Annotation of rpl/lapack/lapack/dggevx.f, revision 1.18
1.8 bertrand 1: *> \brief <b> DGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b>
2: *
3: * =========== DOCUMENTATION ===========
4: *
1.15 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.8 bertrand 7: *
8: *> \htmlonly
1.15 bertrand 9: *> Download DGGEVX + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggevx.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggevx.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggevx.f">
1.8 bertrand 15: *> [TXT]</a>
1.15 bertrand 16: *> \endhtmlonly
1.8 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
22: * ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,
23: * IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,
24: * RCONDV, WORK, LWORK, IWORK, BWORK, INFO )
1.15 bertrand 25: *
1.8 bertrand 26: * .. Scalar Arguments ..
27: * CHARACTER BALANC, JOBVL, JOBVR, SENSE
28: * INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
29: * DOUBLE PRECISION ABNRM, BBNRM
30: * ..
31: * .. Array Arguments ..
32: * LOGICAL BWORK( * )
33: * INTEGER IWORK( * )
34: * DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
35: * $ B( LDB, * ), BETA( * ), LSCALE( * ),
36: * $ RCONDE( * ), RCONDV( * ), RSCALE( * ),
37: * $ VL( LDVL, * ), VR( LDVR, * ), WORK( * )
38: * ..
1.15 bertrand 39: *
1.8 bertrand 40: *
41: *> \par Purpose:
42: * =============
43: *>
44: *> \verbatim
45: *>
46: *> DGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
47: *> the generalized eigenvalues, and optionally, the left and/or right
48: *> generalized eigenvectors.
49: *>
50: *> Optionally also, it computes a balancing transformation to improve
51: *> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
52: *> LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
53: *> the eigenvalues (RCONDE), and reciprocal condition numbers for the
54: *> right eigenvectors (RCONDV).
55: *>
56: *> A generalized eigenvalue for a pair of matrices (A,B) is a scalar
57: *> lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
58: *> singular. It is usually represented as the pair (alpha,beta), as
59: *> there is a reasonable interpretation for beta=0, and even for both
60: *> being zero.
61: *>
62: *> The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
63: *> of (A,B) satisfies
64: *>
65: *> A * v(j) = lambda(j) * B * v(j) .
66: *>
67: *> The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
68: *> of (A,B) satisfies
69: *>
70: *> u(j)**H * A = lambda(j) * u(j)**H * B.
71: *>
72: *> where u(j)**H is the conjugate-transpose of u(j).
73: *>
74: *> \endverbatim
75: *
76: * Arguments:
77: * ==========
78: *
79: *> \param[in] BALANC
80: *> \verbatim
81: *> BALANC is CHARACTER*1
82: *> Specifies the balance option to be performed.
83: *> = 'N': do not diagonally scale or permute;
84: *> = 'P': permute only;
85: *> = 'S': scale only;
86: *> = 'B': both permute and scale.
87: *> Computed reciprocal condition numbers will be for the
88: *> matrices after permuting and/or balancing. Permuting does
89: *> not change condition numbers (in exact arithmetic), but
90: *> balancing does.
91: *> \endverbatim
92: *>
93: *> \param[in] JOBVL
94: *> \verbatim
95: *> JOBVL is CHARACTER*1
96: *> = 'N': do not compute the left generalized eigenvectors;
97: *> = 'V': compute the left generalized eigenvectors.
98: *> \endverbatim
99: *>
100: *> \param[in] JOBVR
101: *> \verbatim
102: *> JOBVR is CHARACTER*1
103: *> = 'N': do not compute the right generalized eigenvectors;
104: *> = 'V': compute the right generalized eigenvectors.
105: *> \endverbatim
106: *>
107: *> \param[in] SENSE
108: *> \verbatim
109: *> SENSE is CHARACTER*1
110: *> Determines which reciprocal condition numbers are computed.
111: *> = 'N': none are computed;
112: *> = 'E': computed for eigenvalues only;
113: *> = 'V': computed for eigenvectors only;
114: *> = 'B': computed for eigenvalues and eigenvectors.
115: *> \endverbatim
116: *>
117: *> \param[in] N
118: *> \verbatim
119: *> N is INTEGER
120: *> The order of the matrices A, B, VL, and VR. N >= 0.
121: *> \endverbatim
122: *>
123: *> \param[in,out] A
124: *> \verbatim
125: *> A is DOUBLE PRECISION array, dimension (LDA, N)
126: *> On entry, the matrix A in the pair (A,B).
127: *> On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
128: *> or both, then A contains the first part of the real Schur
129: *> form of the "balanced" versions of the input A and B.
130: *> \endverbatim
131: *>
132: *> \param[in] LDA
133: *> \verbatim
134: *> LDA is INTEGER
135: *> The leading dimension of A. LDA >= max(1,N).
136: *> \endverbatim
137: *>
138: *> \param[in,out] B
139: *> \verbatim
140: *> B is DOUBLE PRECISION array, dimension (LDB, N)
141: *> On entry, the matrix B in the pair (A,B).
142: *> On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
143: *> or both, then B contains the second part of the real Schur
144: *> form of the "balanced" versions of the input A and B.
145: *> \endverbatim
146: *>
147: *> \param[in] LDB
148: *> \verbatim
149: *> LDB is INTEGER
150: *> The leading dimension of B. LDB >= max(1,N).
151: *> \endverbatim
152: *>
153: *> \param[out] ALPHAR
154: *> \verbatim
155: *> ALPHAR is DOUBLE PRECISION array, dimension (N)
156: *> \endverbatim
157: *>
158: *> \param[out] ALPHAI
159: *> \verbatim
160: *> ALPHAI is DOUBLE PRECISION array, dimension (N)
161: *> \endverbatim
162: *>
163: *> \param[out] BETA
164: *> \verbatim
165: *> BETA is DOUBLE PRECISION array, dimension (N)
166: *> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
167: *> be the generalized eigenvalues. If ALPHAI(j) is zero, then
168: *> the j-th eigenvalue is real; if positive, then the j-th and
169: *> (j+1)-st eigenvalues are a complex conjugate pair, with
170: *> ALPHAI(j+1) negative.
171: *>
172: *> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
173: *> may easily over- or underflow, and BETA(j) may even be zero.
174: *> Thus, the user should avoid naively computing the ratio
175: *> ALPHA/BETA. However, ALPHAR and ALPHAI will be always less
176: *> than and usually comparable with norm(A) in magnitude, and
177: *> BETA always less than and usually comparable with norm(B).
178: *> \endverbatim
179: *>
180: *> \param[out] VL
181: *> \verbatim
182: *> VL is DOUBLE PRECISION array, dimension (LDVL,N)
183: *> If JOBVL = 'V', the left eigenvectors u(j) are stored one
184: *> after another in the columns of VL, in the same order as
185: *> their eigenvalues. If the j-th eigenvalue is real, then
186: *> u(j) = VL(:,j), the j-th column of VL. If the j-th and
187: *> (j+1)-th eigenvalues form a complex conjugate pair, then
188: *> u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
189: *> Each eigenvector will be scaled so the largest component have
190: *> abs(real part) + abs(imag. part) = 1.
191: *> Not referenced if JOBVL = 'N'.
192: *> \endverbatim
193: *>
194: *> \param[in] LDVL
195: *> \verbatim
196: *> LDVL is INTEGER
197: *> The leading dimension of the matrix VL. LDVL >= 1, and
198: *> if JOBVL = 'V', LDVL >= N.
199: *> \endverbatim
200: *>
201: *> \param[out] VR
202: *> \verbatim
203: *> VR is DOUBLE PRECISION array, dimension (LDVR,N)
204: *> If JOBVR = 'V', the right eigenvectors v(j) are stored one
205: *> after another in the columns of VR, in the same order as
206: *> their eigenvalues. If the j-th eigenvalue is real, then
207: *> v(j) = VR(:,j), the j-th column of VR. If the j-th and
208: *> (j+1)-th eigenvalues form a complex conjugate pair, then
209: *> v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
210: *> Each eigenvector will be scaled so the largest component have
211: *> abs(real part) + abs(imag. part) = 1.
212: *> Not referenced if JOBVR = 'N'.
213: *> \endverbatim
214: *>
215: *> \param[in] LDVR
216: *> \verbatim
217: *> LDVR is INTEGER
218: *> The leading dimension of the matrix VR. LDVR >= 1, and
219: *> if JOBVR = 'V', LDVR >= N.
220: *> \endverbatim
221: *>
222: *> \param[out] ILO
223: *> \verbatim
224: *> ILO is INTEGER
225: *> \endverbatim
226: *>
227: *> \param[out] IHI
228: *> \verbatim
229: *> IHI is INTEGER
230: *> ILO and IHI are integer values such that on exit
231: *> A(i,j) = 0 and B(i,j) = 0 if i > j and
232: *> j = 1,...,ILO-1 or i = IHI+1,...,N.
233: *> If BALANC = 'N' or 'S', ILO = 1 and IHI = N.
234: *> \endverbatim
235: *>
236: *> \param[out] LSCALE
237: *> \verbatim
238: *> LSCALE is DOUBLE PRECISION array, dimension (N)
239: *> Details of the permutations and scaling factors applied
240: *> to the left side of A and B. If PL(j) is the index of the
241: *> row interchanged with row j, and DL(j) is the scaling
242: *> factor applied to row j, then
243: *> LSCALE(j) = PL(j) for j = 1,...,ILO-1
244: *> = DL(j) for j = ILO,...,IHI
245: *> = PL(j) for j = IHI+1,...,N.
246: *> The order in which the interchanges are made is N to IHI+1,
247: *> then 1 to ILO-1.
248: *> \endverbatim
249: *>
250: *> \param[out] RSCALE
251: *> \verbatim
252: *> RSCALE is DOUBLE PRECISION array, dimension (N)
253: *> Details of the permutations and scaling factors applied
254: *> to the right side of A and B. If PR(j) is the index of the
255: *> column interchanged with column j, and DR(j) is the scaling
256: *> factor applied to column j, then
257: *> RSCALE(j) = PR(j) for j = 1,...,ILO-1
258: *> = DR(j) for j = ILO,...,IHI
259: *> = PR(j) for j = IHI+1,...,N
260: *> The order in which the interchanges are made is N to IHI+1,
261: *> then 1 to ILO-1.
262: *> \endverbatim
263: *>
264: *> \param[out] ABNRM
265: *> \verbatim
266: *> ABNRM is DOUBLE PRECISION
267: *> The one-norm of the balanced matrix A.
268: *> \endverbatim
269: *>
270: *> \param[out] BBNRM
271: *> \verbatim
272: *> BBNRM is DOUBLE PRECISION
273: *> The one-norm of the balanced matrix B.
274: *> \endverbatim
275: *>
276: *> \param[out] RCONDE
277: *> \verbatim
278: *> RCONDE is DOUBLE PRECISION array, dimension (N)
279: *> If SENSE = 'E' or 'B', the reciprocal condition numbers of
280: *> the eigenvalues, stored in consecutive elements of the array.
281: *> For a complex conjugate pair of eigenvalues two consecutive
282: *> elements of RCONDE are set to the same value. Thus RCONDE(j),
283: *> RCONDV(j), and the j-th columns of VL and VR all correspond
284: *> to the j-th eigenpair.
285: *> If SENSE = 'N or 'V', RCONDE is not referenced.
286: *> \endverbatim
287: *>
288: *> \param[out] RCONDV
289: *> \verbatim
290: *> RCONDV is DOUBLE PRECISION array, dimension (N)
291: *> If SENSE = 'V' or 'B', the estimated reciprocal condition
292: *> numbers of the eigenvectors, stored in consecutive elements
293: *> of the array. For a complex eigenvector two consecutive
294: *> elements of RCONDV are set to the same value. If the
295: *> eigenvalues cannot be reordered to compute RCONDV(j),
296: *> RCONDV(j) is set to 0; this can only occur when the true
297: *> value would be very small anyway.
298: *> If SENSE = 'N' or 'E', RCONDV is not referenced.
299: *> \endverbatim
300: *>
301: *> \param[out] WORK
302: *> \verbatim
303: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
304: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
305: *> \endverbatim
306: *>
307: *> \param[in] LWORK
308: *> \verbatim
309: *> LWORK is INTEGER
310: *> The dimension of the array WORK. LWORK >= max(1,2*N).
311: *> If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
312: *> LWORK >= max(1,6*N).
313: *> If SENSE = 'E' or 'B', LWORK >= max(1,10*N).
314: *> If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.
315: *>
316: *> If LWORK = -1, then a workspace query is assumed; the routine
317: *> only calculates the optimal size of the WORK array, returns
318: *> this value as the first entry of the WORK array, and no error
319: *> message related to LWORK is issued by XERBLA.
320: *> \endverbatim
321: *>
322: *> \param[out] IWORK
323: *> \verbatim
324: *> IWORK is INTEGER array, dimension (N+6)
325: *> If SENSE = 'E', IWORK is not referenced.
326: *> \endverbatim
327: *>
328: *> \param[out] BWORK
329: *> \verbatim
330: *> BWORK is LOGICAL array, dimension (N)
331: *> If SENSE = 'N', BWORK is not referenced.
332: *> \endverbatim
333: *>
334: *> \param[out] INFO
335: *> \verbatim
336: *> INFO is INTEGER
337: *> = 0: successful exit
338: *> < 0: if INFO = -i, the i-th argument had an illegal value.
339: *> = 1,...,N:
340: *> The QZ iteration failed. No eigenvectors have been
341: *> calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
342: *> should be correct for j=INFO+1,...,N.
343: *> > N: =N+1: other than QZ iteration failed in DHGEQZ.
344: *> =N+2: error return from DTGEVC.
345: *> \endverbatim
346: *
347: * Authors:
348: * ========
349: *
1.15 bertrand 350: *> \author Univ. of Tennessee
351: *> \author Univ. of California Berkeley
352: *> \author Univ. of Colorado Denver
353: *> \author NAG Ltd.
1.8 bertrand 354: *
355: *> \ingroup doubleGEeigen
356: *
357: *> \par Further Details:
358: * =====================
359: *>
360: *> \verbatim
361: *>
362: *> Balancing a matrix pair (A,B) includes, first, permuting rows and
363: *> columns to isolate eigenvalues, second, applying diagonal similarity
364: *> transformation to the rows and columns to make the rows and columns
365: *> as close in norm as possible. The computed reciprocal condition
366: *> numbers correspond to the balanced matrix. Permuting rows and columns
367: *> will not change the condition numbers (in exact arithmetic) but
368: *> diagonal scaling will. For further explanation of balancing, see
369: *> section 4.11.1.2 of LAPACK Users' Guide.
370: *>
371: *> An approximate error bound on the chordal distance between the i-th
372: *> computed generalized eigenvalue w and the corresponding exact
373: *> eigenvalue lambda is
374: *>
375: *> chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
376: *>
377: *> An approximate error bound for the angle between the i-th computed
378: *> eigenvector VL(i) or VR(i) is given by
379: *>
380: *> EPS * norm(ABNRM, BBNRM) / DIF(i).
381: *>
382: *> For further explanation of the reciprocal condition numbers RCONDE
383: *> and RCONDV, see section 4.11 of LAPACK User's Guide.
384: *> \endverbatim
385: *>
386: * =====================================================================
1.1 bertrand 387: SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
388: $ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,
389: $ IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,
390: $ RCONDV, WORK, LWORK, IWORK, BWORK, INFO )
391: *
1.18 ! bertrand 392: * -- LAPACK driver routine --
1.1 bertrand 393: * -- LAPACK is a software package provided by Univ. of Tennessee, --
394: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
395: *
396: * .. Scalar Arguments ..
397: CHARACTER BALANC, JOBVL, JOBVR, SENSE
398: INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
399: DOUBLE PRECISION ABNRM, BBNRM
400: * ..
401: * .. Array Arguments ..
402: LOGICAL BWORK( * )
403: INTEGER IWORK( * )
404: DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
405: $ B( LDB, * ), BETA( * ), LSCALE( * ),
406: $ RCONDE( * ), RCONDV( * ), RSCALE( * ),
407: $ VL( LDVL, * ), VR( LDVR, * ), WORK( * )
408: * ..
409: *
410: * =====================================================================
411: *
412: * .. Parameters ..
413: DOUBLE PRECISION ZERO, ONE
414: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
415: * ..
416: * .. Local Scalars ..
417: LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY, NOSCL,
418: $ PAIR, WANTSB, WANTSE, WANTSN, WANTSV
419: CHARACTER CHTEMP
420: INTEGER I, ICOLS, IERR, IJOBVL, IJOBVR, IN, IROWS,
421: $ ITAU, IWRK, IWRK1, J, JC, JR, M, MAXWRK,
422: $ MINWRK, MM
423: DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
424: $ SMLNUM, TEMP
425: * ..
426: * .. Local Arrays ..
427: LOGICAL LDUMMA( 1 )
428: * ..
429: * .. External Subroutines ..
430: EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLABAD,
431: $ DLACPY, DLASCL, DLASET, DORGQR, DORMQR, DTGEVC,
1.15 bertrand 432: $ DTGSNA, XERBLA
1.1 bertrand 433: * ..
434: * .. External Functions ..
435: LOGICAL LSAME
436: INTEGER ILAENV
437: DOUBLE PRECISION DLAMCH, DLANGE
438: EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
439: * ..
440: * .. Intrinsic Functions ..
441: INTRINSIC ABS, MAX, SQRT
442: * ..
443: * .. Executable Statements ..
444: *
445: * Decode the input arguments
446: *
447: IF( LSAME( JOBVL, 'N' ) ) THEN
448: IJOBVL = 1
449: ILVL = .FALSE.
450: ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
451: IJOBVL = 2
452: ILVL = .TRUE.
453: ELSE
454: IJOBVL = -1
455: ILVL = .FALSE.
456: END IF
457: *
458: IF( LSAME( JOBVR, 'N' ) ) THEN
459: IJOBVR = 1
460: ILVR = .FALSE.
461: ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
462: IJOBVR = 2
463: ILVR = .TRUE.
464: ELSE
465: IJOBVR = -1
466: ILVR = .FALSE.
467: END IF
468: ILV = ILVL .OR. ILVR
469: *
470: NOSCL = LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'P' )
471: WANTSN = LSAME( SENSE, 'N' )
472: WANTSE = LSAME( SENSE, 'E' )
473: WANTSV = LSAME( SENSE, 'V' )
474: WANTSB = LSAME( SENSE, 'B' )
475: *
476: * Test the input arguments
477: *
478: INFO = 0
479: LQUERY = ( LWORK.EQ.-1 )
480: IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC,
481: $ 'S' ) .OR. LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) )
482: $ THEN
483: INFO = -1
484: ELSE IF( IJOBVL.LE.0 ) THEN
485: INFO = -2
486: ELSE IF( IJOBVR.LE.0 ) THEN
487: INFO = -3
488: ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSB .OR. WANTSV ) )
489: $ THEN
490: INFO = -4
491: ELSE IF( N.LT.0 ) THEN
492: INFO = -5
493: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
494: INFO = -7
495: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
496: INFO = -9
497: ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
498: INFO = -14
499: ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
500: INFO = -16
501: END IF
502: *
503: * Compute workspace
504: * (Note: Comments in the code beginning "Workspace:" describe the
505: * minimal amount of workspace needed at that point in the code,
506: * as well as the preferred amount for good performance.
507: * NB refers to the optimal block size for the immediately
508: * following subroutine, as returned by ILAENV. The workspace is
509: * computed assuming ILO = 1 and IHI = N, the worst case.)
510: *
511: IF( INFO.EQ.0 ) THEN
512: IF( N.EQ.0 ) THEN
513: MINWRK = 1
514: MAXWRK = 1
515: ELSE
516: IF( NOSCL .AND. .NOT.ILV ) THEN
517: MINWRK = 2*N
518: ELSE
519: MINWRK = 6*N
520: END IF
521: IF( WANTSE .OR. WANTSB ) THEN
522: MINWRK = 10*N
523: END IF
524: IF( WANTSV .OR. WANTSB ) THEN
525: MINWRK = MAX( MINWRK, 2*N*( N + 4 ) + 16 )
526: END IF
527: MAXWRK = MINWRK
528: MAXWRK = MAX( MAXWRK,
529: $ N + N*ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 ) )
530: MAXWRK = MAX( MAXWRK,
531: $ N + N*ILAENV( 1, 'DORMQR', ' ', N, 1, N, 0 ) )
532: IF( ILVL ) THEN
533: MAXWRK = MAX( MAXWRK, N +
534: $ N*ILAENV( 1, 'DORGQR', ' ', N, 1, N, 0 ) )
535: END IF
536: END IF
537: WORK( 1 ) = MAXWRK
538: *
539: IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
540: INFO = -26
541: END IF
542: END IF
543: *
544: IF( INFO.NE.0 ) THEN
545: CALL XERBLA( 'DGGEVX', -INFO )
546: RETURN
547: ELSE IF( LQUERY ) THEN
548: RETURN
549: END IF
550: *
551: * Quick return if possible
552: *
553: IF( N.EQ.0 )
554: $ RETURN
555: *
556: *
557: * Get machine constants
558: *
559: EPS = DLAMCH( 'P' )
560: SMLNUM = DLAMCH( 'S' )
561: BIGNUM = ONE / SMLNUM
562: CALL DLABAD( SMLNUM, BIGNUM )
563: SMLNUM = SQRT( SMLNUM ) / EPS
564: BIGNUM = ONE / SMLNUM
565: *
566: * Scale A if max element outside range [SMLNUM,BIGNUM]
567: *
568: ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
569: ILASCL = .FALSE.
570: IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
571: ANRMTO = SMLNUM
572: ILASCL = .TRUE.
573: ELSE IF( ANRM.GT.BIGNUM ) THEN
574: ANRMTO = BIGNUM
575: ILASCL = .TRUE.
576: END IF
577: IF( ILASCL )
578: $ CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
579: *
580: * Scale B if max element outside range [SMLNUM,BIGNUM]
581: *
582: BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
583: ILBSCL = .FALSE.
584: IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
585: BNRMTO = SMLNUM
586: ILBSCL = .TRUE.
587: ELSE IF( BNRM.GT.BIGNUM ) THEN
588: BNRMTO = BIGNUM
589: ILBSCL = .TRUE.
590: END IF
591: IF( ILBSCL )
592: $ CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
593: *
594: * Permute and/or balance the matrix pair (A,B)
595: * (Workspace: need 6*N if BALANC = 'S' or 'B', 1 otherwise)
596: *
597: CALL DGGBAL( BALANC, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE,
598: $ WORK, IERR )
599: *
600: * Compute ABNRM and BBNRM
601: *
602: ABNRM = DLANGE( '1', N, N, A, LDA, WORK( 1 ) )
603: IF( ILASCL ) THEN
604: WORK( 1 ) = ABNRM
605: CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, 1, 1, WORK( 1 ), 1,
606: $ IERR )
607: ABNRM = WORK( 1 )
608: END IF
609: *
610: BBNRM = DLANGE( '1', N, N, B, LDB, WORK( 1 ) )
611: IF( ILBSCL ) THEN
612: WORK( 1 ) = BBNRM
613: CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, 1, 1, WORK( 1 ), 1,
614: $ IERR )
615: BBNRM = WORK( 1 )
616: END IF
617: *
618: * Reduce B to triangular form (QR decomposition of B)
619: * (Workspace: need N, prefer N*NB )
620: *
621: IROWS = IHI + 1 - ILO
622: IF( ILV .OR. .NOT.WANTSN ) THEN
623: ICOLS = N + 1 - ILO
624: ELSE
625: ICOLS = IROWS
626: END IF
627: ITAU = 1
628: IWRK = ITAU + IROWS
629: CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
630: $ WORK( IWRK ), LWORK+1-IWRK, IERR )
631: *
632: * Apply the orthogonal transformation to A
633: * (Workspace: need N, prefer N*NB)
634: *
635: CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
636: $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
637: $ LWORK+1-IWRK, IERR )
638: *
639: * Initialize VL and/or VR
640: * (Workspace: need N, prefer N*NB)
641: *
642: IF( ILVL ) THEN
643: CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
644: IF( IROWS.GT.1 ) THEN
645: CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
646: $ VL( ILO+1, ILO ), LDVL )
647: END IF
648: CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
649: $ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
650: END IF
651: *
652: IF( ILVR )
653: $ CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
654: *
655: * Reduce to generalized Hessenberg form
656: * (Workspace: none needed)
657: *
658: IF( ILV .OR. .NOT.WANTSN ) THEN
659: *
660: * Eigenvectors requested -- work on whole matrix.
661: *
662: CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
663: $ LDVL, VR, LDVR, IERR )
664: ELSE
665: CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
666: $ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR )
667: END IF
668: *
669: * Perform QZ algorithm (Compute eigenvalues, and optionally, the
670: * Schur forms and Schur vectors)
671: * (Workspace: need N)
672: *
673: IF( ILV .OR. .NOT.WANTSN ) THEN
674: CHTEMP = 'S'
675: ELSE
676: CHTEMP = 'E'
677: END IF
678: *
679: CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
680: $ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK,
681: $ LWORK, IERR )
682: IF( IERR.NE.0 ) THEN
683: IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
684: INFO = IERR
685: ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
686: INFO = IERR - N
687: ELSE
688: INFO = N + 1
689: END IF
690: GO TO 130
691: END IF
692: *
693: * Compute Eigenvectors and estimate condition numbers if desired
694: * (Workspace: DTGEVC: need 6*N
695: * DTGSNA: need 2*N*(N+2)+16 if SENSE = 'V' or 'B',
696: * need N otherwise )
697: *
698: IF( ILV .OR. .NOT.WANTSN ) THEN
699: IF( ILV ) THEN
700: IF( ILVL ) THEN
701: IF( ILVR ) THEN
702: CHTEMP = 'B'
703: ELSE
704: CHTEMP = 'L'
705: END IF
706: ELSE
707: CHTEMP = 'R'
708: END IF
709: *
710: CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL,
711: $ LDVL, VR, LDVR, N, IN, WORK, IERR )
712: IF( IERR.NE.0 ) THEN
713: INFO = N + 2
714: GO TO 130
715: END IF
716: END IF
717: *
718: IF( .NOT.WANTSN ) THEN
719: *
720: * compute eigenvectors (DTGEVC) and estimate condition
721: * numbers (DTGSNA). Note that the definition of the condition
722: * number is not invariant under transformation (u,v) to
723: * (Q*u, Z*v), where (u,v) are eigenvectors of the generalized
724: * Schur form (S,T), Q and Z are orthogonal matrices. In order
725: * to avoid using extra 2*N*N workspace, we have to recalculate
726: * eigenvectors and estimate one condition numbers at a time.
727: *
728: PAIR = .FALSE.
729: DO 20 I = 1, N
730: *
731: IF( PAIR ) THEN
732: PAIR = .FALSE.
733: GO TO 20
734: END IF
735: MM = 1
736: IF( I.LT.N ) THEN
737: IF( A( I+1, I ).NE.ZERO ) THEN
738: PAIR = .TRUE.
739: MM = 2
740: END IF
741: END IF
742: *
743: DO 10 J = 1, N
744: BWORK( J ) = .FALSE.
745: 10 CONTINUE
746: IF( MM.EQ.1 ) THEN
747: BWORK( I ) = .TRUE.
748: ELSE IF( MM.EQ.2 ) THEN
749: BWORK( I ) = .TRUE.
750: BWORK( I+1 ) = .TRUE.
751: END IF
752: *
753: IWRK = MM*N + 1
754: IWRK1 = IWRK + MM*N
755: *
756: * Compute a pair of left and right eigenvectors.
757: * (compute workspace: need up to 4*N + 6*N)
758: *
759: IF( WANTSE .OR. WANTSB ) THEN
760: CALL DTGEVC( 'B', 'S', BWORK, N, A, LDA, B, LDB,
761: $ WORK( 1 ), N, WORK( IWRK ), N, MM, M,
762: $ WORK( IWRK1 ), IERR )
763: IF( IERR.NE.0 ) THEN
764: INFO = N + 2
765: GO TO 130
766: END IF
767: END IF
768: *
769: CALL DTGSNA( SENSE, 'S', BWORK, N, A, LDA, B, LDB,
770: $ WORK( 1 ), N, WORK( IWRK ), N, RCONDE( I ),
771: $ RCONDV( I ), MM, M, WORK( IWRK1 ),
772: $ LWORK-IWRK1+1, IWORK, IERR )
773: *
774: 20 CONTINUE
775: END IF
776: END IF
777: *
778: * Undo balancing on VL and VR and normalization
779: * (Workspace: none needed)
780: *
781: IF( ILVL ) THEN
782: CALL DGGBAK( BALANC, 'L', N, ILO, IHI, LSCALE, RSCALE, N, VL,
783: $ LDVL, IERR )
784: *
785: DO 70 JC = 1, N
786: IF( ALPHAI( JC ).LT.ZERO )
787: $ GO TO 70
788: TEMP = ZERO
789: IF( ALPHAI( JC ).EQ.ZERO ) THEN
790: DO 30 JR = 1, N
791: TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
792: 30 CONTINUE
793: ELSE
794: DO 40 JR = 1, N
795: TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
796: $ ABS( VL( JR, JC+1 ) ) )
797: 40 CONTINUE
798: END IF
799: IF( TEMP.LT.SMLNUM )
800: $ GO TO 70
801: TEMP = ONE / TEMP
802: IF( ALPHAI( JC ).EQ.ZERO ) THEN
803: DO 50 JR = 1, N
804: VL( JR, JC ) = VL( JR, JC )*TEMP
805: 50 CONTINUE
806: ELSE
807: DO 60 JR = 1, N
808: VL( JR, JC ) = VL( JR, JC )*TEMP
809: VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
810: 60 CONTINUE
811: END IF
812: 70 CONTINUE
813: END IF
814: IF( ILVR ) THEN
815: CALL DGGBAK( BALANC, 'R', N, ILO, IHI, LSCALE, RSCALE, N, VR,
816: $ LDVR, IERR )
817: DO 120 JC = 1, N
818: IF( ALPHAI( JC ).LT.ZERO )
819: $ GO TO 120
820: TEMP = ZERO
821: IF( ALPHAI( JC ).EQ.ZERO ) THEN
822: DO 80 JR = 1, N
823: TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
824: 80 CONTINUE
825: ELSE
826: DO 90 JR = 1, N
827: TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
828: $ ABS( VR( JR, JC+1 ) ) )
829: 90 CONTINUE
830: END IF
831: IF( TEMP.LT.SMLNUM )
832: $ GO TO 120
833: TEMP = ONE / TEMP
834: IF( ALPHAI( JC ).EQ.ZERO ) THEN
835: DO 100 JR = 1, N
836: VR( JR, JC ) = VR( JR, JC )*TEMP
837: 100 CONTINUE
838: ELSE
839: DO 110 JR = 1, N
840: VR( JR, JC ) = VR( JR, JC )*TEMP
841: VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
842: 110 CONTINUE
843: END IF
844: 120 CONTINUE
845: END IF
846: *
847: * Undo scaling if necessary
848: *
1.10 bertrand 849: 130 CONTINUE
850: *
1.1 bertrand 851: IF( ILASCL ) THEN
852: CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR )
853: CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR )
854: END IF
855: *
856: IF( ILBSCL ) THEN
857: CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
858: END IF
859: *
860: WORK( 1 ) = MAXWRK
861: RETURN
862: *
863: * End of DGGEVX
864: *
865: END
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