1: *> \brief <b> ZGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b>
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZGGEV + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zggev.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zggev.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zggev.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA,
22: * VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER JOBVL, JOBVR
26: * INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION RWORK( * )
30: * COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
31: * $ BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
32: * $ WORK( * )
33: * ..
34: *
35: *
36: *> \par Purpose:
37: * =============
38: *>
39: *> \verbatim
40: *>
41: *> ZGGEV computes for a pair of N-by-N complex nonsymmetric matrices
42: *> (A,B), the generalized eigenvalues, and optionally, the left and/or
43: *> right generalized eigenvectors.
44: *>
45: *> A generalized eigenvalue for a pair of matrices (A,B) is a scalar
46: *> lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
47: *> singular. It is usually represented as the pair (alpha,beta), as
48: *> there is a reasonable interpretation for beta=0, and even for both
49: *> being zero.
50: *>
51: *> The right generalized eigenvector v(j) corresponding to the
52: *> generalized eigenvalue lambda(j) of (A,B) satisfies
53: *>
54: *> A * v(j) = lambda(j) * B * v(j).
55: *>
56: *> The left generalized eigenvector u(j) corresponding to the
57: *> generalized eigenvalues lambda(j) of (A,B) satisfies
58: *>
59: *> u(j)**H * A = lambda(j) * u(j)**H * B
60: *>
61: *> where u(j)**H is the conjugate-transpose of u(j).
62: *> \endverbatim
63: *
64: * Arguments:
65: * ==========
66: *
67: *> \param[in] JOBVL
68: *> \verbatim
69: *> JOBVL is CHARACTER*1
70: *> = 'N': do not compute the left generalized eigenvectors;
71: *> = 'V': compute the left generalized eigenvectors.
72: *> \endverbatim
73: *>
74: *> \param[in] JOBVR
75: *> \verbatim
76: *> JOBVR is CHARACTER*1
77: *> = 'N': do not compute the right generalized eigenvectors;
78: *> = 'V': compute the right generalized eigenvectors.
79: *> \endverbatim
80: *>
81: *> \param[in] N
82: *> \verbatim
83: *> N is INTEGER
84: *> The order of the matrices A, B, VL, and VR. N >= 0.
85: *> \endverbatim
86: *>
87: *> \param[in,out] A
88: *> \verbatim
89: *> A is COMPLEX*16 array, dimension (LDA, N)
90: *> On entry, the matrix A in the pair (A,B).
91: *> On exit, A has been overwritten.
92: *> \endverbatim
93: *>
94: *> \param[in] LDA
95: *> \verbatim
96: *> LDA is INTEGER
97: *> The leading dimension of A. LDA >= max(1,N).
98: *> \endverbatim
99: *>
100: *> \param[in,out] B
101: *> \verbatim
102: *> B is COMPLEX*16 array, dimension (LDB, N)
103: *> On entry, the matrix B in the pair (A,B).
104: *> On exit, B has been overwritten.
105: *> \endverbatim
106: *>
107: *> \param[in] LDB
108: *> \verbatim
109: *> LDB is INTEGER
110: *> The leading dimension of B. LDB >= max(1,N).
111: *> \endverbatim
112: *>
113: *> \param[out] ALPHA
114: *> \verbatim
115: *> ALPHA is COMPLEX*16 array, dimension (N)
116: *> \endverbatim
117: *>
118: *> \param[out] BETA
119: *> \verbatim
120: *> BETA is COMPLEX*16 array, dimension (N)
121: *> On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
122: *> generalized eigenvalues.
123: *>
124: *> Note: the quotients ALPHA(j)/BETA(j) may easily over- or
125: *> underflow, and BETA(j) may even be zero. Thus, the user
126: *> should avoid naively computing the ratio alpha/beta.
127: *> However, ALPHA will be always less than and usually
128: *> comparable with norm(A) in magnitude, and BETA always less
129: *> than and usually comparable with norm(B).
130: *> \endverbatim
131: *>
132: *> \param[out] VL
133: *> \verbatim
134: *> VL is COMPLEX*16 array, dimension (LDVL,N)
135: *> If JOBVL = 'V', the left generalized eigenvectors u(j) are
136: *> stored one after another in the columns of VL, in the same
137: *> order as their eigenvalues.
138: *> Each eigenvector is scaled so the largest component has
139: *> abs(real part) + abs(imag. part) = 1.
140: *> Not referenced if JOBVL = 'N'.
141: *> \endverbatim
142: *>
143: *> \param[in] LDVL
144: *> \verbatim
145: *> LDVL is INTEGER
146: *> The leading dimension of the matrix VL. LDVL >= 1, and
147: *> if JOBVL = 'V', LDVL >= N.
148: *> \endverbatim
149: *>
150: *> \param[out] VR
151: *> \verbatim
152: *> VR is COMPLEX*16 array, dimension (LDVR,N)
153: *> If JOBVR = 'V', the right generalized eigenvectors v(j) are
154: *> stored one after another in the columns of VR, in the same
155: *> order as their eigenvalues.
156: *> Each eigenvector is scaled so the largest component has
157: *> abs(real part) + abs(imag. part) = 1.
158: *> Not referenced if JOBVR = 'N'.
159: *> \endverbatim
160: *>
161: *> \param[in] LDVR
162: *> \verbatim
163: *> LDVR is INTEGER
164: *> The leading dimension of the matrix VR. LDVR >= 1, and
165: *> if JOBVR = 'V', LDVR >= N.
166: *> \endverbatim
167: *>
168: *> \param[out] WORK
169: *> \verbatim
170: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
171: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
172: *> \endverbatim
173: *>
174: *> \param[in] LWORK
175: *> \verbatim
176: *> LWORK is INTEGER
177: *> The dimension of the array WORK. LWORK >= max(1,2*N).
178: *> For good performance, LWORK must generally be larger.
179: *>
180: *> If LWORK = -1, then a workspace query is assumed; the routine
181: *> only calculates the optimal size of the WORK array, returns
182: *> this value as the first entry of the WORK array, and no error
183: *> message related to LWORK is issued by XERBLA.
184: *> \endverbatim
185: *>
186: *> \param[out] RWORK
187: *> \verbatim
188: *> RWORK is DOUBLE PRECISION array, dimension (8*N)
189: *> \endverbatim
190: *>
191: *> \param[out] INFO
192: *> \verbatim
193: *> INFO is INTEGER
194: *> = 0: successful exit
195: *> < 0: if INFO = -i, the i-th argument had an illegal value.
196: *> =1,...,N:
197: *> The QZ iteration failed. No eigenvectors have been
198: *> calculated, but ALPHA(j) and BETA(j) should be
199: *> correct for j=INFO+1,...,N.
200: *> > N: =N+1: other then QZ iteration failed in ZHGEQZ,
201: *> =N+2: error return from ZTGEVC.
202: *> \endverbatim
203: *
204: * Authors:
205: * ========
206: *
207: *> \author Univ. of Tennessee
208: *> \author Univ. of California Berkeley
209: *> \author Univ. of Colorado Denver
210: *> \author NAG Ltd.
211: *
212: *> \ingroup complex16GEeigen
213: *
214: * =====================================================================
215: SUBROUTINE ZGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA,
216: $ VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
217: *
218: * -- LAPACK driver routine --
219: * -- LAPACK is a software package provided by Univ. of Tennessee, --
220: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
221: *
222: * .. Scalar Arguments ..
223: CHARACTER JOBVL, JOBVR
224: INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
225: * ..
226: * .. Array Arguments ..
227: DOUBLE PRECISION RWORK( * )
228: COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
229: $ BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
230: $ WORK( * )
231: * ..
232: *
233: * =====================================================================
234: *
235: * .. Parameters ..
236: DOUBLE PRECISION ZERO, ONE
237: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
238: COMPLEX*16 CZERO, CONE
239: PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ),
240: $ CONE = ( 1.0D0, 0.0D0 ) )
241: * ..
242: * .. Local Scalars ..
243: LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY
244: CHARACTER CHTEMP
245: INTEGER ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT, ILO,
246: $ IN, IRIGHT, IROWS, IRWRK, ITAU, IWRK, JC, JR,
247: $ LWKMIN, LWKOPT
248: DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
249: $ SMLNUM, TEMP
250: COMPLEX*16 X
251: * ..
252: * .. Local Arrays ..
253: LOGICAL LDUMMA( 1 )
254: * ..
255: * .. External Subroutines ..
256: EXTERNAL DLABAD, XERBLA, ZGEQRF, ZGGBAK, ZGGBAL, ZGGHRD,
257: $ ZHGEQZ, ZLACPY, ZLASCL, ZLASET, ZTGEVC, ZUNGQR,
258: $ ZUNMQR
259: * ..
260: * .. External Functions ..
261: LOGICAL LSAME
262: INTEGER ILAENV
263: DOUBLE PRECISION DLAMCH, ZLANGE
264: EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE
265: * ..
266: * .. Intrinsic Functions ..
267: INTRINSIC ABS, DBLE, DIMAG, MAX, SQRT
268: * ..
269: * .. Statement Functions ..
270: DOUBLE PRECISION ABS1
271: * ..
272: * .. Statement Function definitions ..
273: ABS1( X ) = ABS( DBLE( X ) ) + ABS( DIMAG( X ) )
274: * ..
275: * .. Executable Statements ..
276: *
277: * Decode the input arguments
278: *
279: IF( LSAME( JOBVL, 'N' ) ) THEN
280: IJOBVL = 1
281: ILVL = .FALSE.
282: ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
283: IJOBVL = 2
284: ILVL = .TRUE.
285: ELSE
286: IJOBVL = -1
287: ILVL = .FALSE.
288: END IF
289: *
290: IF( LSAME( JOBVR, 'N' ) ) THEN
291: IJOBVR = 1
292: ILVR = .FALSE.
293: ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
294: IJOBVR = 2
295: ILVR = .TRUE.
296: ELSE
297: IJOBVR = -1
298: ILVR = .FALSE.
299: END IF
300: ILV = ILVL .OR. ILVR
301: *
302: * Test the input arguments
303: *
304: INFO = 0
305: LQUERY = ( LWORK.EQ.-1 )
306: IF( IJOBVL.LE.0 ) THEN
307: INFO = -1
308: ELSE IF( IJOBVR.LE.0 ) THEN
309: INFO = -2
310: ELSE IF( N.LT.0 ) THEN
311: INFO = -3
312: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
313: INFO = -5
314: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
315: INFO = -7
316: ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
317: INFO = -11
318: ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
319: INFO = -13
320: END IF
321: *
322: * Compute workspace
323: * (Note: Comments in the code beginning "Workspace:" describe the
324: * minimal amount of workspace needed at that point in the code,
325: * as well as the preferred amount for good performance.
326: * NB refers to the optimal block size for the immediately
327: * following subroutine, as returned by ILAENV. The workspace is
328: * computed assuming ILO = 1 and IHI = N, the worst case.)
329: *
330: IF( INFO.EQ.0 ) THEN
331: LWKMIN = MAX( 1, 2*N )
332: LWKOPT = MAX( 1, N + N*ILAENV( 1, 'ZGEQRF', ' ', N, 1, N, 0 ) )
333: LWKOPT = MAX( LWKOPT, N +
334: $ N*ILAENV( 1, 'ZUNMQR', ' ', N, 1, N, 0 ) )
335: IF( ILVL ) THEN
336: LWKOPT = MAX( LWKOPT, N +
337: $ N*ILAENV( 1, 'ZUNGQR', ' ', N, 1, N, -1 ) )
338: END IF
339: WORK( 1 ) = LWKOPT
340: *
341: IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY )
342: $ INFO = -15
343: END IF
344: *
345: IF( INFO.NE.0 ) THEN
346: CALL XERBLA( 'ZGGEV ', -INFO )
347: RETURN
348: ELSE IF( LQUERY ) THEN
349: RETURN
350: END IF
351: *
352: * Quick return if possible
353: *
354: IF( N.EQ.0 )
355: $ RETURN
356: *
357: * Get machine constants
358: *
359: EPS = DLAMCH( 'E' )*DLAMCH( 'B' )
360: SMLNUM = DLAMCH( 'S' )
361: BIGNUM = ONE / SMLNUM
362: CALL DLABAD( SMLNUM, BIGNUM )
363: SMLNUM = SQRT( SMLNUM ) / EPS
364: BIGNUM = ONE / SMLNUM
365: *
366: * Scale A if max element outside range [SMLNUM,BIGNUM]
367: *
368: ANRM = ZLANGE( 'M', N, N, A, LDA, RWORK )
369: ILASCL = .FALSE.
370: IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
371: ANRMTO = SMLNUM
372: ILASCL = .TRUE.
373: ELSE IF( ANRM.GT.BIGNUM ) THEN
374: ANRMTO = BIGNUM
375: ILASCL = .TRUE.
376: END IF
377: IF( ILASCL )
378: $ CALL ZLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
379: *
380: * Scale B if max element outside range [SMLNUM,BIGNUM]
381: *
382: BNRM = ZLANGE( 'M', N, N, B, LDB, RWORK )
383: ILBSCL = .FALSE.
384: IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
385: BNRMTO = SMLNUM
386: ILBSCL = .TRUE.
387: ELSE IF( BNRM.GT.BIGNUM ) THEN
388: BNRMTO = BIGNUM
389: ILBSCL = .TRUE.
390: END IF
391: IF( ILBSCL )
392: $ CALL ZLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
393: *
394: * Permute the matrices A, B to isolate eigenvalues if possible
395: * (Real Workspace: need 6*N)
396: *
397: ILEFT = 1
398: IRIGHT = N + 1
399: IRWRK = IRIGHT + N
400: CALL ZGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, RWORK( ILEFT ),
401: $ RWORK( IRIGHT ), RWORK( IRWRK ), IERR )
402: *
403: * Reduce B to triangular form (QR decomposition of B)
404: * (Complex Workspace: need N, prefer N*NB)
405: *
406: IROWS = IHI + 1 - ILO
407: IF( ILV ) THEN
408: ICOLS = N + 1 - ILO
409: ELSE
410: ICOLS = IROWS
411: END IF
412: ITAU = 1
413: IWRK = ITAU + IROWS
414: CALL ZGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
415: $ WORK( IWRK ), LWORK+1-IWRK, IERR )
416: *
417: * Apply the orthogonal transformation to matrix A
418: * (Complex Workspace: need N, prefer N*NB)
419: *
420: CALL ZUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
421: $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
422: $ LWORK+1-IWRK, IERR )
423: *
424: * Initialize VL
425: * (Complex Workspace: need N, prefer N*NB)
426: *
427: IF( ILVL ) THEN
428: CALL ZLASET( 'Full', N, N, CZERO, CONE, VL, LDVL )
429: IF( IROWS.GT.1 ) THEN
430: CALL ZLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
431: $ VL( ILO+1, ILO ), LDVL )
432: END IF
433: CALL ZUNGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
434: $ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
435: END IF
436: *
437: * Initialize VR
438: *
439: IF( ILVR )
440: $ CALL ZLASET( 'Full', N, N, CZERO, CONE, VR, LDVR )
441: *
442: * Reduce to generalized Hessenberg form
443: *
444: IF( ILV ) THEN
445: *
446: * Eigenvectors requested -- work on whole matrix.
447: *
448: CALL ZGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
449: $ LDVL, VR, LDVR, IERR )
450: ELSE
451: CALL ZGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
452: $ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR )
453: END IF
454: *
455: * Perform QZ algorithm (Compute eigenvalues, and optionally, the
456: * Schur form and Schur vectors)
457: * (Complex Workspace: need N)
458: * (Real Workspace: need N)
459: *
460: IWRK = ITAU
461: IF( ILV ) THEN
462: CHTEMP = 'S'
463: ELSE
464: CHTEMP = 'E'
465: END IF
466: CALL ZHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
467: $ ALPHA, BETA, VL, LDVL, VR, LDVR, WORK( IWRK ),
468: $ LWORK+1-IWRK, RWORK( IRWRK ), IERR )
469: IF( IERR.NE.0 ) THEN
470: IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
471: INFO = IERR
472: ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
473: INFO = IERR - N
474: ELSE
475: INFO = N + 1
476: END IF
477: GO TO 70
478: END IF
479: *
480: * Compute Eigenvectors
481: * (Real Workspace: need 2*N)
482: * (Complex Workspace: need 2*N)
483: *
484: IF( ILV ) THEN
485: IF( ILVL ) THEN
486: IF( ILVR ) THEN
487: CHTEMP = 'B'
488: ELSE
489: CHTEMP = 'L'
490: END IF
491: ELSE
492: CHTEMP = 'R'
493: END IF
494: *
495: CALL ZTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
496: $ VR, LDVR, N, IN, WORK( IWRK ), RWORK( IRWRK ),
497: $ IERR )
498: IF( IERR.NE.0 ) THEN
499: INFO = N + 2
500: GO TO 70
501: END IF
502: *
503: * Undo balancing on VL and VR and normalization
504: * (Workspace: none needed)
505: *
506: IF( ILVL ) THEN
507: CALL ZGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ),
508: $ RWORK( IRIGHT ), N, VL, LDVL, IERR )
509: DO 30 JC = 1, N
510: TEMP = ZERO
511: DO 10 JR = 1, N
512: TEMP = MAX( TEMP, ABS1( VL( JR, JC ) ) )
513: 10 CONTINUE
514: IF( TEMP.LT.SMLNUM )
515: $ GO TO 30
516: TEMP = ONE / TEMP
517: DO 20 JR = 1, N
518: VL( JR, JC ) = VL( JR, JC )*TEMP
519: 20 CONTINUE
520: 30 CONTINUE
521: END IF
522: IF( ILVR ) THEN
523: CALL ZGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ),
524: $ RWORK( IRIGHT ), N, VR, LDVR, IERR )
525: DO 60 JC = 1, N
526: TEMP = ZERO
527: DO 40 JR = 1, N
528: TEMP = MAX( TEMP, ABS1( VR( JR, JC ) ) )
529: 40 CONTINUE
530: IF( TEMP.LT.SMLNUM )
531: $ GO TO 60
532: TEMP = ONE / TEMP
533: DO 50 JR = 1, N
534: VR( JR, JC ) = VR( JR, JC )*TEMP
535: 50 CONTINUE
536: 60 CONTINUE
537: END IF
538: END IF
539: *
540: * Undo scaling if necessary
541: *
542: 70 CONTINUE
543: *
544: IF( ILASCL )
545: $ CALL ZLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHA, N, IERR )
546: *
547: IF( ILBSCL )
548: $ CALL ZLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
549: *
550: WORK( 1 ) = LWKOPT
551: RETURN
552: *
553: * End of ZGGEV
554: *
555: END
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