1 SUBROUTINE DGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
2 $ BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
3 *
4 * -- LAPACK driver routine (version 3.2) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * November 2006
8 *
9 * .. Scalar Arguments ..
10 CHARACTER JOBVL, JOBVR
11 INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
12 * ..
13 * .. Array Arguments ..
14 DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
15 $ B( LDB, * ), BETA( * ), VL( LDVL, * ),
16 $ VR( LDVR, * ), WORK( * )
17 * ..
18 *
19 * Purpose
20 * =======
21 *
22 * DGGEV computes for a pair of N-by-N real nonsymmetric matrices (A,B)
23 * the generalized eigenvalues, and optionally, the left and/or right
24 * generalized eigenvectors.
25 *
26 * A generalized eigenvalue for a pair of matrices (A,B) is a scalar
27 * lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
28 * singular. It is usually represented as the pair (alpha,beta), as
29 * there is a reasonable interpretation for beta=0, and even for both
30 * being zero.
31 *
32 * The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
33 * of (A,B) satisfies
34 *
35 * A * v(j) = lambda(j) * B * v(j).
36 *
37 * The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
38 * of (A,B) satisfies
39 *
40 * u(j)**H * A = lambda(j) * u(j)**H * B .
41 *
42 * where u(j)**H is the conjugate-transpose of u(j).
43 *
44 *
45 * Arguments
46 * =========
47 *
48 * JOBVL (input) CHARACTER*1
49 * = 'N': do not compute the left generalized eigenvectors;
50 * = 'V': compute the left generalized eigenvectors.
51 *
52 * JOBVR (input) CHARACTER*1
53 * = 'N': do not compute the right generalized eigenvectors;
54 * = 'V': compute the right generalized eigenvectors.
55 *
56 * N (input) INTEGER
57 * The order of the matrices A, B, VL, and VR. N >= 0.
58 *
59 * A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
60 * On entry, the matrix A in the pair (A,B).
61 * On exit, A has been overwritten.
62 *
63 * LDA (input) INTEGER
64 * The leading dimension of A. LDA >= max(1,N).
65 *
66 * B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
67 * On entry, the matrix B in the pair (A,B).
68 * On exit, B has been overwritten.
69 *
70 * LDB (input) INTEGER
71 * The leading dimension of B. LDB >= max(1,N).
72 *
73 * ALPHAR (output) DOUBLE PRECISION array, dimension (N)
74 * ALPHAI (output) DOUBLE PRECISION array, dimension (N)
75 * BETA (output) DOUBLE PRECISION array, dimension (N)
76 * On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
77 * be the generalized eigenvalues. If ALPHAI(j) is zero, then
78 * the j-th eigenvalue is real; if positive, then the j-th and
79 * (j+1)-st eigenvalues are a complex conjugate pair, with
80 * ALPHAI(j+1) negative.
81 *
82 * Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
83 * may easily over- or underflow, and BETA(j) may even be zero.
84 * Thus, the user should avoid naively computing the ratio
85 * alpha/beta. However, ALPHAR and ALPHAI will be always less
86 * than and usually comparable with norm(A) in magnitude, and
87 * BETA always less than and usually comparable with norm(B).
88 *
89 * VL (output) DOUBLE PRECISION array, dimension (LDVL,N)
90 * If JOBVL = 'V', the left eigenvectors u(j) are stored one
91 * after another in the columns of VL, in the same order as
92 * their eigenvalues. If the j-th eigenvalue is real, then
93 * u(j) = VL(:,j), the j-th column of VL. If the j-th and
94 * (j+1)-th eigenvalues form a complex conjugate pair, then
95 * u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
96 * Each eigenvector is scaled so the largest component has
97 * abs(real part)+abs(imag. part)=1.
98 * Not referenced if JOBVL = 'N'.
99 *
100 * LDVL (input) INTEGER
101 * The leading dimension of the matrix VL. LDVL >= 1, and
102 * if JOBVL = 'V', LDVL >= N.
103 *
104 * VR (output) DOUBLE PRECISION array, dimension (LDVR,N)
105 * If JOBVR = 'V', the right eigenvectors v(j) are stored one
106 * after another in the columns of VR, in the same order as
107 * their eigenvalues. If the j-th eigenvalue is real, then
108 * v(j) = VR(:,j), the j-th column of VR. If the j-th and
109 * (j+1)-th eigenvalues form a complex conjugate pair, then
110 * v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
111 * Each eigenvector is scaled so the largest component has
112 * abs(real part)+abs(imag. part)=1.
113 * Not referenced if JOBVR = 'N'.
114 *
115 * LDVR (input) INTEGER
116 * The leading dimension of the matrix VR. LDVR >= 1, and
117 * if JOBVR = 'V', LDVR >= N.
118 *
119 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
120 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
121 *
122 * LWORK (input) INTEGER
123 * The dimension of the array WORK. LWORK >= max(1,8*N).
124 * For good performance, LWORK must generally be larger.
125 *
126 * If LWORK = -1, then a workspace query is assumed; the routine
127 * only calculates the optimal size of the WORK array, returns
128 * this value as the first entry of the WORK array, and no error
129 * message related to LWORK is issued by XERBLA.
130 *
131 * INFO (output) INTEGER
132 * = 0: successful exit
133 * < 0: if INFO = -i, the i-th argument had an illegal value.
134 * = 1,...,N:
135 * The QZ iteration failed. No eigenvectors have been
136 * calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
137 * should be correct for j=INFO+1,...,N.
138 * > N: =N+1: other than QZ iteration failed in DHGEQZ.
139 * =N+2: error return from DTGEVC.
140 *
141 * =====================================================================
142 *
143 * .. Parameters ..
144 DOUBLE PRECISION ZERO, ONE
145 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
146 * ..
147 * .. Local Scalars ..
148 LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY
149 CHARACTER CHTEMP
150 INTEGER ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT, ILO,
151 $ IN, IRIGHT, IROWS, ITAU, IWRK, JC, JR, MAXWRK,
152 $ MINWRK
153 DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
154 $ SMLNUM, TEMP
155 * ..
156 * .. Local Arrays ..
157 LOGICAL LDUMMA( 1 )
158 * ..
159 * .. External Subroutines ..
160 EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLABAD,
161 $ DLACPY,DLASCL, DLASET, DORGQR, DORMQR, DTGEVC,
162 $ XERBLA
163 * ..
164 * .. External Functions ..
165 LOGICAL LSAME
166 INTEGER ILAENV
167 DOUBLE PRECISION DLAMCH, DLANGE
168 EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
169 * ..
170 * .. Intrinsic Functions ..
171 INTRINSIC ABS, MAX, SQRT
172 * ..
173 * .. Executable Statements ..
174 *
175 * Decode the input arguments
176 *
177 IF( LSAME( JOBVL, 'N' ) ) THEN
178 IJOBVL = 1
179 ILVL = .FALSE.
180 ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
181 IJOBVL = 2
182 ILVL = .TRUE.
183 ELSE
184 IJOBVL = -1
185 ILVL = .FALSE.
186 END IF
187 *
188 IF( LSAME( JOBVR, 'N' ) ) THEN
189 IJOBVR = 1
190 ILVR = .FALSE.
191 ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
192 IJOBVR = 2
193 ILVR = .TRUE.
194 ELSE
195 IJOBVR = -1
196 ILVR = .FALSE.
197 END IF
198 ILV = ILVL .OR. ILVR
199 *
200 * Test the input arguments
201 *
202 INFO = 0
203 LQUERY = ( LWORK.EQ.-1 )
204 IF( IJOBVL.LE.0 ) THEN
205 INFO = -1
206 ELSE IF( IJOBVR.LE.0 ) THEN
207 INFO = -2
208 ELSE IF( N.LT.0 ) THEN
209 INFO = -3
210 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
211 INFO = -5
212 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
213 INFO = -7
214 ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
215 INFO = -12
216 ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
217 INFO = -14
218 END IF
219 *
220 * Compute workspace
221 * (Note: Comments in the code beginning "Workspace:" describe the
222 * minimal amount of workspace needed at that point in the code,
223 * as well as the preferred amount for good performance.
224 * NB refers to the optimal block size for the immediately
225 * following subroutine, as returned by ILAENV. The workspace is
226 * computed assuming ILO = 1 and IHI = N, the worst case.)
227 *
228 IF( INFO.EQ.0 ) THEN
229 MINWRK = MAX( 1, 8*N )
230 MAXWRK = MAX( 1, N*( 7 +
231 $ ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 ) ) )
232 MAXWRK = MAX( MAXWRK, N*( 7 +
233 $ ILAENV( 1, 'DORMQR', ' ', N, 1, N, 0 ) ) )
234 IF( ILVL ) THEN
235 MAXWRK = MAX( MAXWRK, N*( 7 +
236 $ ILAENV( 1, 'DORGQR', ' ', N, 1, N, -1 ) ) )
237 END IF
238 WORK( 1 ) = MAXWRK
239 *
240 IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY )
241 $ INFO = -16
242 END IF
243 *
244 IF( INFO.NE.0 ) THEN
245 CALL XERBLA( 'DGGEV ', -INFO )
246 RETURN
247 ELSE IF( LQUERY ) THEN
248 RETURN
249 END IF
250 *
251 * Quick return if possible
252 *
253 IF( N.EQ.0 )
254 $ RETURN
255 *
256 * Get machine constants
257 *
258 EPS = DLAMCH( 'P' )
259 SMLNUM = DLAMCH( 'S' )
260 BIGNUM = ONE / SMLNUM
261 CALL DLABAD( SMLNUM, BIGNUM )
262 SMLNUM = SQRT( SMLNUM ) / EPS
263 BIGNUM = ONE / SMLNUM
264 *
265 * Scale A if max element outside range [SMLNUM,BIGNUM]
266 *
267 ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
268 ILASCL = .FALSE.
269 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
270 ANRMTO = SMLNUM
271 ILASCL = .TRUE.
272 ELSE IF( ANRM.GT.BIGNUM ) THEN
273 ANRMTO = BIGNUM
274 ILASCL = .TRUE.
275 END IF
276 IF( ILASCL )
277 $ CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
278 *
279 * Scale B if max element outside range [SMLNUM,BIGNUM]
280 *
281 BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
282 ILBSCL = .FALSE.
283 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
284 BNRMTO = SMLNUM
285 ILBSCL = .TRUE.
286 ELSE IF( BNRM.GT.BIGNUM ) THEN
287 BNRMTO = BIGNUM
288 ILBSCL = .TRUE.
289 END IF
290 IF( ILBSCL )
291 $ CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
292 *
293 * Permute the matrices A, B to isolate eigenvalues if possible
294 * (Workspace: need 6*N)
295 *
296 ILEFT = 1
297 IRIGHT = N + 1
298 IWRK = IRIGHT + N
299 CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
300 $ WORK( IRIGHT ), WORK( IWRK ), IERR )
301 *
302 * Reduce B to triangular form (QR decomposition of B)
303 * (Workspace: need N, prefer N*NB)
304 *
305 IROWS = IHI + 1 - ILO
306 IF( ILV ) THEN
307 ICOLS = N + 1 - ILO
308 ELSE
309 ICOLS = IROWS
310 END IF
311 ITAU = IWRK
312 IWRK = ITAU + IROWS
313 CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
314 $ WORK( IWRK ), LWORK+1-IWRK, IERR )
315 *
316 * Apply the orthogonal transformation to matrix A
317 * (Workspace: need N, prefer N*NB)
318 *
319 CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
320 $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
321 $ LWORK+1-IWRK, IERR )
322 *
323 * Initialize VL
324 * (Workspace: need N, prefer N*NB)
325 *
326 IF( ILVL ) THEN
327 CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
328 IF( IROWS.GT.1 ) THEN
329 CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
330 $ VL( ILO+1, ILO ), LDVL )
331 END IF
332 CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
333 $ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
334 END IF
335 *
336 * Initialize VR
337 *
338 IF( ILVR )
339 $ CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
340 *
341 * Reduce to generalized Hessenberg form
342 * (Workspace: none needed)
343 *
344 IF( ILV ) THEN
345 *
346 * Eigenvectors requested -- work on whole matrix.
347 *
348 CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
349 $ LDVL, VR, LDVR, IERR )
350 ELSE
351 CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
352 $ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR )
353 END IF
354 *
355 * Perform QZ algorithm (Compute eigenvalues, and optionally, the
356 * Schur forms and Schur vectors)
357 * (Workspace: need N)
358 *
359 IWRK = ITAU
360 IF( ILV ) THEN
361 CHTEMP = 'S'
362 ELSE
363 CHTEMP = 'E'
364 END IF
365 CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
366 $ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR,
367 $ WORK( IWRK ), LWORK+1-IWRK, IERR )
368 IF( IERR.NE.0 ) THEN
369 IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
370 INFO = IERR
371 ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
372 INFO = IERR - N
373 ELSE
374 INFO = N + 1
375 END IF
376 GO TO 110
377 END IF
378 *
379 * Compute Eigenvectors
380 * (Workspace: need 6*N)
381 *
382 IF( ILV ) THEN
383 IF( ILVL ) THEN
384 IF( ILVR ) THEN
385 CHTEMP = 'B'
386 ELSE
387 CHTEMP = 'L'
388 END IF
389 ELSE
390 CHTEMP = 'R'
391 END IF
392 CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
393 $ VR, LDVR, N, IN, WORK( IWRK ), IERR )
394 IF( IERR.NE.0 ) THEN
395 INFO = N + 2
396 GO TO 110
397 END IF
398 *
399 * Undo balancing on VL and VR and normalization
400 * (Workspace: none needed)
401 *
402 IF( ILVL ) THEN
403 CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
404 $ WORK( IRIGHT ), N, VL, LDVL, IERR )
405 DO 50 JC = 1, N
406 IF( ALPHAI( JC ).LT.ZERO )
407 $ GO TO 50
408 TEMP = ZERO
409 IF( ALPHAI( JC ).EQ.ZERO ) THEN
410 DO 10 JR = 1, N
411 TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
412 10 CONTINUE
413 ELSE
414 DO 20 JR = 1, N
415 TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
416 $ ABS( VL( JR, JC+1 ) ) )
417 20 CONTINUE
418 END IF
419 IF( TEMP.LT.SMLNUM )
420 $ GO TO 50
421 TEMP = ONE / TEMP
422 IF( ALPHAI( JC ).EQ.ZERO ) THEN
423 DO 30 JR = 1, N
424 VL( JR, JC ) = VL( JR, JC )*TEMP
425 30 CONTINUE
426 ELSE
427 DO 40 JR = 1, N
428 VL( JR, JC ) = VL( JR, JC )*TEMP
429 VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
430 40 CONTINUE
431 END IF
432 50 CONTINUE
433 END IF
434 IF( ILVR ) THEN
435 CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
436 $ WORK( IRIGHT ), N, VR, LDVR, IERR )
437 DO 100 JC = 1, N
438 IF( ALPHAI( JC ).LT.ZERO )
439 $ GO TO 100
440 TEMP = ZERO
441 IF( ALPHAI( JC ).EQ.ZERO ) THEN
442 DO 60 JR = 1, N
443 TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
444 60 CONTINUE
445 ELSE
446 DO 70 JR = 1, N
447 TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
448 $ ABS( VR( JR, JC+1 ) ) )
449 70 CONTINUE
450 END IF
451 IF( TEMP.LT.SMLNUM )
452 $ GO TO 100
453 TEMP = ONE / TEMP
454 IF( ALPHAI( JC ).EQ.ZERO ) THEN
455 DO 80 JR = 1, N
456 VR( JR, JC ) = VR( JR, JC )*TEMP
457 80 CONTINUE
458 ELSE
459 DO 90 JR = 1, N
460 VR( JR, JC ) = VR( JR, JC )*TEMP
461 VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
462 90 CONTINUE
463 END IF
464 100 CONTINUE
465 END IF
466 *
467 * End of eigenvector calculation
468 *
469 END IF
470 *
471 * Undo scaling if necessary
472 *
473 IF( ILASCL ) THEN
474 CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR )
475 CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR )
476 END IF
477 *
478 IF( ILBSCL ) THEN
479 CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
480 END IF
481 *
482 110 CONTINUE
483 *
484 WORK( 1 ) = MAXWRK
485 *
486 RETURN
487 *
488 * End of DGGEV
489 *
490 END
2 $ BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
3 *
4 * -- LAPACK driver routine (version 3.2) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * November 2006
8 *
9 * .. Scalar Arguments ..
10 CHARACTER JOBVL, JOBVR
11 INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
12 * ..
13 * .. Array Arguments ..
14 DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
15 $ B( LDB, * ), BETA( * ), VL( LDVL, * ),
16 $ VR( LDVR, * ), WORK( * )
17 * ..
18 *
19 * Purpose
20 * =======
21 *
22 * DGGEV computes for a pair of N-by-N real nonsymmetric matrices (A,B)
23 * the generalized eigenvalues, and optionally, the left and/or right
24 * generalized eigenvectors.
25 *
26 * A generalized eigenvalue for a pair of matrices (A,B) is a scalar
27 * lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
28 * singular. It is usually represented as the pair (alpha,beta), as
29 * there is a reasonable interpretation for beta=0, and even for both
30 * being zero.
31 *
32 * The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
33 * of (A,B) satisfies
34 *
35 * A * v(j) = lambda(j) * B * v(j).
36 *
37 * The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
38 * of (A,B) satisfies
39 *
40 * u(j)**H * A = lambda(j) * u(j)**H * B .
41 *
42 * where u(j)**H is the conjugate-transpose of u(j).
43 *
44 *
45 * Arguments
46 * =========
47 *
48 * JOBVL (input) CHARACTER*1
49 * = 'N': do not compute the left generalized eigenvectors;
50 * = 'V': compute the left generalized eigenvectors.
51 *
52 * JOBVR (input) CHARACTER*1
53 * = 'N': do not compute the right generalized eigenvectors;
54 * = 'V': compute the right generalized eigenvectors.
55 *
56 * N (input) INTEGER
57 * The order of the matrices A, B, VL, and VR. N >= 0.
58 *
59 * A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
60 * On entry, the matrix A in the pair (A,B).
61 * On exit, A has been overwritten.
62 *
63 * LDA (input) INTEGER
64 * The leading dimension of A. LDA >= max(1,N).
65 *
66 * B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
67 * On entry, the matrix B in the pair (A,B).
68 * On exit, B has been overwritten.
69 *
70 * LDB (input) INTEGER
71 * The leading dimension of B. LDB >= max(1,N).
72 *
73 * ALPHAR (output) DOUBLE PRECISION array, dimension (N)
74 * ALPHAI (output) DOUBLE PRECISION array, dimension (N)
75 * BETA (output) DOUBLE PRECISION array, dimension (N)
76 * On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
77 * be the generalized eigenvalues. If ALPHAI(j) is zero, then
78 * the j-th eigenvalue is real; if positive, then the j-th and
79 * (j+1)-st eigenvalues are a complex conjugate pair, with
80 * ALPHAI(j+1) negative.
81 *
82 * Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
83 * may easily over- or underflow, and BETA(j) may even be zero.
84 * Thus, the user should avoid naively computing the ratio
85 * alpha/beta. However, ALPHAR and ALPHAI will be always less
86 * than and usually comparable with norm(A) in magnitude, and
87 * BETA always less than and usually comparable with norm(B).
88 *
89 * VL (output) DOUBLE PRECISION array, dimension (LDVL,N)
90 * If JOBVL = 'V', the left eigenvectors u(j) are stored one
91 * after another in the columns of VL, in the same order as
92 * their eigenvalues. If the j-th eigenvalue is real, then
93 * u(j) = VL(:,j), the j-th column of VL. If the j-th and
94 * (j+1)-th eigenvalues form a complex conjugate pair, then
95 * u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
96 * Each eigenvector is scaled so the largest component has
97 * abs(real part)+abs(imag. part)=1.
98 * Not referenced if JOBVL = 'N'.
99 *
100 * LDVL (input) INTEGER
101 * The leading dimension of the matrix VL. LDVL >= 1, and
102 * if JOBVL = 'V', LDVL >= N.
103 *
104 * VR (output) DOUBLE PRECISION array, dimension (LDVR,N)
105 * If JOBVR = 'V', the right eigenvectors v(j) are stored one
106 * after another in the columns of VR, in the same order as
107 * their eigenvalues. If the j-th eigenvalue is real, then
108 * v(j) = VR(:,j), the j-th column of VR. If the j-th and
109 * (j+1)-th eigenvalues form a complex conjugate pair, then
110 * v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
111 * Each eigenvector is scaled so the largest component has
112 * abs(real part)+abs(imag. part)=1.
113 * Not referenced if JOBVR = 'N'.
114 *
115 * LDVR (input) INTEGER
116 * The leading dimension of the matrix VR. LDVR >= 1, and
117 * if JOBVR = 'V', LDVR >= N.
118 *
119 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
120 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
121 *
122 * LWORK (input) INTEGER
123 * The dimension of the array WORK. LWORK >= max(1,8*N).
124 * For good performance, LWORK must generally be larger.
125 *
126 * If LWORK = -1, then a workspace query is assumed; the routine
127 * only calculates the optimal size of the WORK array, returns
128 * this value as the first entry of the WORK array, and no error
129 * message related to LWORK is issued by XERBLA.
130 *
131 * INFO (output) INTEGER
132 * = 0: successful exit
133 * < 0: if INFO = -i, the i-th argument had an illegal value.
134 * = 1,...,N:
135 * The QZ iteration failed. No eigenvectors have been
136 * calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
137 * should be correct for j=INFO+1,...,N.
138 * > N: =N+1: other than QZ iteration failed in DHGEQZ.
139 * =N+2: error return from DTGEVC.
140 *
141 * =====================================================================
142 *
143 * .. Parameters ..
144 DOUBLE PRECISION ZERO, ONE
145 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
146 * ..
147 * .. Local Scalars ..
148 LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY
149 CHARACTER CHTEMP
150 INTEGER ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT, ILO,
151 $ IN, IRIGHT, IROWS, ITAU, IWRK, JC, JR, MAXWRK,
152 $ MINWRK
153 DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
154 $ SMLNUM, TEMP
155 * ..
156 * .. Local Arrays ..
157 LOGICAL LDUMMA( 1 )
158 * ..
159 * .. External Subroutines ..
160 EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLABAD,
161 $ DLACPY,DLASCL, DLASET, DORGQR, DORMQR, DTGEVC,
162 $ XERBLA
163 * ..
164 * .. External Functions ..
165 LOGICAL LSAME
166 INTEGER ILAENV
167 DOUBLE PRECISION DLAMCH, DLANGE
168 EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
169 * ..
170 * .. Intrinsic Functions ..
171 INTRINSIC ABS, MAX, SQRT
172 * ..
173 * .. Executable Statements ..
174 *
175 * Decode the input arguments
176 *
177 IF( LSAME( JOBVL, 'N' ) ) THEN
178 IJOBVL = 1
179 ILVL = .FALSE.
180 ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
181 IJOBVL = 2
182 ILVL = .TRUE.
183 ELSE
184 IJOBVL = -1
185 ILVL = .FALSE.
186 END IF
187 *
188 IF( LSAME( JOBVR, 'N' ) ) THEN
189 IJOBVR = 1
190 ILVR = .FALSE.
191 ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
192 IJOBVR = 2
193 ILVR = .TRUE.
194 ELSE
195 IJOBVR = -1
196 ILVR = .FALSE.
197 END IF
198 ILV = ILVL .OR. ILVR
199 *
200 * Test the input arguments
201 *
202 INFO = 0
203 LQUERY = ( LWORK.EQ.-1 )
204 IF( IJOBVL.LE.0 ) THEN
205 INFO = -1
206 ELSE IF( IJOBVR.LE.0 ) THEN
207 INFO = -2
208 ELSE IF( N.LT.0 ) THEN
209 INFO = -3
210 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
211 INFO = -5
212 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
213 INFO = -7
214 ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
215 INFO = -12
216 ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
217 INFO = -14
218 END IF
219 *
220 * Compute workspace
221 * (Note: Comments in the code beginning "Workspace:" describe the
222 * minimal amount of workspace needed at that point in the code,
223 * as well as the preferred amount for good performance.
224 * NB refers to the optimal block size for the immediately
225 * following subroutine, as returned by ILAENV. The workspace is
226 * computed assuming ILO = 1 and IHI = N, the worst case.)
227 *
228 IF( INFO.EQ.0 ) THEN
229 MINWRK = MAX( 1, 8*N )
230 MAXWRK = MAX( 1, N*( 7 +
231 $ ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 ) ) )
232 MAXWRK = MAX( MAXWRK, N*( 7 +
233 $ ILAENV( 1, 'DORMQR', ' ', N, 1, N, 0 ) ) )
234 IF( ILVL ) THEN
235 MAXWRK = MAX( MAXWRK, N*( 7 +
236 $ ILAENV( 1, 'DORGQR', ' ', N, 1, N, -1 ) ) )
237 END IF
238 WORK( 1 ) = MAXWRK
239 *
240 IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY )
241 $ INFO = -16
242 END IF
243 *
244 IF( INFO.NE.0 ) THEN
245 CALL XERBLA( 'DGGEV ', -INFO )
246 RETURN
247 ELSE IF( LQUERY ) THEN
248 RETURN
249 END IF
250 *
251 * Quick return if possible
252 *
253 IF( N.EQ.0 )
254 $ RETURN
255 *
256 * Get machine constants
257 *
258 EPS = DLAMCH( 'P' )
259 SMLNUM = DLAMCH( 'S' )
260 BIGNUM = ONE / SMLNUM
261 CALL DLABAD( SMLNUM, BIGNUM )
262 SMLNUM = SQRT( SMLNUM ) / EPS
263 BIGNUM = ONE / SMLNUM
264 *
265 * Scale A if max element outside range [SMLNUM,BIGNUM]
266 *
267 ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
268 ILASCL = .FALSE.
269 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
270 ANRMTO = SMLNUM
271 ILASCL = .TRUE.
272 ELSE IF( ANRM.GT.BIGNUM ) THEN
273 ANRMTO = BIGNUM
274 ILASCL = .TRUE.
275 END IF
276 IF( ILASCL )
277 $ CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
278 *
279 * Scale B if max element outside range [SMLNUM,BIGNUM]
280 *
281 BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
282 ILBSCL = .FALSE.
283 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
284 BNRMTO = SMLNUM
285 ILBSCL = .TRUE.
286 ELSE IF( BNRM.GT.BIGNUM ) THEN
287 BNRMTO = BIGNUM
288 ILBSCL = .TRUE.
289 END IF
290 IF( ILBSCL )
291 $ CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
292 *
293 * Permute the matrices A, B to isolate eigenvalues if possible
294 * (Workspace: need 6*N)
295 *
296 ILEFT = 1
297 IRIGHT = N + 1
298 IWRK = IRIGHT + N
299 CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
300 $ WORK( IRIGHT ), WORK( IWRK ), IERR )
301 *
302 * Reduce B to triangular form (QR decomposition of B)
303 * (Workspace: need N, prefer N*NB)
304 *
305 IROWS = IHI + 1 - ILO
306 IF( ILV ) THEN
307 ICOLS = N + 1 - ILO
308 ELSE
309 ICOLS = IROWS
310 END IF
311 ITAU = IWRK
312 IWRK = ITAU + IROWS
313 CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
314 $ WORK( IWRK ), LWORK+1-IWRK, IERR )
315 *
316 * Apply the orthogonal transformation to matrix A
317 * (Workspace: need N, prefer N*NB)
318 *
319 CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
320 $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
321 $ LWORK+1-IWRK, IERR )
322 *
323 * Initialize VL
324 * (Workspace: need N, prefer N*NB)
325 *
326 IF( ILVL ) THEN
327 CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
328 IF( IROWS.GT.1 ) THEN
329 CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
330 $ VL( ILO+1, ILO ), LDVL )
331 END IF
332 CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
333 $ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
334 END IF
335 *
336 * Initialize VR
337 *
338 IF( ILVR )
339 $ CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
340 *
341 * Reduce to generalized Hessenberg form
342 * (Workspace: none needed)
343 *
344 IF( ILV ) THEN
345 *
346 * Eigenvectors requested -- work on whole matrix.
347 *
348 CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
349 $ LDVL, VR, LDVR, IERR )
350 ELSE
351 CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
352 $ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR )
353 END IF
354 *
355 * Perform QZ algorithm (Compute eigenvalues, and optionally, the
356 * Schur forms and Schur vectors)
357 * (Workspace: need N)
358 *
359 IWRK = ITAU
360 IF( ILV ) THEN
361 CHTEMP = 'S'
362 ELSE
363 CHTEMP = 'E'
364 END IF
365 CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
366 $ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR,
367 $ WORK( IWRK ), LWORK+1-IWRK, IERR )
368 IF( IERR.NE.0 ) THEN
369 IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
370 INFO = IERR
371 ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
372 INFO = IERR - N
373 ELSE
374 INFO = N + 1
375 END IF
376 GO TO 110
377 END IF
378 *
379 * Compute Eigenvectors
380 * (Workspace: need 6*N)
381 *
382 IF( ILV ) THEN
383 IF( ILVL ) THEN
384 IF( ILVR ) THEN
385 CHTEMP = 'B'
386 ELSE
387 CHTEMP = 'L'
388 END IF
389 ELSE
390 CHTEMP = 'R'
391 END IF
392 CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
393 $ VR, LDVR, N, IN, WORK( IWRK ), IERR )
394 IF( IERR.NE.0 ) THEN
395 INFO = N + 2
396 GO TO 110
397 END IF
398 *
399 * Undo balancing on VL and VR and normalization
400 * (Workspace: none needed)
401 *
402 IF( ILVL ) THEN
403 CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
404 $ WORK( IRIGHT ), N, VL, LDVL, IERR )
405 DO 50 JC = 1, N
406 IF( ALPHAI( JC ).LT.ZERO )
407 $ GO TO 50
408 TEMP = ZERO
409 IF( ALPHAI( JC ).EQ.ZERO ) THEN
410 DO 10 JR = 1, N
411 TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
412 10 CONTINUE
413 ELSE
414 DO 20 JR = 1, N
415 TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
416 $ ABS( VL( JR, JC+1 ) ) )
417 20 CONTINUE
418 END IF
419 IF( TEMP.LT.SMLNUM )
420 $ GO TO 50
421 TEMP = ONE / TEMP
422 IF( ALPHAI( JC ).EQ.ZERO ) THEN
423 DO 30 JR = 1, N
424 VL( JR, JC ) = VL( JR, JC )*TEMP
425 30 CONTINUE
426 ELSE
427 DO 40 JR = 1, N
428 VL( JR, JC ) = VL( JR, JC )*TEMP
429 VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
430 40 CONTINUE
431 END IF
432 50 CONTINUE
433 END IF
434 IF( ILVR ) THEN
435 CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
436 $ WORK( IRIGHT ), N, VR, LDVR, IERR )
437 DO 100 JC = 1, N
438 IF( ALPHAI( JC ).LT.ZERO )
439 $ GO TO 100
440 TEMP = ZERO
441 IF( ALPHAI( JC ).EQ.ZERO ) THEN
442 DO 60 JR = 1, N
443 TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
444 60 CONTINUE
445 ELSE
446 DO 70 JR = 1, N
447 TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
448 $ ABS( VR( JR, JC+1 ) ) )
449 70 CONTINUE
450 END IF
451 IF( TEMP.LT.SMLNUM )
452 $ GO TO 100
453 TEMP = ONE / TEMP
454 IF( ALPHAI( JC ).EQ.ZERO ) THEN
455 DO 80 JR = 1, N
456 VR( JR, JC ) = VR( JR, JC )*TEMP
457 80 CONTINUE
458 ELSE
459 DO 90 JR = 1, N
460 VR( JR, JC ) = VR( JR, JC )*TEMP
461 VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
462 90 CONTINUE
463 END IF
464 100 CONTINUE
465 END IF
466 *
467 * End of eigenvector calculation
468 *
469 END IF
470 *
471 * Undo scaling if necessary
472 *
473 IF( ILASCL ) THEN
474 CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR )
475 CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR )
476 END IF
477 *
478 IF( ILBSCL ) THEN
479 CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
480 END IF
481 *
482 110 CONTINUE
483 *
484 WORK( 1 ) = MAXWRK
485 *
486 RETURN
487 *
488 * End of DGGEV
489 *
490 END