Annotation of rpl/lapack/lapack/dgeevx.f, revision 1.13
1.9 bertrand 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 DGEEVX + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgeevx.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgeevx.f">
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
1.12 bertrand 56: *> u(j)**H * A = lambda(j) * u(j)**H
57: *> where u(j)**H denotes the conjugate-transpose of u(j).
1.9 bertrand 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: *
1.12 bertrand 297: *> \date September 2012
1.9 bertrand 298: *
299: *> \ingroup doubleGEeigen
300: *
301: * =====================================================================
1.1 bertrand 302: SUBROUTINE DGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI,
303: $ VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM,
304: $ RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )
305: *
1.12 bertrand 306: * -- LAPACK driver routine (version 3.4.2) --
1.1 bertrand 307: * -- LAPACK is a software package provided by Univ. of Tennessee, --
308: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.12 bertrand 309: * September 2012
1.1 bertrand 310: *
311: * .. Scalar Arguments ..
312: CHARACTER BALANC, JOBVL, JOBVR, SENSE
313: INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
314: DOUBLE PRECISION ABNRM
315: * ..
316: * .. Array Arguments ..
317: INTEGER IWORK( * )
318: DOUBLE PRECISION A( LDA, * ), RCONDE( * ), RCONDV( * ),
319: $ SCALE( * ), VL( LDVL, * ), VR( LDVR, * ),
320: $ WI( * ), WORK( * ), WR( * )
321: * ..
322: *
323: * =====================================================================
324: *
325: * .. Parameters ..
326: DOUBLE PRECISION ZERO, ONE
327: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
328: * ..
329: * .. Local Scalars ..
330: LOGICAL LQUERY, SCALEA, WANTVL, WANTVR, WNTSNB, WNTSNE,
331: $ WNTSNN, WNTSNV
332: CHARACTER JOB, SIDE
333: INTEGER HSWORK, I, ICOND, IERR, ITAU, IWRK, K, MAXWRK,
334: $ MINWRK, NOUT
335: DOUBLE PRECISION ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM,
336: $ SN
337: * ..
338: * .. Local Arrays ..
339: LOGICAL SELECT( 1 )
340: DOUBLE PRECISION DUM( 1 )
341: * ..
342: * .. External Subroutines ..
343: EXTERNAL DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLABAD, DLACPY,
344: $ DLARTG, DLASCL, DORGHR, DROT, DSCAL, DTREVC,
345: $ DTRSNA, XERBLA
346: * ..
347: * .. External Functions ..
348: LOGICAL LSAME
349: INTEGER IDAMAX, ILAENV
350: DOUBLE PRECISION DLAMCH, DLANGE, DLAPY2, DNRM2
351: EXTERNAL LSAME, IDAMAX, ILAENV, DLAMCH, DLANGE, DLAPY2,
352: $ DNRM2
353: * ..
354: * .. Intrinsic Functions ..
355: INTRINSIC MAX, SQRT
356: * ..
357: * .. Executable Statements ..
358: *
359: * Test the input arguments
360: *
361: INFO = 0
362: LQUERY = ( LWORK.EQ.-1 )
363: WANTVL = LSAME( JOBVL, 'V' )
364: WANTVR = LSAME( JOBVR, 'V' )
365: WNTSNN = LSAME( SENSE, 'N' )
366: WNTSNE = LSAME( SENSE, 'E' )
367: WNTSNV = LSAME( SENSE, 'V' )
368: WNTSNB = LSAME( SENSE, 'B' )
369: IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC,
370: $ 'S' ) .OR. LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) )
371: $ THEN
372: INFO = -1
373: ELSE IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
374: INFO = -2
375: ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
376: INFO = -3
377: ELSE IF( .NOT.( WNTSNN .OR. WNTSNE .OR. WNTSNB .OR. WNTSNV ) .OR.
378: $ ( ( WNTSNE .OR. WNTSNB ) .AND. .NOT.( WANTVL .AND.
379: $ WANTVR ) ) ) THEN
380: INFO = -4
381: ELSE IF( N.LT.0 ) THEN
382: INFO = -5
383: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
384: INFO = -7
385: ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
386: INFO = -11
387: ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
388: INFO = -13
389: END IF
390: *
391: * Compute workspace
392: * (Note: Comments in the code beginning "Workspace:" describe the
393: * minimal amount of workspace needed at that point in the code,
394: * as well as the preferred amount for good performance.
395: * NB refers to the optimal block size for the immediately
396: * following subroutine, as returned by ILAENV.
397: * HSWORK refers to the workspace preferred by DHSEQR, as
398: * calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
399: * the worst case.)
400: *
401: IF( INFO.EQ.0 ) THEN
402: IF( N.EQ.0 ) THEN
403: MINWRK = 1
404: MAXWRK = 1
405: ELSE
406: MAXWRK = N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
407: *
408: IF( WANTVL ) THEN
409: CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL, LDVL,
410: $ WORK, -1, INFO )
411: ELSE IF( WANTVR ) THEN
412: CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR, LDVR,
413: $ WORK, -1, INFO )
414: ELSE
415: IF( WNTSNN ) THEN
416: CALL DHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR,
417: $ LDVR, WORK, -1, INFO )
418: ELSE
419: CALL DHSEQR( 'S', 'N', N, 1, N, A, LDA, WR, WI, VR,
420: $ LDVR, WORK, -1, INFO )
421: END IF
422: END IF
423: HSWORK = WORK( 1 )
424: *
425: IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN
426: MINWRK = 2*N
427: IF( .NOT.WNTSNN )
428: $ MINWRK = MAX( MINWRK, N*N+6*N )
429: MAXWRK = MAX( MAXWRK, HSWORK )
430: IF( .NOT.WNTSNN )
431: $ MAXWRK = MAX( MAXWRK, N*N + 6*N )
432: ELSE
433: MINWRK = 3*N
434: IF( ( .NOT.WNTSNN ) .AND. ( .NOT.WNTSNE ) )
435: $ MINWRK = MAX( MINWRK, N*N + 6*N )
436: MAXWRK = MAX( MAXWRK, HSWORK )
437: MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'DORGHR',
438: $ ' ', N, 1, N, -1 ) )
439: IF( ( .NOT.WNTSNN ) .AND. ( .NOT.WNTSNE ) )
440: $ MAXWRK = MAX( MAXWRK, N*N + 6*N )
441: MAXWRK = MAX( MAXWRK, 3*N )
442: END IF
443: MAXWRK = MAX( MAXWRK, MINWRK )
444: END IF
445: WORK( 1 ) = MAXWRK
446: *
447: IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
448: INFO = -21
449: END IF
450: END IF
451: *
452: IF( INFO.NE.0 ) THEN
453: CALL XERBLA( 'DGEEVX', -INFO )
454: RETURN
455: ELSE IF( LQUERY ) THEN
456: RETURN
457: END IF
458: *
459: * Quick return if possible
460: *
461: IF( N.EQ.0 )
462: $ RETURN
463: *
464: * Get machine constants
465: *
466: EPS = DLAMCH( 'P' )
467: SMLNUM = DLAMCH( 'S' )
468: BIGNUM = ONE / SMLNUM
469: CALL DLABAD( SMLNUM, BIGNUM )
470: SMLNUM = SQRT( SMLNUM ) / EPS
471: BIGNUM = ONE / SMLNUM
472: *
473: * Scale A if max element outside range [SMLNUM,BIGNUM]
474: *
475: ICOND = 0
476: ANRM = DLANGE( 'M', N, N, A, LDA, DUM )
477: SCALEA = .FALSE.
478: IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
479: SCALEA = .TRUE.
480: CSCALE = SMLNUM
481: ELSE IF( ANRM.GT.BIGNUM ) THEN
482: SCALEA = .TRUE.
483: CSCALE = BIGNUM
484: END IF
485: IF( SCALEA )
486: $ CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
487: *
488: * Balance the matrix and compute ABNRM
489: *
490: CALL DGEBAL( BALANC, N, A, LDA, ILO, IHI, SCALE, IERR )
491: ABNRM = DLANGE( '1', N, N, A, LDA, DUM )
492: IF( SCALEA ) THEN
493: DUM( 1 ) = ABNRM
494: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR )
495: ABNRM = DUM( 1 )
496: END IF
497: *
498: * Reduce to upper Hessenberg form
499: * (Workspace: need 2*N, prefer N+N*NB)
500: *
501: ITAU = 1
502: IWRK = ITAU + N
503: CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
504: $ LWORK-IWRK+1, IERR )
505: *
506: IF( WANTVL ) THEN
507: *
508: * Want left eigenvectors
509: * Copy Householder vectors to VL
510: *
511: SIDE = 'L'
512: CALL DLACPY( 'L', N, N, A, LDA, VL, LDVL )
513: *
514: * Generate orthogonal matrix in VL
515: * (Workspace: need 2*N-1, prefer N+(N-1)*NB)
516: *
517: CALL DORGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ),
518: $ LWORK-IWRK+1, IERR )
519: *
520: * Perform QR iteration, accumulating Schur vectors in VL
521: * (Workspace: need 1, prefer HSWORK (see comments) )
522: *
523: IWRK = ITAU
524: CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VL, LDVL,
525: $ WORK( IWRK ), LWORK-IWRK+1, INFO )
526: *
527: IF( WANTVR ) THEN
528: *
529: * Want left and right eigenvectors
530: * Copy Schur vectors to VR
531: *
532: SIDE = 'B'
533: CALL DLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
534: END IF
535: *
536: ELSE IF( WANTVR ) THEN
537: *
538: * Want right eigenvectors
539: * Copy Householder vectors to VR
540: *
541: SIDE = 'R'
542: CALL DLACPY( 'L', N, N, A, LDA, VR, LDVR )
543: *
544: * Generate orthogonal matrix in VR
545: * (Workspace: need 2*N-1, prefer N+(N-1)*NB)
546: *
547: CALL DORGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ),
548: $ LWORK-IWRK+1, IERR )
549: *
550: * Perform QR iteration, accumulating Schur vectors in VR
551: * (Workspace: need 1, prefer HSWORK (see comments) )
552: *
553: IWRK = ITAU
554: CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
555: $ WORK( IWRK ), LWORK-IWRK+1, INFO )
556: *
557: ELSE
558: *
559: * Compute eigenvalues only
560: * If condition numbers desired, compute Schur form
561: *
562: IF( WNTSNN ) THEN
563: JOB = 'E'
564: ELSE
565: JOB = 'S'
566: END IF
567: *
568: * (Workspace: need 1, prefer HSWORK (see comments) )
569: *
570: IWRK = ITAU
571: CALL DHSEQR( JOB, 'N', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
572: $ WORK( IWRK ), LWORK-IWRK+1, INFO )
573: END IF
574: *
575: * If INFO > 0 from DHSEQR, then quit
576: *
577: IF( INFO.GT.0 )
578: $ GO TO 50
579: *
580: IF( WANTVL .OR. WANTVR ) THEN
581: *
582: * Compute left and/or right eigenvectors
583: * (Workspace: need 3*N)
584: *
585: CALL DTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
586: $ N, NOUT, WORK( IWRK ), IERR )
587: END IF
588: *
589: * Compute condition numbers if desired
590: * (Workspace: need N*N+6*N unless SENSE = 'E')
591: *
592: IF( .NOT.WNTSNN ) THEN
593: CALL DTRSNA( SENSE, 'A', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
594: $ RCONDE, RCONDV, N, NOUT, WORK( IWRK ), N, IWORK,
595: $ ICOND )
596: END IF
597: *
598: IF( WANTVL ) THEN
599: *
600: * Undo balancing of left eigenvectors
601: *
602: CALL DGEBAK( BALANC, 'L', N, ILO, IHI, SCALE, N, VL, LDVL,
603: $ IERR )
604: *
605: * Normalize left eigenvectors and make largest component real
606: *
607: DO 20 I = 1, N
608: IF( WI( I ).EQ.ZERO ) THEN
609: SCL = ONE / DNRM2( N, VL( 1, I ), 1 )
610: CALL DSCAL( N, SCL, VL( 1, I ), 1 )
611: ELSE IF( WI( I ).GT.ZERO ) THEN
612: SCL = ONE / DLAPY2( DNRM2( N, VL( 1, I ), 1 ),
613: $ DNRM2( N, VL( 1, I+1 ), 1 ) )
614: CALL DSCAL( N, SCL, VL( 1, I ), 1 )
615: CALL DSCAL( N, SCL, VL( 1, I+1 ), 1 )
616: DO 10 K = 1, N
617: WORK( K ) = VL( K, I )**2 + VL( K, I+1 )**2
618: 10 CONTINUE
619: K = IDAMAX( N, WORK, 1 )
620: CALL DLARTG( VL( K, I ), VL( K, I+1 ), CS, SN, R )
621: CALL DROT( N, VL( 1, I ), 1, VL( 1, I+1 ), 1, CS, SN )
622: VL( K, I+1 ) = ZERO
623: END IF
624: 20 CONTINUE
625: END IF
626: *
627: IF( WANTVR ) THEN
628: *
629: * Undo balancing of right eigenvectors
630: *
631: CALL DGEBAK( BALANC, 'R', N, ILO, IHI, SCALE, N, VR, LDVR,
632: $ IERR )
633: *
634: * Normalize right eigenvectors and make largest component real
635: *
636: DO 40 I = 1, N
637: IF( WI( I ).EQ.ZERO ) THEN
638: SCL = ONE / DNRM2( N, VR( 1, I ), 1 )
639: CALL DSCAL( N, SCL, VR( 1, I ), 1 )
640: ELSE IF( WI( I ).GT.ZERO ) THEN
641: SCL = ONE / DLAPY2( DNRM2( N, VR( 1, I ), 1 ),
642: $ DNRM2( N, VR( 1, I+1 ), 1 ) )
643: CALL DSCAL( N, SCL, VR( 1, I ), 1 )
644: CALL DSCAL( N, SCL, VR( 1, I+1 ), 1 )
645: DO 30 K = 1, N
646: WORK( K ) = VR( K, I )**2 + VR( K, I+1 )**2
647: 30 CONTINUE
648: K = IDAMAX( N, WORK, 1 )
649: CALL DLARTG( VR( K, I ), VR( K, I+1 ), CS, SN, R )
650: CALL DROT( N, VR( 1, I ), 1, VR( 1, I+1 ), 1, CS, SN )
651: VR( K, I+1 ) = ZERO
652: END IF
653: 40 CONTINUE
654: END IF
655: *
656: * Undo scaling if necessary
657: *
658: 50 CONTINUE
659: IF( SCALEA ) THEN
660: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WR( INFO+1 ),
661: $ MAX( N-INFO, 1 ), IERR )
662: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WI( INFO+1 ),
663: $ MAX( N-INFO, 1 ), IERR )
664: IF( INFO.EQ.0 ) THEN
665: IF( ( WNTSNV .OR. WNTSNB ) .AND. ICOND.EQ.0 )
666: $ CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, RCONDV, N,
667: $ IERR )
668: ELSE
669: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WR, N,
670: $ IERR )
671: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N,
672: $ IERR )
673: END IF
674: END IF
675: *
676: WORK( 1 ) = MAXWRK
677: RETURN
678: *
679: * End of DGEEVX
680: *
681: END
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