1 SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
2 $ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,
3 $ IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,
4 $ RCONDV, WORK, LWORK, IWORK, BWORK, INFO )
5 *
6 * -- LAPACK driver routine (version 3.2) --
7 * -- LAPACK is a software package provided by Univ. of Tennessee, --
8 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
9 * November 2006
10 *
11 * .. Scalar Arguments ..
12 CHARACTER BALANC, JOBVL, JOBVR, SENSE
13 INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
14 DOUBLE PRECISION ABNRM, BBNRM
15 * ..
16 * .. Array Arguments ..
17 LOGICAL BWORK( * )
18 INTEGER IWORK( * )
19 DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
20 $ B( LDB, * ), BETA( * ), LSCALE( * ),
21 $ RCONDE( * ), RCONDV( * ), RSCALE( * ),
22 $ VL( LDVL, * ), VR( LDVR, * ), WORK( * )
23 * ..
24 *
25 * Purpose
26 * =======
27 *
28 * DGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
29 * the generalized eigenvalues, and optionally, the left and/or right
30 * generalized eigenvectors.
31 *
32 * Optionally also, it computes a balancing transformation to improve
33 * the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
34 * LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
35 * the eigenvalues (RCONDE), and reciprocal condition numbers for the
36 * right eigenvectors (RCONDV).
37 *
38 * A generalized eigenvalue for a pair of matrices (A,B) is a scalar
39 * lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
40 * singular. It is usually represented as the pair (alpha,beta), as
41 * there is a reasonable interpretation for beta=0, and even for both
42 * being zero.
43 *
44 * The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
45 * of (A,B) satisfies
46 *
47 * A * v(j) = lambda(j) * B * v(j) .
48 *
49 * The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
50 * of (A,B) satisfies
51 *
52 * u(j)**H * A = lambda(j) * u(j)**H * B.
53 *
54 * where u(j)**H is the conjugate-transpose of u(j).
55 *
56 *
57 * Arguments
58 * =========
59 *
60 * BALANC (input) CHARACTER*1
61 * Specifies the balance option to be performed.
62 * = 'N': do not diagonally scale or permute;
63 * = 'P': permute only;
64 * = 'S': scale only;
65 * = 'B': both permute and scale.
66 * Computed reciprocal condition numbers will be for the
67 * matrices after permuting and/or balancing. Permuting does
68 * not change condition numbers (in exact arithmetic), but
69 * balancing does.
70 *
71 * JOBVL (input) CHARACTER*1
72 * = 'N': do not compute the left generalized eigenvectors;
73 * = 'V': compute the left generalized eigenvectors.
74 *
75 * JOBVR (input) CHARACTER*1
76 * = 'N': do not compute the right generalized eigenvectors;
77 * = 'V': compute the right generalized eigenvectors.
78 *
79 * SENSE (input) CHARACTER*1
80 * Determines which reciprocal condition numbers are computed.
81 * = 'N': none are computed;
82 * = 'E': computed for eigenvalues only;
83 * = 'V': computed for eigenvectors only;
84 * = 'B': computed for eigenvalues and eigenvectors.
85 *
86 * N (input) INTEGER
87 * The order of the matrices A, B, VL, and VR. N >= 0.
88 *
89 * A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
90 * On entry, the matrix A in the pair (A,B).
91 * On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
92 * or both, then A contains the first part of the real Schur
93 * form of the "balanced" versions of the input A and B.
94 *
95 * LDA (input) INTEGER
96 * The leading dimension of A. LDA >= max(1,N).
97 *
98 * B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
99 * On entry, the matrix B in the pair (A,B).
100 * On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
101 * or both, then B contains the second part of the real Schur
102 * form of the "balanced" versions of the input A and B.
103 *
104 * LDB (input) INTEGER
105 * The leading dimension of B. LDB >= max(1,N).
106 *
107 * ALPHAR (output) DOUBLE PRECISION array, dimension (N)
108 * ALPHAI (output) DOUBLE PRECISION array, dimension (N)
109 * BETA (output) DOUBLE PRECISION array, dimension (N)
110 * On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
111 * be the generalized eigenvalues. If ALPHAI(j) is zero, then
112 * the j-th eigenvalue is real; if positive, then the j-th and
113 * (j+1)-st eigenvalues are a complex conjugate pair, with
114 * ALPHAI(j+1) negative.
115 *
116 * Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
117 * may easily over- or underflow, and BETA(j) may even be zero.
118 * Thus, the user should avoid naively computing the ratio
119 * ALPHA/BETA. However, ALPHAR and ALPHAI will be always less
120 * than and usually comparable with norm(A) in magnitude, and
121 * BETA always less than and usually comparable with norm(B).
122 *
123 * VL (output) DOUBLE PRECISION array, dimension (LDVL,N)
124 * If JOBVL = 'V', the left eigenvectors u(j) are stored one
125 * after another in the columns of VL, in the same order as
126 * their eigenvalues. If the j-th eigenvalue is real, then
127 * u(j) = VL(:,j), the j-th column of VL. If the j-th and
128 * (j+1)-th eigenvalues form a complex conjugate pair, then
129 * u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
130 * Each eigenvector will be scaled so the largest component have
131 * abs(real part) + abs(imag. part) = 1.
132 * Not referenced if JOBVL = 'N'.
133 *
134 * LDVL (input) INTEGER
135 * The leading dimension of the matrix VL. LDVL >= 1, and
136 * if JOBVL = 'V', LDVL >= N.
137 *
138 * VR (output) DOUBLE PRECISION array, dimension (LDVR,N)
139 * If JOBVR = 'V', the right eigenvectors v(j) are stored one
140 * after another in the columns of VR, in the same order as
141 * their eigenvalues. If the j-th eigenvalue is real, then
142 * v(j) = VR(:,j), the j-th column of VR. If the j-th and
143 * (j+1)-th eigenvalues form a complex conjugate pair, then
144 * v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
145 * Each eigenvector will be scaled so the largest component have
146 * abs(real part) + abs(imag. part) = 1.
147 * Not referenced if JOBVR = 'N'.
148 *
149 * LDVR (input) INTEGER
150 * The leading dimension of the matrix VR. LDVR >= 1, and
151 * if JOBVR = 'V', LDVR >= N.
152 *
153 * ILO (output) INTEGER
154 * IHI (output) INTEGER
155 * ILO and IHI are integer values such that on exit
156 * A(i,j) = 0 and B(i,j) = 0 if i > j and
157 * j = 1,...,ILO-1 or i = IHI+1,...,N.
158 * If BALANC = 'N' or 'S', ILO = 1 and IHI = N.
159 *
160 * LSCALE (output) DOUBLE PRECISION array, dimension (N)
161 * Details of the permutations and scaling factors applied
162 * to the left side of A and B. If PL(j) is the index of the
163 * row interchanged with row j, and DL(j) is the scaling
164 * factor applied to row j, then
165 * LSCALE(j) = PL(j) for j = 1,...,ILO-1
166 * = DL(j) for j = ILO,...,IHI
167 * = PL(j) for j = IHI+1,...,N.
168 * The order in which the interchanges are made is N to IHI+1,
169 * then 1 to ILO-1.
170 *
171 * RSCALE (output) DOUBLE PRECISION array, dimension (N)
172 * Details of the permutations and scaling factors applied
173 * to the right side of A and B. If PR(j) is the index of the
174 * column interchanged with column j, and DR(j) is the scaling
175 * factor applied to column j, then
176 * RSCALE(j) = PR(j) for j = 1,...,ILO-1
177 * = DR(j) for j = ILO,...,IHI
178 * = PR(j) for j = IHI+1,...,N
179 * The order in which the interchanges are made is N to IHI+1,
180 * then 1 to ILO-1.
181 *
182 * ABNRM (output) DOUBLE PRECISION
183 * The one-norm of the balanced matrix A.
184 *
185 * BBNRM (output) DOUBLE PRECISION
186 * The one-norm of the balanced matrix B.
187 *
188 * RCONDE (output) DOUBLE PRECISION array, dimension (N)
189 * If SENSE = 'E' or 'B', the reciprocal condition numbers of
190 * the eigenvalues, stored in consecutive elements of the array.
191 * For a complex conjugate pair of eigenvalues two consecutive
192 * elements of RCONDE are set to the same value. Thus RCONDE(j),
193 * RCONDV(j), and the j-th columns of VL and VR all correspond
194 * to the j-th eigenpair.
195 * If SENSE = 'N or 'V', RCONDE is not referenced.
196 *
197 * RCONDV (output) DOUBLE PRECISION array, dimension (N)
198 * If SENSE = 'V' or 'B', the estimated reciprocal condition
199 * numbers of the eigenvectors, stored in consecutive elements
200 * of the array. For a complex eigenvector two consecutive
201 * elements of RCONDV are set to the same value. If the
202 * eigenvalues cannot be reordered to compute RCONDV(j),
203 * RCONDV(j) is set to 0; this can only occur when the true
204 * value would be very small anyway.
205 * If SENSE = 'N' or 'E', RCONDV is not referenced.
206 *
207 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
208 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
209 *
210 * LWORK (input) INTEGER
211 * The dimension of the array WORK. LWORK >= max(1,2*N).
212 * If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
213 * LWORK >= max(1,6*N).
214 * If SENSE = 'E' or 'B', LWORK >= max(1,10*N).
215 * If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.
216 *
217 * If LWORK = -1, then a workspace query is assumed; the routine
218 * only calculates the optimal size of the WORK array, returns
219 * this value as the first entry of the WORK array, and no error
220 * message related to LWORK is issued by XERBLA.
221 *
222 * IWORK (workspace) INTEGER array, dimension (N+6)
223 * If SENSE = 'E', IWORK is not referenced.
224 *
225 * BWORK (workspace) LOGICAL array, dimension (N)
226 * If SENSE = 'N', BWORK is not referenced.
227 *
228 * INFO (output) INTEGER
229 * = 0: successful exit
230 * < 0: if INFO = -i, the i-th argument had an illegal value.
231 * = 1,...,N:
232 * The QZ iteration failed. No eigenvectors have been
233 * calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
234 * should be correct for j=INFO+1,...,N.
235 * > N: =N+1: other than QZ iteration failed in DHGEQZ.
236 * =N+2: error return from DTGEVC.
237 *
238 * Further Details
239 * ===============
240 *
241 * Balancing a matrix pair (A,B) includes, first, permuting rows and
242 * columns to isolate eigenvalues, second, applying diagonal similarity
243 * transformation to the rows and columns to make the rows and columns
244 * as close in norm as possible. The computed reciprocal condition
245 * numbers correspond to the balanced matrix. Permuting rows and columns
246 * will not change the condition numbers (in exact arithmetic) but
247 * diagonal scaling will. For further explanation of balancing, see
248 * section 4.11.1.2 of LAPACK Users' Guide.
249 *
250 * An approximate error bound on the chordal distance between the i-th
251 * computed generalized eigenvalue w and the corresponding exact
252 * eigenvalue lambda is
253 *
254 * chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
255 *
256 * An approximate error bound for the angle between the i-th computed
257 * eigenvector VL(i) or VR(i) is given by
258 *
259 * EPS * norm(ABNRM, BBNRM) / DIF(i).
260 *
261 * For further explanation of the reciprocal condition numbers RCONDE
262 * and RCONDV, see section 4.11 of LAPACK User's Guide.
263 *
264 * =====================================================================
265 *
266 * .. Parameters ..
267 DOUBLE PRECISION ZERO, ONE
268 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
269 * ..
270 * .. Local Scalars ..
271 LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY, NOSCL,
272 $ PAIR, WANTSB, WANTSE, WANTSN, WANTSV
273 CHARACTER CHTEMP
274 INTEGER I, ICOLS, IERR, IJOBVL, IJOBVR, IN, IROWS,
275 $ ITAU, IWRK, IWRK1, J, JC, JR, M, MAXWRK,
276 $ MINWRK, MM
277 DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
278 $ SMLNUM, TEMP
279 * ..
280 * .. Local Arrays ..
281 LOGICAL LDUMMA( 1 )
282 * ..
283 * .. External Subroutines ..
284 EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLABAD,
285 $ DLACPY, DLASCL, DLASET, DORGQR, DORMQR, DTGEVC,
286 $ DTGSNA, XERBLA
287 * ..
288 * .. External Functions ..
289 LOGICAL LSAME
290 INTEGER ILAENV
291 DOUBLE PRECISION DLAMCH, DLANGE
292 EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
293 * ..
294 * .. Intrinsic Functions ..
295 INTRINSIC ABS, MAX, SQRT
296 * ..
297 * .. Executable Statements ..
298 *
299 * Decode the input arguments
300 *
301 IF( LSAME( JOBVL, 'N' ) ) THEN
302 IJOBVL = 1
303 ILVL = .FALSE.
304 ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
305 IJOBVL = 2
306 ILVL = .TRUE.
307 ELSE
308 IJOBVL = -1
309 ILVL = .FALSE.
310 END IF
311 *
312 IF( LSAME( JOBVR, 'N' ) ) THEN
313 IJOBVR = 1
314 ILVR = .FALSE.
315 ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
316 IJOBVR = 2
317 ILVR = .TRUE.
318 ELSE
319 IJOBVR = -1
320 ILVR = .FALSE.
321 END IF
322 ILV = ILVL .OR. ILVR
323 *
324 NOSCL = LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'P' )
325 WANTSN = LSAME( SENSE, 'N' )
326 WANTSE = LSAME( SENSE, 'E' )
327 WANTSV = LSAME( SENSE, 'V' )
328 WANTSB = LSAME( SENSE, 'B' )
329 *
330 * Test the input arguments
331 *
332 INFO = 0
333 LQUERY = ( LWORK.EQ.-1 )
334 IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC,
335 $ 'S' ) .OR. LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) )
336 $ THEN
337 INFO = -1
338 ELSE IF( IJOBVL.LE.0 ) THEN
339 INFO = -2
340 ELSE IF( IJOBVR.LE.0 ) THEN
341 INFO = -3
342 ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSB .OR. WANTSV ) )
343 $ THEN
344 INFO = -4
345 ELSE IF( N.LT.0 ) THEN
346 INFO = -5
347 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
348 INFO = -7
349 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
350 INFO = -9
351 ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
352 INFO = -14
353 ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
354 INFO = -16
355 END IF
356 *
357 * Compute workspace
358 * (Note: Comments in the code beginning "Workspace:" describe the
359 * minimal amount of workspace needed at that point in the code,
360 * as well as the preferred amount for good performance.
361 * NB refers to the optimal block size for the immediately
362 * following subroutine, as returned by ILAENV. The workspace is
363 * computed assuming ILO = 1 and IHI = N, the worst case.)
364 *
365 IF( INFO.EQ.0 ) THEN
366 IF( N.EQ.0 ) THEN
367 MINWRK = 1
368 MAXWRK = 1
369 ELSE
370 IF( NOSCL .AND. .NOT.ILV ) THEN
371 MINWRK = 2*N
372 ELSE
373 MINWRK = 6*N
374 END IF
375 IF( WANTSE .OR. WANTSB ) THEN
376 MINWRK = 10*N
377 END IF
378 IF( WANTSV .OR. WANTSB ) THEN
379 MINWRK = MAX( MINWRK, 2*N*( N + 4 ) + 16 )
380 END IF
381 MAXWRK = MINWRK
382 MAXWRK = MAX( MAXWRK,
383 $ N + N*ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 ) )
384 MAXWRK = MAX( MAXWRK,
385 $ N + N*ILAENV( 1, 'DORMQR', ' ', N, 1, N, 0 ) )
386 IF( ILVL ) THEN
387 MAXWRK = MAX( MAXWRK, N +
388 $ N*ILAENV( 1, 'DORGQR', ' ', N, 1, N, 0 ) )
389 END IF
390 END IF
391 WORK( 1 ) = MAXWRK
392 *
393 IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
394 INFO = -26
395 END IF
396 END IF
397 *
398 IF( INFO.NE.0 ) THEN
399 CALL XERBLA( 'DGGEVX', -INFO )
400 RETURN
401 ELSE IF( LQUERY ) THEN
402 RETURN
403 END IF
404 *
405 * Quick return if possible
406 *
407 IF( N.EQ.0 )
408 $ RETURN
409 *
410 *
411 * Get machine constants
412 *
413 EPS = DLAMCH( 'P' )
414 SMLNUM = DLAMCH( 'S' )
415 BIGNUM = ONE / SMLNUM
416 CALL DLABAD( SMLNUM, BIGNUM )
417 SMLNUM = SQRT( SMLNUM ) / EPS
418 BIGNUM = ONE / SMLNUM
419 *
420 * Scale A if max element outside range [SMLNUM,BIGNUM]
421 *
422 ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
423 ILASCL = .FALSE.
424 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
425 ANRMTO = SMLNUM
426 ILASCL = .TRUE.
427 ELSE IF( ANRM.GT.BIGNUM ) THEN
428 ANRMTO = BIGNUM
429 ILASCL = .TRUE.
430 END IF
431 IF( ILASCL )
432 $ CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
433 *
434 * Scale B if max element outside range [SMLNUM,BIGNUM]
435 *
436 BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
437 ILBSCL = .FALSE.
438 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
439 BNRMTO = SMLNUM
440 ILBSCL = .TRUE.
441 ELSE IF( BNRM.GT.BIGNUM ) THEN
442 BNRMTO = BIGNUM
443 ILBSCL = .TRUE.
444 END IF
445 IF( ILBSCL )
446 $ CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
447 *
448 * Permute and/or balance the matrix pair (A,B)
449 * (Workspace: need 6*N if BALANC = 'S' or 'B', 1 otherwise)
450 *
451 CALL DGGBAL( BALANC, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE,
452 $ WORK, IERR )
453 *
454 * Compute ABNRM and BBNRM
455 *
456 ABNRM = DLANGE( '1', N, N, A, LDA, WORK( 1 ) )
457 IF( ILASCL ) THEN
458 WORK( 1 ) = ABNRM
459 CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, 1, 1, WORK( 1 ), 1,
460 $ IERR )
461 ABNRM = WORK( 1 )
462 END IF
463 *
464 BBNRM = DLANGE( '1', N, N, B, LDB, WORK( 1 ) )
465 IF( ILBSCL ) THEN
466 WORK( 1 ) = BBNRM
467 CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, 1, 1, WORK( 1 ), 1,
468 $ IERR )
469 BBNRM = WORK( 1 )
470 END IF
471 *
472 * Reduce B to triangular form (QR decomposition of B)
473 * (Workspace: need N, prefer N*NB )
474 *
475 IROWS = IHI + 1 - ILO
476 IF( ILV .OR. .NOT.WANTSN ) THEN
477 ICOLS = N + 1 - ILO
478 ELSE
479 ICOLS = IROWS
480 END IF
481 ITAU = 1
482 IWRK = ITAU + IROWS
483 CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
484 $ WORK( IWRK ), LWORK+1-IWRK, IERR )
485 *
486 * Apply the orthogonal transformation to A
487 * (Workspace: need N, prefer N*NB)
488 *
489 CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
490 $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
491 $ LWORK+1-IWRK, IERR )
492 *
493 * Initialize VL and/or VR
494 * (Workspace: need N, prefer N*NB)
495 *
496 IF( ILVL ) THEN
497 CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
498 IF( IROWS.GT.1 ) THEN
499 CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
500 $ VL( ILO+1, ILO ), LDVL )
501 END IF
502 CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
503 $ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
504 END IF
505 *
506 IF( ILVR )
507 $ CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
508 *
509 * Reduce to generalized Hessenberg form
510 * (Workspace: none needed)
511 *
512 IF( ILV .OR. .NOT.WANTSN ) THEN
513 *
514 * Eigenvectors requested -- work on whole matrix.
515 *
516 CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
517 $ LDVL, VR, LDVR, IERR )
518 ELSE
519 CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
520 $ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR )
521 END IF
522 *
523 * Perform QZ algorithm (Compute eigenvalues, and optionally, the
524 * Schur forms and Schur vectors)
525 * (Workspace: need N)
526 *
527 IF( ILV .OR. .NOT.WANTSN ) THEN
528 CHTEMP = 'S'
529 ELSE
530 CHTEMP = 'E'
531 END IF
532 *
533 CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
534 $ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK,
535 $ LWORK, IERR )
536 IF( IERR.NE.0 ) THEN
537 IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
538 INFO = IERR
539 ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
540 INFO = IERR - N
541 ELSE
542 INFO = N + 1
543 END IF
544 GO TO 130
545 END IF
546 *
547 * Compute Eigenvectors and estimate condition numbers if desired
548 * (Workspace: DTGEVC: need 6*N
549 * DTGSNA: need 2*N*(N+2)+16 if SENSE = 'V' or 'B',
550 * need N otherwise )
551 *
552 IF( ILV .OR. .NOT.WANTSN ) THEN
553 IF( ILV ) THEN
554 IF( ILVL ) THEN
555 IF( ILVR ) THEN
556 CHTEMP = 'B'
557 ELSE
558 CHTEMP = 'L'
559 END IF
560 ELSE
561 CHTEMP = 'R'
562 END IF
563 *
564 CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL,
565 $ LDVL, VR, LDVR, N, IN, WORK, IERR )
566 IF( IERR.NE.0 ) THEN
567 INFO = N + 2
568 GO TO 130
569 END IF
570 END IF
571 *
572 IF( .NOT.WANTSN ) THEN
573 *
574 * compute eigenvectors (DTGEVC) and estimate condition
575 * numbers (DTGSNA). Note that the definition of the condition
576 * number is not invariant under transformation (u,v) to
577 * (Q*u, Z*v), where (u,v) are eigenvectors of the generalized
578 * Schur form (S,T), Q and Z are orthogonal matrices. In order
579 * to avoid using extra 2*N*N workspace, we have to recalculate
580 * eigenvectors and estimate one condition numbers at a time.
581 *
582 PAIR = .FALSE.
583 DO 20 I = 1, N
584 *
585 IF( PAIR ) THEN
586 PAIR = .FALSE.
587 GO TO 20
588 END IF
589 MM = 1
590 IF( I.LT.N ) THEN
591 IF( A( I+1, I ).NE.ZERO ) THEN
592 PAIR = .TRUE.
593 MM = 2
594 END IF
595 END IF
596 *
597 DO 10 J = 1, N
598 BWORK( J ) = .FALSE.
599 10 CONTINUE
600 IF( MM.EQ.1 ) THEN
601 BWORK( I ) = .TRUE.
602 ELSE IF( MM.EQ.2 ) THEN
603 BWORK( I ) = .TRUE.
604 BWORK( I+1 ) = .TRUE.
605 END IF
606 *
607 IWRK = MM*N + 1
608 IWRK1 = IWRK + MM*N
609 *
610 * Compute a pair of left and right eigenvectors.
611 * (compute workspace: need up to 4*N + 6*N)
612 *
613 IF( WANTSE .OR. WANTSB ) THEN
614 CALL DTGEVC( 'B', 'S', BWORK, N, A, LDA, B, LDB,
615 $ WORK( 1 ), N, WORK( IWRK ), N, MM, M,
616 $ WORK( IWRK1 ), IERR )
617 IF( IERR.NE.0 ) THEN
618 INFO = N + 2
619 GO TO 130
620 END IF
621 END IF
622 *
623 CALL DTGSNA( SENSE, 'S', BWORK, N, A, LDA, B, LDB,
624 $ WORK( 1 ), N, WORK( IWRK ), N, RCONDE( I ),
625 $ RCONDV( I ), MM, M, WORK( IWRK1 ),
626 $ LWORK-IWRK1+1, IWORK, IERR )
627 *
628 20 CONTINUE
629 END IF
630 END IF
631 *
632 * Undo balancing on VL and VR and normalization
633 * (Workspace: none needed)
634 *
635 IF( ILVL ) THEN
636 CALL DGGBAK( BALANC, 'L', N, ILO, IHI, LSCALE, RSCALE, N, VL,
637 $ LDVL, IERR )
638 *
639 DO 70 JC = 1, N
640 IF( ALPHAI( JC ).LT.ZERO )
641 $ GO TO 70
642 TEMP = ZERO
643 IF( ALPHAI( JC ).EQ.ZERO ) THEN
644 DO 30 JR = 1, N
645 TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
646 30 CONTINUE
647 ELSE
648 DO 40 JR = 1, N
649 TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
650 $ ABS( VL( JR, JC+1 ) ) )
651 40 CONTINUE
652 END IF
653 IF( TEMP.LT.SMLNUM )
654 $ GO TO 70
655 TEMP = ONE / TEMP
656 IF( ALPHAI( JC ).EQ.ZERO ) THEN
657 DO 50 JR = 1, N
658 VL( JR, JC ) = VL( JR, JC )*TEMP
659 50 CONTINUE
660 ELSE
661 DO 60 JR = 1, N
662 VL( JR, JC ) = VL( JR, JC )*TEMP
663 VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
664 60 CONTINUE
665 END IF
666 70 CONTINUE
667 END IF
668 IF( ILVR ) THEN
669 CALL DGGBAK( BALANC, 'R', N, ILO, IHI, LSCALE, RSCALE, N, VR,
670 $ LDVR, IERR )
671 DO 120 JC = 1, N
672 IF( ALPHAI( JC ).LT.ZERO )
673 $ GO TO 120
674 TEMP = ZERO
675 IF( ALPHAI( JC ).EQ.ZERO ) THEN
676 DO 80 JR = 1, N
677 TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
678 80 CONTINUE
679 ELSE
680 DO 90 JR = 1, N
681 TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
682 $ ABS( VR( JR, JC+1 ) ) )
683 90 CONTINUE
684 END IF
685 IF( TEMP.LT.SMLNUM )
686 $ GO TO 120
687 TEMP = ONE / TEMP
688 IF( ALPHAI( JC ).EQ.ZERO ) THEN
689 DO 100 JR = 1, N
690 VR( JR, JC ) = VR( JR, JC )*TEMP
691 100 CONTINUE
692 ELSE
693 DO 110 JR = 1, N
694 VR( JR, JC ) = VR( JR, JC )*TEMP
695 VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
696 110 CONTINUE
697 END IF
698 120 CONTINUE
699 END IF
700 *
701 * Undo scaling if necessary
702 *
703 IF( ILASCL ) THEN
704 CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR )
705 CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR )
706 END IF
707 *
708 IF( ILBSCL ) THEN
709 CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
710 END IF
711 *
712 130 CONTINUE
713 WORK( 1 ) = MAXWRK
714 *
715 RETURN
716 *
717 * End of DGGEVX
718 *
719 END
2 $ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,
3 $ IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,
4 $ RCONDV, WORK, LWORK, IWORK, BWORK, INFO )
5 *
6 * -- LAPACK driver routine (version 3.2) --
7 * -- LAPACK is a software package provided by Univ. of Tennessee, --
8 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
9 * November 2006
10 *
11 * .. Scalar Arguments ..
12 CHARACTER BALANC, JOBVL, JOBVR, SENSE
13 INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
14 DOUBLE PRECISION ABNRM, BBNRM
15 * ..
16 * .. Array Arguments ..
17 LOGICAL BWORK( * )
18 INTEGER IWORK( * )
19 DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
20 $ B( LDB, * ), BETA( * ), LSCALE( * ),
21 $ RCONDE( * ), RCONDV( * ), RSCALE( * ),
22 $ VL( LDVL, * ), VR( LDVR, * ), WORK( * )
23 * ..
24 *
25 * Purpose
26 * =======
27 *
28 * DGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
29 * the generalized eigenvalues, and optionally, the left and/or right
30 * generalized eigenvectors.
31 *
32 * Optionally also, it computes a balancing transformation to improve
33 * the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
34 * LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
35 * the eigenvalues (RCONDE), and reciprocal condition numbers for the
36 * right eigenvectors (RCONDV).
37 *
38 * A generalized eigenvalue for a pair of matrices (A,B) is a scalar
39 * lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
40 * singular. It is usually represented as the pair (alpha,beta), as
41 * there is a reasonable interpretation for beta=0, and even for both
42 * being zero.
43 *
44 * The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
45 * of (A,B) satisfies
46 *
47 * A * v(j) = lambda(j) * B * v(j) .
48 *
49 * The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
50 * of (A,B) satisfies
51 *
52 * u(j)**H * A = lambda(j) * u(j)**H * B.
53 *
54 * where u(j)**H is the conjugate-transpose of u(j).
55 *
56 *
57 * Arguments
58 * =========
59 *
60 * BALANC (input) CHARACTER*1
61 * Specifies the balance option to be performed.
62 * = 'N': do not diagonally scale or permute;
63 * = 'P': permute only;
64 * = 'S': scale only;
65 * = 'B': both permute and scale.
66 * Computed reciprocal condition numbers will be for the
67 * matrices after permuting and/or balancing. Permuting does
68 * not change condition numbers (in exact arithmetic), but
69 * balancing does.
70 *
71 * JOBVL (input) CHARACTER*1
72 * = 'N': do not compute the left generalized eigenvectors;
73 * = 'V': compute the left generalized eigenvectors.
74 *
75 * JOBVR (input) CHARACTER*1
76 * = 'N': do not compute the right generalized eigenvectors;
77 * = 'V': compute the right generalized eigenvectors.
78 *
79 * SENSE (input) CHARACTER*1
80 * Determines which reciprocal condition numbers are computed.
81 * = 'N': none are computed;
82 * = 'E': computed for eigenvalues only;
83 * = 'V': computed for eigenvectors only;
84 * = 'B': computed for eigenvalues and eigenvectors.
85 *
86 * N (input) INTEGER
87 * The order of the matrices A, B, VL, and VR. N >= 0.
88 *
89 * A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
90 * On entry, the matrix A in the pair (A,B).
91 * On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
92 * or both, then A contains the first part of the real Schur
93 * form of the "balanced" versions of the input A and B.
94 *
95 * LDA (input) INTEGER
96 * The leading dimension of A. LDA >= max(1,N).
97 *
98 * B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
99 * On entry, the matrix B in the pair (A,B).
100 * On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
101 * or both, then B contains the second part of the real Schur
102 * form of the "balanced" versions of the input A and B.
103 *
104 * LDB (input) INTEGER
105 * The leading dimension of B. LDB >= max(1,N).
106 *
107 * ALPHAR (output) DOUBLE PRECISION array, dimension (N)
108 * ALPHAI (output) DOUBLE PRECISION array, dimension (N)
109 * BETA (output) DOUBLE PRECISION array, dimension (N)
110 * On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
111 * be the generalized eigenvalues. If ALPHAI(j) is zero, then
112 * the j-th eigenvalue is real; if positive, then the j-th and
113 * (j+1)-st eigenvalues are a complex conjugate pair, with
114 * ALPHAI(j+1) negative.
115 *
116 * Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
117 * may easily over- or underflow, and BETA(j) may even be zero.
118 * Thus, the user should avoid naively computing the ratio
119 * ALPHA/BETA. However, ALPHAR and ALPHAI will be always less
120 * than and usually comparable with norm(A) in magnitude, and
121 * BETA always less than and usually comparable with norm(B).
122 *
123 * VL (output) DOUBLE PRECISION array, dimension (LDVL,N)
124 * If JOBVL = 'V', the left eigenvectors u(j) are stored one
125 * after another in the columns of VL, in the same order as
126 * their eigenvalues. If the j-th eigenvalue is real, then
127 * u(j) = VL(:,j), the j-th column of VL. If the j-th and
128 * (j+1)-th eigenvalues form a complex conjugate pair, then
129 * u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
130 * Each eigenvector will be scaled so the largest component have
131 * abs(real part) + abs(imag. part) = 1.
132 * Not referenced if JOBVL = 'N'.
133 *
134 * LDVL (input) INTEGER
135 * The leading dimension of the matrix VL. LDVL >= 1, and
136 * if JOBVL = 'V', LDVL >= N.
137 *
138 * VR (output) DOUBLE PRECISION array, dimension (LDVR,N)
139 * If JOBVR = 'V', the right eigenvectors v(j) are stored one
140 * after another in the columns of VR, in the same order as
141 * their eigenvalues. If the j-th eigenvalue is real, then
142 * v(j) = VR(:,j), the j-th column of VR. If the j-th and
143 * (j+1)-th eigenvalues form a complex conjugate pair, then
144 * v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
145 * Each eigenvector will be scaled so the largest component have
146 * abs(real part) + abs(imag. part) = 1.
147 * Not referenced if JOBVR = 'N'.
148 *
149 * LDVR (input) INTEGER
150 * The leading dimension of the matrix VR. LDVR >= 1, and
151 * if JOBVR = 'V', LDVR >= N.
152 *
153 * ILO (output) INTEGER
154 * IHI (output) INTEGER
155 * ILO and IHI are integer values such that on exit
156 * A(i,j) = 0 and B(i,j) = 0 if i > j and
157 * j = 1,...,ILO-1 or i = IHI+1,...,N.
158 * If BALANC = 'N' or 'S', ILO = 1 and IHI = N.
159 *
160 * LSCALE (output) DOUBLE PRECISION array, dimension (N)
161 * Details of the permutations and scaling factors applied
162 * to the left side of A and B. If PL(j) is the index of the
163 * row interchanged with row j, and DL(j) is the scaling
164 * factor applied to row j, then
165 * LSCALE(j) = PL(j) for j = 1,...,ILO-1
166 * = DL(j) for j = ILO,...,IHI
167 * = PL(j) for j = IHI+1,...,N.
168 * The order in which the interchanges are made is N to IHI+1,
169 * then 1 to ILO-1.
170 *
171 * RSCALE (output) DOUBLE PRECISION array, dimension (N)
172 * Details of the permutations and scaling factors applied
173 * to the right side of A and B. If PR(j) is the index of the
174 * column interchanged with column j, and DR(j) is the scaling
175 * factor applied to column j, then
176 * RSCALE(j) = PR(j) for j = 1,...,ILO-1
177 * = DR(j) for j = ILO,...,IHI
178 * = PR(j) for j = IHI+1,...,N
179 * The order in which the interchanges are made is N to IHI+1,
180 * then 1 to ILO-1.
181 *
182 * ABNRM (output) DOUBLE PRECISION
183 * The one-norm of the balanced matrix A.
184 *
185 * BBNRM (output) DOUBLE PRECISION
186 * The one-norm of the balanced matrix B.
187 *
188 * RCONDE (output) DOUBLE PRECISION array, dimension (N)
189 * If SENSE = 'E' or 'B', the reciprocal condition numbers of
190 * the eigenvalues, stored in consecutive elements of the array.
191 * For a complex conjugate pair of eigenvalues two consecutive
192 * elements of RCONDE are set to the same value. Thus RCONDE(j),
193 * RCONDV(j), and the j-th columns of VL and VR all correspond
194 * to the j-th eigenpair.
195 * If SENSE = 'N or 'V', RCONDE is not referenced.
196 *
197 * RCONDV (output) DOUBLE PRECISION array, dimension (N)
198 * If SENSE = 'V' or 'B', the estimated reciprocal condition
199 * numbers of the eigenvectors, stored in consecutive elements
200 * of the array. For a complex eigenvector two consecutive
201 * elements of RCONDV are set to the same value. If the
202 * eigenvalues cannot be reordered to compute RCONDV(j),
203 * RCONDV(j) is set to 0; this can only occur when the true
204 * value would be very small anyway.
205 * If SENSE = 'N' or 'E', RCONDV is not referenced.
206 *
207 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
208 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
209 *
210 * LWORK (input) INTEGER
211 * The dimension of the array WORK. LWORK >= max(1,2*N).
212 * If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
213 * LWORK >= max(1,6*N).
214 * If SENSE = 'E' or 'B', LWORK >= max(1,10*N).
215 * If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.
216 *
217 * If LWORK = -1, then a workspace query is assumed; the routine
218 * only calculates the optimal size of the WORK array, returns
219 * this value as the first entry of the WORK array, and no error
220 * message related to LWORK is issued by XERBLA.
221 *
222 * IWORK (workspace) INTEGER array, dimension (N+6)
223 * If SENSE = 'E', IWORK is not referenced.
224 *
225 * BWORK (workspace) LOGICAL array, dimension (N)
226 * If SENSE = 'N', BWORK is not referenced.
227 *
228 * INFO (output) INTEGER
229 * = 0: successful exit
230 * < 0: if INFO = -i, the i-th argument had an illegal value.
231 * = 1,...,N:
232 * The QZ iteration failed. No eigenvectors have been
233 * calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
234 * should be correct for j=INFO+1,...,N.
235 * > N: =N+1: other than QZ iteration failed in DHGEQZ.
236 * =N+2: error return from DTGEVC.
237 *
238 * Further Details
239 * ===============
240 *
241 * Balancing a matrix pair (A,B) includes, first, permuting rows and
242 * columns to isolate eigenvalues, second, applying diagonal similarity
243 * transformation to the rows and columns to make the rows and columns
244 * as close in norm as possible. The computed reciprocal condition
245 * numbers correspond to the balanced matrix. Permuting rows and columns
246 * will not change the condition numbers (in exact arithmetic) but
247 * diagonal scaling will. For further explanation of balancing, see
248 * section 4.11.1.2 of LAPACK Users' Guide.
249 *
250 * An approximate error bound on the chordal distance between the i-th
251 * computed generalized eigenvalue w and the corresponding exact
252 * eigenvalue lambda is
253 *
254 * chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
255 *
256 * An approximate error bound for the angle between the i-th computed
257 * eigenvector VL(i) or VR(i) is given by
258 *
259 * EPS * norm(ABNRM, BBNRM) / DIF(i).
260 *
261 * For further explanation of the reciprocal condition numbers RCONDE
262 * and RCONDV, see section 4.11 of LAPACK User's Guide.
263 *
264 * =====================================================================
265 *
266 * .. Parameters ..
267 DOUBLE PRECISION ZERO, ONE
268 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
269 * ..
270 * .. Local Scalars ..
271 LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY, NOSCL,
272 $ PAIR, WANTSB, WANTSE, WANTSN, WANTSV
273 CHARACTER CHTEMP
274 INTEGER I, ICOLS, IERR, IJOBVL, IJOBVR, IN, IROWS,
275 $ ITAU, IWRK, IWRK1, J, JC, JR, M, MAXWRK,
276 $ MINWRK, MM
277 DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
278 $ SMLNUM, TEMP
279 * ..
280 * .. Local Arrays ..
281 LOGICAL LDUMMA( 1 )
282 * ..
283 * .. External Subroutines ..
284 EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLABAD,
285 $ DLACPY, DLASCL, DLASET, DORGQR, DORMQR, DTGEVC,
286 $ DTGSNA, XERBLA
287 * ..
288 * .. External Functions ..
289 LOGICAL LSAME
290 INTEGER ILAENV
291 DOUBLE PRECISION DLAMCH, DLANGE
292 EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
293 * ..
294 * .. Intrinsic Functions ..
295 INTRINSIC ABS, MAX, SQRT
296 * ..
297 * .. Executable Statements ..
298 *
299 * Decode the input arguments
300 *
301 IF( LSAME( JOBVL, 'N' ) ) THEN
302 IJOBVL = 1
303 ILVL = .FALSE.
304 ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
305 IJOBVL = 2
306 ILVL = .TRUE.
307 ELSE
308 IJOBVL = -1
309 ILVL = .FALSE.
310 END IF
311 *
312 IF( LSAME( JOBVR, 'N' ) ) THEN
313 IJOBVR = 1
314 ILVR = .FALSE.
315 ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
316 IJOBVR = 2
317 ILVR = .TRUE.
318 ELSE
319 IJOBVR = -1
320 ILVR = .FALSE.
321 END IF
322 ILV = ILVL .OR. ILVR
323 *
324 NOSCL = LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'P' )
325 WANTSN = LSAME( SENSE, 'N' )
326 WANTSE = LSAME( SENSE, 'E' )
327 WANTSV = LSAME( SENSE, 'V' )
328 WANTSB = LSAME( SENSE, 'B' )
329 *
330 * Test the input arguments
331 *
332 INFO = 0
333 LQUERY = ( LWORK.EQ.-1 )
334 IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC,
335 $ 'S' ) .OR. LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) )
336 $ THEN
337 INFO = -1
338 ELSE IF( IJOBVL.LE.0 ) THEN
339 INFO = -2
340 ELSE IF( IJOBVR.LE.0 ) THEN
341 INFO = -3
342 ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSB .OR. WANTSV ) )
343 $ THEN
344 INFO = -4
345 ELSE IF( N.LT.0 ) THEN
346 INFO = -5
347 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
348 INFO = -7
349 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
350 INFO = -9
351 ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
352 INFO = -14
353 ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
354 INFO = -16
355 END IF
356 *
357 * Compute workspace
358 * (Note: Comments in the code beginning "Workspace:" describe the
359 * minimal amount of workspace needed at that point in the code,
360 * as well as the preferred amount for good performance.
361 * NB refers to the optimal block size for the immediately
362 * following subroutine, as returned by ILAENV. The workspace is
363 * computed assuming ILO = 1 and IHI = N, the worst case.)
364 *
365 IF( INFO.EQ.0 ) THEN
366 IF( N.EQ.0 ) THEN
367 MINWRK = 1
368 MAXWRK = 1
369 ELSE
370 IF( NOSCL .AND. .NOT.ILV ) THEN
371 MINWRK = 2*N
372 ELSE
373 MINWRK = 6*N
374 END IF
375 IF( WANTSE .OR. WANTSB ) THEN
376 MINWRK = 10*N
377 END IF
378 IF( WANTSV .OR. WANTSB ) THEN
379 MINWRK = MAX( MINWRK, 2*N*( N + 4 ) + 16 )
380 END IF
381 MAXWRK = MINWRK
382 MAXWRK = MAX( MAXWRK,
383 $ N + N*ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 ) )
384 MAXWRK = MAX( MAXWRK,
385 $ N + N*ILAENV( 1, 'DORMQR', ' ', N, 1, N, 0 ) )
386 IF( ILVL ) THEN
387 MAXWRK = MAX( MAXWRK, N +
388 $ N*ILAENV( 1, 'DORGQR', ' ', N, 1, N, 0 ) )
389 END IF
390 END IF
391 WORK( 1 ) = MAXWRK
392 *
393 IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
394 INFO = -26
395 END IF
396 END IF
397 *
398 IF( INFO.NE.0 ) THEN
399 CALL XERBLA( 'DGGEVX', -INFO )
400 RETURN
401 ELSE IF( LQUERY ) THEN
402 RETURN
403 END IF
404 *
405 * Quick return if possible
406 *
407 IF( N.EQ.0 )
408 $ RETURN
409 *
410 *
411 * Get machine constants
412 *
413 EPS = DLAMCH( 'P' )
414 SMLNUM = DLAMCH( 'S' )
415 BIGNUM = ONE / SMLNUM
416 CALL DLABAD( SMLNUM, BIGNUM )
417 SMLNUM = SQRT( SMLNUM ) / EPS
418 BIGNUM = ONE / SMLNUM
419 *
420 * Scale A if max element outside range [SMLNUM,BIGNUM]
421 *
422 ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
423 ILASCL = .FALSE.
424 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
425 ANRMTO = SMLNUM
426 ILASCL = .TRUE.
427 ELSE IF( ANRM.GT.BIGNUM ) THEN
428 ANRMTO = BIGNUM
429 ILASCL = .TRUE.
430 END IF
431 IF( ILASCL )
432 $ CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
433 *
434 * Scale B if max element outside range [SMLNUM,BIGNUM]
435 *
436 BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
437 ILBSCL = .FALSE.
438 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
439 BNRMTO = SMLNUM
440 ILBSCL = .TRUE.
441 ELSE IF( BNRM.GT.BIGNUM ) THEN
442 BNRMTO = BIGNUM
443 ILBSCL = .TRUE.
444 END IF
445 IF( ILBSCL )
446 $ CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
447 *
448 * Permute and/or balance the matrix pair (A,B)
449 * (Workspace: need 6*N if BALANC = 'S' or 'B', 1 otherwise)
450 *
451 CALL DGGBAL( BALANC, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE,
452 $ WORK, IERR )
453 *
454 * Compute ABNRM and BBNRM
455 *
456 ABNRM = DLANGE( '1', N, N, A, LDA, WORK( 1 ) )
457 IF( ILASCL ) THEN
458 WORK( 1 ) = ABNRM
459 CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, 1, 1, WORK( 1 ), 1,
460 $ IERR )
461 ABNRM = WORK( 1 )
462 END IF
463 *
464 BBNRM = DLANGE( '1', N, N, B, LDB, WORK( 1 ) )
465 IF( ILBSCL ) THEN
466 WORK( 1 ) = BBNRM
467 CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, 1, 1, WORK( 1 ), 1,
468 $ IERR )
469 BBNRM = WORK( 1 )
470 END IF
471 *
472 * Reduce B to triangular form (QR decomposition of B)
473 * (Workspace: need N, prefer N*NB )
474 *
475 IROWS = IHI + 1 - ILO
476 IF( ILV .OR. .NOT.WANTSN ) THEN
477 ICOLS = N + 1 - ILO
478 ELSE
479 ICOLS = IROWS
480 END IF
481 ITAU = 1
482 IWRK = ITAU + IROWS
483 CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
484 $ WORK( IWRK ), LWORK+1-IWRK, IERR )
485 *
486 * Apply the orthogonal transformation to A
487 * (Workspace: need N, prefer N*NB)
488 *
489 CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
490 $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
491 $ LWORK+1-IWRK, IERR )
492 *
493 * Initialize VL and/or VR
494 * (Workspace: need N, prefer N*NB)
495 *
496 IF( ILVL ) THEN
497 CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
498 IF( IROWS.GT.1 ) THEN
499 CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
500 $ VL( ILO+1, ILO ), LDVL )
501 END IF
502 CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
503 $ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
504 END IF
505 *
506 IF( ILVR )
507 $ CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
508 *
509 * Reduce to generalized Hessenberg form
510 * (Workspace: none needed)
511 *
512 IF( ILV .OR. .NOT.WANTSN ) THEN
513 *
514 * Eigenvectors requested -- work on whole matrix.
515 *
516 CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
517 $ LDVL, VR, LDVR, IERR )
518 ELSE
519 CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
520 $ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR )
521 END IF
522 *
523 * Perform QZ algorithm (Compute eigenvalues, and optionally, the
524 * Schur forms and Schur vectors)
525 * (Workspace: need N)
526 *
527 IF( ILV .OR. .NOT.WANTSN ) THEN
528 CHTEMP = 'S'
529 ELSE
530 CHTEMP = 'E'
531 END IF
532 *
533 CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
534 $ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK,
535 $ LWORK, IERR )
536 IF( IERR.NE.0 ) THEN
537 IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
538 INFO = IERR
539 ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
540 INFO = IERR - N
541 ELSE
542 INFO = N + 1
543 END IF
544 GO TO 130
545 END IF
546 *
547 * Compute Eigenvectors and estimate condition numbers if desired
548 * (Workspace: DTGEVC: need 6*N
549 * DTGSNA: need 2*N*(N+2)+16 if SENSE = 'V' or 'B',
550 * need N otherwise )
551 *
552 IF( ILV .OR. .NOT.WANTSN ) THEN
553 IF( ILV ) THEN
554 IF( ILVL ) THEN
555 IF( ILVR ) THEN
556 CHTEMP = 'B'
557 ELSE
558 CHTEMP = 'L'
559 END IF
560 ELSE
561 CHTEMP = 'R'
562 END IF
563 *
564 CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL,
565 $ LDVL, VR, LDVR, N, IN, WORK, IERR )
566 IF( IERR.NE.0 ) THEN
567 INFO = N + 2
568 GO TO 130
569 END IF
570 END IF
571 *
572 IF( .NOT.WANTSN ) THEN
573 *
574 * compute eigenvectors (DTGEVC) and estimate condition
575 * numbers (DTGSNA). Note that the definition of the condition
576 * number is not invariant under transformation (u,v) to
577 * (Q*u, Z*v), where (u,v) are eigenvectors of the generalized
578 * Schur form (S,T), Q and Z are orthogonal matrices. In order
579 * to avoid using extra 2*N*N workspace, we have to recalculate
580 * eigenvectors and estimate one condition numbers at a time.
581 *
582 PAIR = .FALSE.
583 DO 20 I = 1, N
584 *
585 IF( PAIR ) THEN
586 PAIR = .FALSE.
587 GO TO 20
588 END IF
589 MM = 1
590 IF( I.LT.N ) THEN
591 IF( A( I+1, I ).NE.ZERO ) THEN
592 PAIR = .TRUE.
593 MM = 2
594 END IF
595 END IF
596 *
597 DO 10 J = 1, N
598 BWORK( J ) = .FALSE.
599 10 CONTINUE
600 IF( MM.EQ.1 ) THEN
601 BWORK( I ) = .TRUE.
602 ELSE IF( MM.EQ.2 ) THEN
603 BWORK( I ) = .TRUE.
604 BWORK( I+1 ) = .TRUE.
605 END IF
606 *
607 IWRK = MM*N + 1
608 IWRK1 = IWRK + MM*N
609 *
610 * Compute a pair of left and right eigenvectors.
611 * (compute workspace: need up to 4*N + 6*N)
612 *
613 IF( WANTSE .OR. WANTSB ) THEN
614 CALL DTGEVC( 'B', 'S', BWORK, N, A, LDA, B, LDB,
615 $ WORK( 1 ), N, WORK( IWRK ), N, MM, M,
616 $ WORK( IWRK1 ), IERR )
617 IF( IERR.NE.0 ) THEN
618 INFO = N + 2
619 GO TO 130
620 END IF
621 END IF
622 *
623 CALL DTGSNA( SENSE, 'S', BWORK, N, A, LDA, B, LDB,
624 $ WORK( 1 ), N, WORK( IWRK ), N, RCONDE( I ),
625 $ RCONDV( I ), MM, M, WORK( IWRK1 ),
626 $ LWORK-IWRK1+1, IWORK, IERR )
627 *
628 20 CONTINUE
629 END IF
630 END IF
631 *
632 * Undo balancing on VL and VR and normalization
633 * (Workspace: none needed)
634 *
635 IF( ILVL ) THEN
636 CALL DGGBAK( BALANC, 'L', N, ILO, IHI, LSCALE, RSCALE, N, VL,
637 $ LDVL, IERR )
638 *
639 DO 70 JC = 1, N
640 IF( ALPHAI( JC ).LT.ZERO )
641 $ GO TO 70
642 TEMP = ZERO
643 IF( ALPHAI( JC ).EQ.ZERO ) THEN
644 DO 30 JR = 1, N
645 TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
646 30 CONTINUE
647 ELSE
648 DO 40 JR = 1, N
649 TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
650 $ ABS( VL( JR, JC+1 ) ) )
651 40 CONTINUE
652 END IF
653 IF( TEMP.LT.SMLNUM )
654 $ GO TO 70
655 TEMP = ONE / TEMP
656 IF( ALPHAI( JC ).EQ.ZERO ) THEN
657 DO 50 JR = 1, N
658 VL( JR, JC ) = VL( JR, JC )*TEMP
659 50 CONTINUE
660 ELSE
661 DO 60 JR = 1, N
662 VL( JR, JC ) = VL( JR, JC )*TEMP
663 VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
664 60 CONTINUE
665 END IF
666 70 CONTINUE
667 END IF
668 IF( ILVR ) THEN
669 CALL DGGBAK( BALANC, 'R', N, ILO, IHI, LSCALE, RSCALE, N, VR,
670 $ LDVR, IERR )
671 DO 120 JC = 1, N
672 IF( ALPHAI( JC ).LT.ZERO )
673 $ GO TO 120
674 TEMP = ZERO
675 IF( ALPHAI( JC ).EQ.ZERO ) THEN
676 DO 80 JR = 1, N
677 TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
678 80 CONTINUE
679 ELSE
680 DO 90 JR = 1, N
681 TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
682 $ ABS( VR( JR, JC+1 ) ) )
683 90 CONTINUE
684 END IF
685 IF( TEMP.LT.SMLNUM )
686 $ GO TO 120
687 TEMP = ONE / TEMP
688 IF( ALPHAI( JC ).EQ.ZERO ) THEN
689 DO 100 JR = 1, N
690 VR( JR, JC ) = VR( JR, JC )*TEMP
691 100 CONTINUE
692 ELSE
693 DO 110 JR = 1, N
694 VR( JR, JC ) = VR( JR, JC )*TEMP
695 VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
696 110 CONTINUE
697 END IF
698 120 CONTINUE
699 END IF
700 *
701 * Undo scaling if necessary
702 *
703 IF( ILASCL ) THEN
704 CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR )
705 CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR )
706 END IF
707 *
708 IF( ILBSCL ) THEN
709 CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
710 END IF
711 *
712 130 CONTINUE
713 WORK( 1 ) = MAXWRK
714 *
715 RETURN
716 *
717 * End of DGGEVX
718 *
719 END