *
358: * Decode the input arguments
359: *
360: IF( LSAME( JOBVL, 'N' ) ) THEN
361: IJOBVL = 1
362: ILVL = .FALSE.
363: ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
364: IJOBVL = 2
365: ILVL = .TRUE.
366: ELSE
367: IJOBVL = -1
368: ILVL = .FALSE.
369: END IF
370: *
371: IF( LSAME( JOBVR, 'N' ) ) THEN
372: IJOBVR = 1
373: ILVR = .FALSE.
374: ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
375: IJOBVR = 2
376: ILVR = .TRUE.
377: ELSE
378: IJOBVR = -1
379: ILVR = .FALSE.
380: END IF
381: ILV = ILVL .OR. ILVR
382: *
383: * Test the input arguments
384: *
385: LWKMIN = MAX( 8*N, 1 )
386: LWKOPT = LWKMIN
387: WORK( 1 ) = LWKOPT
388: LQUERY = ( LWORK.EQ.-1 )
389: INFO = 0
390: IF( IJOBVL.LE.0 ) THEN
391: INFO = -1
392: ELSE IF( IJOBVR.LE.0 ) THEN
393: INFO = -2
394: ELSE IF( N.LT.0 ) THEN
395: INFO = -3
396: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
397: INFO = -5
398: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
399: INFO = -7
400: ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
401: INFO = -12
402: ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
403: INFO = -14
404: ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
405: INFO = -16
406: END IF
407: *
408: IF( INFO.EQ.0 ) THEN
409: NB1 = ILAENV( 1, 'DGEQRF', ' ', N, N, -1, -1 )
410: NB2 = ILAENV( 1, 'DORMQR', ' ', N, N, N, -1 )
411: NB3 = ILAENV( 1, 'DORGQR', ' ', N, N, N, -1 )
412: NB = MAX( NB1, NB2, NB3 )
413: LOPT = 2*N + MAX( 6*N, N*( NB+1 ) )
414: WORK( 1 ) = LOPT
415: END IF
416: *
417: IF( INFO.NE.0 ) THEN
418: CALL XERBLA( 'DGEGV ', -INFO )
419: RETURN
420: ELSE IF( LQUERY ) THEN
421: RETURN
422: END IF
423: *
424: * Quick return if possible
425: *
426: IF( N.EQ.0 )
427: $ RETURN
428: *
429: * Get machine constants
430: *
431: EPS = DLAMCH( 'E' )*DLAMCH( 'B' )
432: SAFMIN = DLAMCH( 'S' )
433: SAFMIN = SAFMIN + SAFMIN
434: SAFMAX = ONE / SAFMIN
435: ONEPLS = ONE + ( 4*EPS )
436: *
437: * Scale A
438: *
439: ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
440: ANRM1 = ANRM
441: ANRM2 = ONE
442: IF( ANRM.LT.ONE ) THEN
443: IF( SAFMAX*ANRM.LT.ONE ) THEN
444: ANRM1 = SAFMIN
445: ANRM2 = SAFMAX*ANRM
446: END IF
447: END IF
448: *
449: IF( ANRM.GT.ZERO ) THEN
450: CALL DLASCL( 'G', -1, -1, ANRM, ONE, N, N, A, LDA, IINFO )
451: IF( IINFO.NE.0 ) THEN
452: INFO = N + 10
453: RETURN
454: END IF
455: END IF
456: *
457: * Scale B
458: *
459: BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
460: BNRM1 = BNRM
461: BNRM2 = ONE
462: IF( BNRM.LT.ONE ) THEN
463: IF( SAFMAX*BNRM.LT.ONE ) THEN
464: BNRM1 = SAFMIN
465: BNRM2 = SAFMAX*BNRM
466: END IF
467: END IF
468: *
469: IF( BNRM.GT.ZERO ) THEN
470: CALL DLASCL( 'G', -1, -1, BNRM, ONE, N, N, B, LDB, IINFO )
471: IF( IINFO.NE.0 ) THEN
472: INFO = N + 10
473: RETURN
474: END IF
475: END IF
476: *
477: * Permute the matrix to make it more nearly triangular
478: * Workspace layout: (8*N words -- "work" requires 6*N words)
479: * left_permutation, right_permutation, work...
480: *
481: ILEFT = 1
482: IRIGHT = N + 1
483: IWORK = IRIGHT + N
484: CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
485: $ WORK( IRIGHT ), WORK( IWORK ), IINFO )
486: IF( IINFO.NE.0 ) THEN
487: INFO = N + 1
488: GO TO 120
489: END IF
490: *
491: * Reduce B to triangular form, and initialize VL and/or VR
492: * Workspace layout: ("work..." must have at least N words)
493: * left_permutation, right_permutation, tau, work...
494: *
495: IROWS = IHI + 1 - ILO
496: IF( ILV ) THEN
497: ICOLS = N + 1 - ILO
498: ELSE
499: ICOLS = IROWS
500: END IF
501: ITAU = IWORK
502: IWORK = ITAU + IROWS
503: CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
504: $ WORK( IWORK ), LWORK+1-IWORK, IINFO )
505: IF( IINFO.GE.0 )
506: $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
507: IF( IINFO.NE.0 ) THEN
508: INFO = N + 2
509: GO TO 120
510: END IF
511: *
512: CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
513: $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
514: $ LWORK+1-IWORK, IINFO )
515: IF( IINFO.GE.0 )
516: $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
517: IF( IINFO.NE.0 ) THEN
518: INFO = N + 3
519: GO TO 120
520: END IF
521: *
522: IF( ILVL ) THEN
523: CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
524: CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
525: $ VL( ILO+1, ILO ), LDVL )
526: CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
527: $ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
528: $ IINFO )
529: IF( IINFO.GE.0 )
530: $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
531: IF( IINFO.NE.0 ) THEN
532: INFO = N + 4
533: GO TO 120
534: END IF
535: END IF
536: *
537: IF( ILVR )
538: $ CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
539: *
540: * Reduce to generalized Hessenberg form
541: *
542: IF( ILV ) THEN
543: *
544: * Eigenvectors requested -- work on whole matrix.
545: *
546: CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
547: $ LDVL, VR, LDVR, IINFO )
548: ELSE
549: CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
550: $ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IINFO )
551: END IF
552: IF( IINFO.NE.0 ) THEN
553: INFO = N + 5
554: GO TO 120
555: END IF
556: *
557: * Perform QZ algorithm
558: * Workspace layout: ("work..." must have at least 1 word)
559: * left_permutation, right_permutation, work...
560: *
561: IWORK = ITAU
562: IF( ILV ) THEN
563: CHTEMP = 'S'
564: ELSE
565: CHTEMP = 'E'
566: END IF
567: CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
568: $ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR,
569: $ WORK( IWORK ), LWORK+1-IWORK, IINFO )
570: IF( IINFO.GE.0 )
571: $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
572: IF( IINFO.NE.0 ) THEN
573: IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
574: INFO = IINFO
575: ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
576: INFO = IINFO - N
577: ELSE
578: INFO = N + 6
579: END IF
580: GO TO 120
581: END IF
582: *
583: IF( ILV ) THEN
584: *
585: * Compute Eigenvectors (DTGEVC requires 6*N words of workspace)
586: *
587: IF( ILVL ) THEN
588: IF( ILVR ) THEN
589: CHTEMP = 'B'
590: ELSE
591: CHTEMP = 'L'
592: END IF
593: ELSE
594: CHTEMP = 'R'
595: END IF
596: *
597: CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
598: $ VR, LDVR, N, IN, WORK( IWORK ), IINFO )
599: IF( IINFO.NE.0 ) THEN
600: INFO = N + 7
601: GO TO 120
602: END IF
603: *
604: * Undo balancing on VL and VR, rescale
605: *
606: IF( ILVL ) THEN
607: CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
608: $ WORK( IRIGHT ), N, VL, LDVL, IINFO )
609: IF( IINFO.NE.0 ) THEN
610: INFO = N + 8
611: GO TO 120
612: END IF
613: DO 50 JC = 1, N
614: IF( ALPHAI( JC ).LT.ZERO )
615: $ GO TO 50
616: TEMP = ZERO
617: IF( ALPHAI( JC ).EQ.ZERO ) THEN
618: DO 10 JR = 1, N
619: TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
620: 10 CONTINUE
621: ELSE
622: DO 20 JR = 1, N
623: TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
624: $ ABS( VL( JR, JC+1 ) ) )
625: 20 CONTINUE
626: END IF
627: IF( TEMP.LT.SAFMIN )
628: $ GO TO 50
629: TEMP = ONE / TEMP
630: IF( ALPHAI( JC ).EQ.ZERO ) THEN
631: DO 30 JR = 1, N
632: VL( JR, JC ) = VL( JR, JC )*TEMP
633: 30 CONTINUE
634: ELSE
635: DO 40 JR = 1, N
636: VL( JR, JC ) = VL( JR, JC )*TEMP
637: VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
638: 40 CONTINUE
639: END IF
640: 50 CONTINUE
641: END IF
642: IF( ILVR ) THEN
643: CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
644: $ WORK( IRIGHT ), N, VR, LDVR, IINFO )
645: IF( IINFO.NE.0 ) THEN
646: INFO = N + 9
647: GO TO 120
648: END IF
649: DO 100 JC = 1, N
650: IF( ALPHAI( JC ).LT.ZERO )
651: $ GO TO 100
652: TEMP = ZERO
653: IF( ALPHAI( JC ).EQ.ZERO ) THEN
654: DO 60 JR = 1, N
655: TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
656: 60 CONTINUE
657: ELSE
658: DO 70 JR = 1, N
659: TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
660: $ ABS( VR( JR, JC+1 ) ) )
661: 70 CONTINUE
662: END IF
663: IF( TEMP.LT.SAFMIN )
664: $ GO TO 100
665: TEMP = ONE / TEMP
666: IF( ALPHAI( JC ).EQ.ZERO ) THEN
667: DO 80 JR = 1, N
668: VR( JR, JC ) = VR( JR, JC )*TEMP
669: 80 CONTINUE
670: ELSE
671: DO 90 JR = 1, N
672: VR( JR, JC ) = VR( JR, JC )*TEMP
673: VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
674: 90 CONTINUE
675: END IF
676: 100 CONTINUE
677: END IF
678: *
679: * End of eigenvector calculation
680: *
681: END IF
682: *
683: * Undo scaling in alpha, beta
684: *
685: * Note: this does not give the alpha and beta for the unscaled
686: * problem.
687: *
688: * Un-scaling is limited to avoid underflow in alpha and beta
689: * if they are significant.
690: *
691: DO 110 JC = 1, N
692: ABSAR = ABS( ALPHAR( JC ) )
693: ABSAI = ABS( ALPHAI( JC ) )
694: ABSB = ABS( BETA( JC ) )
695: SALFAR = ANRM*ALPHAR( JC )
696: SALFAI = ANRM*ALPHAI( JC )
697: SBETA = BNRM*BETA( JC )
698: ILIMIT = .FALSE.
699: SCALE = ONE
700: *
701: * Check for significant underflow in ALPHAI
702: *
703: IF( ABS( SALFAI ).LT.SAFMIN .AND. ABSAI.GE.
704: $ MAX( SAFMIN, EPS*ABSAR, EPS*ABSB ) ) THEN
705: ILIMIT = .TRUE.
706: SCALE = ( ONEPLS*SAFMIN / ANRM1 ) /
707: $ MAX( ONEPLS*SAFMIN, ANRM2*ABSAI )
708: *
709: ELSE IF( SALFAI.EQ.ZERO ) THEN
710: *
711: * If insignificant underflow in ALPHAI, then make the
712: * conjugate eigenvalue real.
713: *
714: IF( ALPHAI( JC ).LT.ZERO .AND. JC.GT.1 ) THEN
715: ALPHAI( JC-1 ) = ZERO
716: ELSE IF( ALPHAI( JC ).GT.ZERO .AND. JC.LT.N ) THEN
717: ALPHAI( JC+1 ) = ZERO
718: END IF
719: END IF
720: *
721: * Check for significant underflow in ALPHAR
722: *
723: IF( ABS( SALFAR ).LT.SAFMIN .AND. ABSAR.GE.
724: $ MAX( SAFMIN, EPS*ABSAI, EPS*ABSB ) ) THEN
725: ILIMIT = .TRUE.
726: SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / ANRM1 ) /
727: $ MAX( ONEPLS*SAFMIN, ANRM2*ABSAR ) )
728: END IF
729: *
730: * Check for significant underflow in BETA
731: *
732: IF( ABS( SBETA ).LT.SAFMIN .AND. ABSB.GE.
733: $ MAX( SAFMIN, EPS*ABSAR, EPS*ABSAI ) ) THEN
734: ILIMIT = .TRUE.
735: SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / BNRM1 ) /
736: $ MAX( ONEPLS*SAFMIN, BNRM2*ABSB ) )
737: END IF
738: *
739: * Check for possible overflow when limiting scaling
740: *
741: IF( ILIMIT ) THEN
742: TEMP = ( SCALE*SAFMIN )*MAX( ABS( SALFAR ), ABS( SALFAI ),
743: $ ABS( SBETA ) )
744: IF( TEMP.GT.ONE )
745: $ SCALE = SCALE / TEMP
746: IF( SCALE.LT.ONE )
747: $ ILIMIT = .FALSE.
748: END IF
749: *
750: * Recompute un-scaled ALPHAR, ALPHAI, BETA if necessary.
751: *
752: IF( ILIMIT ) THEN
753: SALFAR = ( SCALE*ALPHAR( JC ) )*ANRM
754: SALFAI = ( SCALE*ALPHAI( JC ) )*ANRM
755: SBETA = ( SCALE*BETA( JC ) )*BNRM
756: END IF
757: ALPHAR( JC ) = SALFAR
758: ALPHAI( JC ) = SALFAI
759: BETA( JC ) = SBETA
760: 110 CONTINUE
761: *
762: 120 CONTINUE
763: WORK( 1 ) = LWKOPT
764: *
765: RETURN
766: *
767: * End of DGEGV
768: *
769: END
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