1: *> \brief <b> ZGEEVX 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 ZGEEVX + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgeevx.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgeevx.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgeevx.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL,
22: * LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE,
23: * RCONDV, WORK, LWORK, RWORK, 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: * DOUBLE PRECISION RCONDE( * ), RCONDV( * ), RWORK( * ),
32: * $ SCALE( * )
33: * COMPLEX*16 A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
34: * $ W( * ), WORK( * )
35: * ..
36: *
37: *
38: *> \par Purpose:
39: * =============
40: *>
41: *> \verbatim
42: *>
43: *> ZGEEVX computes for an N-by-N complex 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, ie. 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 COMPLEX*16 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 Schur form of the balanced
138: *> version of the 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] W
148: *> \verbatim
149: *> W is COMPLEX*16 array, dimension (N)
150: *> W contains the computed eigenvalues.
151: *> \endverbatim
152: *>
153: *> \param[out] VL
154: *> \verbatim
155: *> VL is COMPLEX*16 array, dimension (LDVL,N)
156: *> If JOBVL = 'V', the left eigenvectors u(j) are stored one
157: *> after another in the columns of VL, in the same order
158: *> as their eigenvalues.
159: *> If JOBVL = 'N', VL is not referenced.
160: *> u(j) = VL(:,j), the j-th column of VL.
161: *> \endverbatim
162: *>
163: *> \param[in] LDVL
164: *> \verbatim
165: *> LDVL is INTEGER
166: *> The leading dimension of the array VL. LDVL >= 1; if
167: *> JOBVL = 'V', LDVL >= N.
168: *> \endverbatim
169: *>
170: *> \param[out] VR
171: *> \verbatim
172: *> VR is COMPLEX*16 array, dimension (LDVR,N)
173: *> If JOBVR = 'V', the right eigenvectors v(j) are stored one
174: *> after another in the columns of VR, in the same order
175: *> as their eigenvalues.
176: *> If JOBVR = 'N', VR is not referenced.
177: *> v(j) = VR(:,j), the j-th column of VR.
178: *> \endverbatim
179: *>
180: *> \param[in] LDVR
181: *> \verbatim
182: *> LDVR is INTEGER
183: *> The leading dimension of the array VR. LDVR >= 1; if
184: *> JOBVR = 'V', LDVR >= N.
185: *> \endverbatim
186: *>
187: *> \param[out] ILO
188: *> \verbatim
189: *> ILO is INTEGER
190: *> \endverbatim
191: *>
192: *> \param[out] IHI
193: *> \verbatim
194: *> IHI is INTEGER
195: *> ILO and IHI are integer values determined when A was
196: *> balanced. The balanced A(i,j) = 0 if I > J and
197: *> J = 1,...,ILO-1 or I = IHI+1,...,N.
198: *> \endverbatim
199: *>
200: *> \param[out] SCALE
201: *> \verbatim
202: *> SCALE is DOUBLE PRECISION array, dimension (N)
203: *> Details of the permutations and scaling factors applied
204: *> when balancing A. If P(j) is the index of the row and column
205: *> interchanged with row and column j, and D(j) is the scaling
206: *> factor applied to row and column j, then
207: *> SCALE(J) = P(J), for J = 1,...,ILO-1
208: *> = D(J), for J = ILO,...,IHI
209: *> = P(J) for J = IHI+1,...,N.
210: *> The order in which the interchanges are made is N to IHI+1,
211: *> then 1 to ILO-1.
212: *> \endverbatim
213: *>
214: *> \param[out] ABNRM
215: *> \verbatim
216: *> ABNRM is DOUBLE PRECISION
217: *> The one-norm of the balanced matrix (the maximum
218: *> of the sum of absolute values of elements of any column).
219: *> \endverbatim
220: *>
221: *> \param[out] RCONDE
222: *> \verbatim
223: *> RCONDE is DOUBLE PRECISION array, dimension (N)
224: *> RCONDE(j) is the reciprocal condition number of the j-th
225: *> eigenvalue.
226: *> \endverbatim
227: *>
228: *> \param[out] RCONDV
229: *> \verbatim
230: *> RCONDV is DOUBLE PRECISION array, dimension (N)
231: *> RCONDV(j) is the reciprocal condition number of the j-th
232: *> right eigenvector.
233: *> \endverbatim
234: *>
235: *> \param[out] WORK
236: *> \verbatim
237: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
238: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
239: *> \endverbatim
240: *>
241: *> \param[in] LWORK
242: *> \verbatim
243: *> LWORK is INTEGER
244: *> The dimension of the array WORK. If SENSE = 'N' or 'E',
245: *> LWORK >= max(1,2*N), and if SENSE = 'V' or 'B',
246: *> LWORK >= N*N+2*N.
247: *> For good performance, LWORK must generally be larger.
248: *>
249: *> If LWORK = -1, then a workspace query is assumed; the routine
250: *> only calculates the optimal size of the WORK array, returns
251: *> this value as the first entry of the WORK array, and no error
252: *> message related to LWORK is issued by XERBLA.
253: *> \endverbatim
254: *>
255: *> \param[out] RWORK
256: *> \verbatim
257: *> RWORK is DOUBLE PRECISION array, dimension (2*N)
258: *> \endverbatim
259: *>
260: *> \param[out] INFO
261: *> \verbatim
262: *> INFO is INTEGER
263: *> = 0: successful exit
264: *> < 0: if INFO = -i, the i-th argument had an illegal value.
265: *> > 0: if INFO = i, the QR algorithm failed to compute all the
266: *> eigenvalues, and no eigenvectors or condition numbers
267: *> have been computed; elements 1:ILO-1 and i+1:N of W
268: *> contain eigenvalues which have converged.
269: *> \endverbatim
270: *
271: * Authors:
272: * ========
273: *
274: *> \author Univ. of Tennessee
275: *> \author Univ. of California Berkeley
276: *> \author Univ. of Colorado Denver
277: *> \author NAG Ltd.
278: *
279: *> \date June 2016
280: *
281: * @precisions fortran z -> c
282: *
283: *> \ingroup complex16GEeigen
284: *
285: * =====================================================================
286: SUBROUTINE ZGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL,
287: $ LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE,
288: $ RCONDV, WORK, LWORK, RWORK, INFO )
289: implicit none
290: *
291: * -- LAPACK driver routine (version 3.7.0) --
292: * -- LAPACK is a software package provided by Univ. of Tennessee, --
293: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
294: * June 2016
295: *
296: * .. Scalar Arguments ..
297: CHARACTER BALANC, JOBVL, JOBVR, SENSE
298: INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
299: DOUBLE PRECISION ABNRM
300: * ..
301: * .. Array Arguments ..
302: DOUBLE PRECISION RCONDE( * ), RCONDV( * ), RWORK( * ),
303: $ SCALE( * )
304: COMPLEX*16 A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
305: $ W( * ), WORK( * )
306: * ..
307: *
308: * =====================================================================
309: *
310: * .. Parameters ..
311: DOUBLE PRECISION ZERO, ONE
312: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
313: * ..
314: * .. Local Scalars ..
315: LOGICAL LQUERY, SCALEA, WANTVL, WANTVR, WNTSNB, WNTSNE,
316: $ WNTSNN, WNTSNV
317: CHARACTER JOB, SIDE
318: INTEGER HSWORK, I, ICOND, IERR, ITAU, IWRK, K,
319: $ LWORK_TREVC, MAXWRK, MINWRK, NOUT
320: DOUBLE PRECISION ANRM, BIGNUM, CSCALE, EPS, SCL, SMLNUM
321: COMPLEX*16 TMP
322: * ..
323: * .. Local Arrays ..
324: LOGICAL SELECT( 1 )
325: DOUBLE PRECISION DUM( 1 )
326: * ..
327: * .. External Subroutines ..
328: EXTERNAL DLABAD, DLASCL, XERBLA, ZDSCAL, ZGEBAK, ZGEBAL,
329: $ ZGEHRD, ZHSEQR, ZLACPY, ZLASCL, ZSCAL, ZTREVC3,
330: $ ZTRSNA, ZUNGHR
331: * ..
332: * .. External Functions ..
333: LOGICAL LSAME
334: INTEGER IDAMAX, ILAENV
335: DOUBLE PRECISION DLAMCH, DZNRM2, ZLANGE
336: EXTERNAL LSAME, IDAMAX, ILAENV, DLAMCH, DZNRM2, ZLANGE
337: * ..
338: * .. Intrinsic Functions ..
339: INTRINSIC DBLE, DCMPLX, CONJG, AIMAG, MAX, SQRT
340: * ..
341: * .. Executable Statements ..
342: *
343: * Test the input arguments
344: *
345: INFO = 0
346: LQUERY = ( LWORK.EQ.-1 )
347: WANTVL = LSAME( JOBVL, 'V' )
348: WANTVR = LSAME( JOBVR, 'V' )
349: WNTSNN = LSAME( SENSE, 'N' )
350: WNTSNE = LSAME( SENSE, 'E' )
351: WNTSNV = LSAME( SENSE, 'V' )
352: WNTSNB = LSAME( SENSE, 'B' )
353: IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'S' ) .OR.
354: $ LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) ) THEN
355: INFO = -1
356: ELSE IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
357: INFO = -2
358: ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
359: INFO = -3
360: ELSE IF( .NOT.( WNTSNN .OR. WNTSNE .OR. WNTSNB .OR. WNTSNV ) .OR.
361: $ ( ( WNTSNE .OR. WNTSNB ) .AND. .NOT.( WANTVL .AND.
362: $ WANTVR ) ) ) THEN
363: INFO = -4
364: ELSE IF( N.LT.0 ) THEN
365: INFO = -5
366: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
367: INFO = -7
368: ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
369: INFO = -10
370: ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
371: INFO = -12
372: END IF
373: *
374: * Compute workspace
375: * (Note: Comments in the code beginning "Workspace:" describe the
376: * minimal amount of workspace needed at that point in the code,
377: * as well as the preferred amount for good performance.
378: * CWorkspace refers to complex workspace, and RWorkspace to real
379: * workspace. NB refers to the optimal block size for the
380: * immediately following subroutine, as returned by ILAENV.
381: * HSWORK refers to the workspace preferred by ZHSEQR, as
382: * calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
383: * the worst case.)
384: *
385: IF( INFO.EQ.0 ) THEN
386: IF( N.EQ.0 ) THEN
387: MINWRK = 1
388: MAXWRK = 1
389: ELSE
390: MAXWRK = N + N*ILAENV( 1, 'ZGEHRD', ' ', N, 1, N, 0 )
391: *
392: IF( WANTVL ) THEN
393: CALL ZTREVC3( 'L', 'B', SELECT, N, A, LDA,
394: $ VL, LDVL, VR, LDVR,
395: $ N, NOUT, WORK, -1, RWORK, -1, IERR )
396: LWORK_TREVC = INT( WORK(1) )
397: MAXWRK = MAX( MAXWRK, LWORK_TREVC )
398: CALL ZHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VL, LDVL,
399: $ WORK, -1, INFO )
400: ELSE IF( WANTVR ) THEN
401: CALL ZTREVC3( 'R', 'B', SELECT, N, A, LDA,
402: $ VL, LDVL, VR, LDVR,
403: $ N, NOUT, WORK, -1, RWORK, -1, IERR )
404: LWORK_TREVC = INT( WORK(1) )
405: MAXWRK = MAX( MAXWRK, LWORK_TREVC )
406: CALL ZHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VR, LDVR,
407: $ WORK, -1, INFO )
408: ELSE
409: IF( WNTSNN ) THEN
410: CALL ZHSEQR( 'E', 'N', N, 1, N, A, LDA, W, VR, LDVR,
411: $ WORK, -1, INFO )
412: ELSE
413: CALL ZHSEQR( 'S', 'N', N, 1, N, A, LDA, W, VR, LDVR,
414: $ WORK, -1, INFO )
415: END IF
416: END IF
417: HSWORK = INT( WORK(1) )
418: *
419: IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN
420: MINWRK = 2*N
421: IF( .NOT.( WNTSNN .OR. WNTSNE ) )
422: $ MINWRK = MAX( MINWRK, N*N + 2*N )
423: MAXWRK = MAX( MAXWRK, HSWORK )
424: IF( .NOT.( WNTSNN .OR. WNTSNE ) )
425: $ MAXWRK = MAX( MAXWRK, N*N + 2*N )
426: ELSE
427: MINWRK = 2*N
428: IF( .NOT.( WNTSNN .OR. WNTSNE ) )
429: $ MINWRK = MAX( MINWRK, N*N + 2*N )
430: MAXWRK = MAX( MAXWRK, HSWORK )
431: MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'ZUNGHR',
432: $ ' ', N, 1, N, -1 ) )
433: IF( .NOT.( WNTSNN .OR. WNTSNE ) )
434: $ MAXWRK = MAX( MAXWRK, N*N + 2*N )
435: MAXWRK = MAX( MAXWRK, 2*N )
436: END IF
437: MAXWRK = MAX( MAXWRK, MINWRK )
438: END IF
439: WORK( 1 ) = MAXWRK
440: *
441: IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
442: INFO = -20
443: END IF
444: END IF
445: *
446: IF( INFO.NE.0 ) THEN
447: CALL XERBLA( 'ZGEEVX', -INFO )
448: RETURN
449: ELSE IF( LQUERY ) THEN
450: RETURN
451: END IF
452: *
453: * Quick return if possible
454: *
455: IF( N.EQ.0 )
456: $ RETURN
457: *
458: * Get machine constants
459: *
460: EPS = DLAMCH( 'P' )
461: SMLNUM = DLAMCH( 'S' )
462: BIGNUM = ONE / SMLNUM
463: CALL DLABAD( SMLNUM, BIGNUM )
464: SMLNUM = SQRT( SMLNUM ) / EPS
465: BIGNUM = ONE / SMLNUM
466: *
467: * Scale A if max element outside range [SMLNUM,BIGNUM]
468: *
469: ICOND = 0
470: ANRM = ZLANGE( 'M', N, N, A, LDA, DUM )
471: SCALEA = .FALSE.
472: IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
473: SCALEA = .TRUE.
474: CSCALE = SMLNUM
475: ELSE IF( ANRM.GT.BIGNUM ) THEN
476: SCALEA = .TRUE.
477: CSCALE = BIGNUM
478: END IF
479: IF( SCALEA )
480: $ CALL ZLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
481: *
482: * Balance the matrix and compute ABNRM
483: *
484: CALL ZGEBAL( BALANC, N, A, LDA, ILO, IHI, SCALE, IERR )
485: ABNRM = ZLANGE( '1', N, N, A, LDA, DUM )
486: IF( SCALEA ) THEN
487: DUM( 1 ) = ABNRM
488: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR )
489: ABNRM = DUM( 1 )
490: END IF
491: *
492: * Reduce to upper Hessenberg form
493: * (CWorkspace: need 2*N, prefer N+N*NB)
494: * (RWorkspace: none)
495: *
496: ITAU = 1
497: IWRK = ITAU + N
498: CALL ZGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
499: $ LWORK-IWRK+1, IERR )
500: *
501: IF( WANTVL ) THEN
502: *
503: * Want left eigenvectors
504: * Copy Householder vectors to VL
505: *
506: SIDE = 'L'
507: CALL ZLACPY( 'L', N, N, A, LDA, VL, LDVL )
508: *
509: * Generate unitary matrix in VL
510: * (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
511: * (RWorkspace: none)
512: *
513: CALL ZUNGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ),
514: $ LWORK-IWRK+1, IERR )
515: *
516: * Perform QR iteration, accumulating Schur vectors in VL
517: * (CWorkspace: need 1, prefer HSWORK (see comments) )
518: * (RWorkspace: none)
519: *
520: IWRK = ITAU
521: CALL ZHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VL, LDVL,
522: $ WORK( IWRK ), LWORK-IWRK+1, INFO )
523: *
524: IF( WANTVR ) THEN
525: *
526: * Want left and right eigenvectors
527: * Copy Schur vectors to VR
528: *
529: SIDE = 'B'
530: CALL ZLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
531: END IF
532: *
533: ELSE IF( WANTVR ) THEN
534: *
535: * Want right eigenvectors
536: * Copy Householder vectors to VR
537: *
538: SIDE = 'R'
539: CALL ZLACPY( 'L', N, N, A, LDA, VR, LDVR )
540: *
541: * Generate unitary matrix in VR
542: * (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
543: * (RWorkspace: none)
544: *
545: CALL ZUNGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ),
546: $ LWORK-IWRK+1, IERR )
547: *
548: * Perform QR iteration, accumulating Schur vectors in VR
549: * (CWorkspace: need 1, prefer HSWORK (see comments) )
550: * (RWorkspace: none)
551: *
552: IWRK = ITAU
553: CALL ZHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VR, LDVR,
554: $ WORK( IWRK ), LWORK-IWRK+1, INFO )
555: *
556: ELSE
557: *
558: * Compute eigenvalues only
559: * If condition numbers desired, compute Schur form
560: *
561: IF( WNTSNN ) THEN
562: JOB = 'E'
563: ELSE
564: JOB = 'S'
565: END IF
566: *
567: * (CWorkspace: need 1, prefer HSWORK (see comments) )
568: * (RWorkspace: none)
569: *
570: IWRK = ITAU
571: CALL ZHSEQR( JOB, 'N', N, ILO, IHI, A, LDA, W, VR, LDVR,
572: $ WORK( IWRK ), LWORK-IWRK+1, INFO )
573: END IF
574: *
575: * If INFO .NE. 0 from ZHSEQR, then quit
576: *
577: IF( INFO.NE.0 )
578: $ GO TO 50
579: *
580: IF( WANTVL .OR. WANTVR ) THEN
581: *
582: * Compute left and/or right eigenvectors
583: * (CWorkspace: need 2*N, prefer N + 2*N*NB)
584: * (RWorkspace: need N)
585: *
586: CALL ZTREVC3( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
587: $ N, NOUT, WORK( IWRK ), LWORK-IWRK+1,
588: $ RWORK, N, IERR )
589: END IF
590: *
591: * Compute condition numbers if desired
592: * (CWorkspace: need N*N+2*N unless SENSE = 'E')
593: * (RWorkspace: need 2*N unless SENSE = 'E')
594: *
595: IF( .NOT.WNTSNN ) THEN
596: CALL ZTRSNA( SENSE, 'A', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
597: $ RCONDE, RCONDV, N, NOUT, WORK( IWRK ), N, RWORK,
598: $ ICOND )
599: END IF
600: *
601: IF( WANTVL ) THEN
602: *
603: * Undo balancing of left eigenvectors
604: *
605: CALL ZGEBAK( BALANC, 'L', N, ILO, IHI, SCALE, N, VL, LDVL,
606: $ IERR )
607: *
608: * Normalize left eigenvectors and make largest component real
609: *
610: DO 20 I = 1, N
611: SCL = ONE / DZNRM2( N, VL( 1, I ), 1 )
612: CALL ZDSCAL( N, SCL, VL( 1, I ), 1 )
613: DO 10 K = 1, N
614: RWORK( K ) = DBLE( VL( K, I ) )**2 +
615: $ AIMAG( VL( K, I ) )**2
616: 10 CONTINUE
617: K = IDAMAX( N, RWORK, 1 )
618: TMP = CONJG( VL( K, I ) ) / SQRT( RWORK( K ) )
619: CALL ZSCAL( N, TMP, VL( 1, I ), 1 )
620: VL( K, I ) = DCMPLX( DBLE( VL( K, I ) ), ZERO )
621: 20 CONTINUE
622: END IF
623: *
624: IF( WANTVR ) THEN
625: *
626: * Undo balancing of right eigenvectors
627: *
628: CALL ZGEBAK( BALANC, 'R', N, ILO, IHI, SCALE, N, VR, LDVR,
629: $ IERR )
630: *
631: * Normalize right eigenvectors and make largest component real
632: *
633: DO 40 I = 1, N
634: SCL = ONE / DZNRM2( N, VR( 1, I ), 1 )
635: CALL ZDSCAL( N, SCL, VR( 1, I ), 1 )
636: DO 30 K = 1, N
637: RWORK( K ) = DBLE( VR( K, I ) )**2 +
638: $ AIMAG( VR( K, I ) )**2
639: 30 CONTINUE
640: K = IDAMAX( N, RWORK, 1 )
641: TMP = CONJG( VR( K, I ) ) / SQRT( RWORK( K ) )
642: CALL ZSCAL( N, TMP, VR( 1, I ), 1 )
643: VR( K, I ) = DCMPLX( DBLE( VR( K, I ) ), ZERO )
644: 40 CONTINUE
645: END IF
646: *
647: * Undo scaling if necessary
648: *
649: 50 CONTINUE
650: IF( SCALEA ) THEN
651: CALL ZLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, W( INFO+1 ),
652: $ MAX( N-INFO, 1 ), IERR )
653: IF( INFO.EQ.0 ) THEN
654: IF( ( WNTSNV .OR. WNTSNB ) .AND. ICOND.EQ.0 )
655: $ CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, RCONDV, N,
656: $ IERR )
657: ELSE
658: CALL ZLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, W, N, IERR )
659: END IF
660: END IF
661: *
662: WORK( 1 ) = MAXWRK
663: RETURN
664: *
665: * End of ZGEEVX
666: *
667: END
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