Annotation of rpl/lapack/lapack/dggev.f, revision 1.3
1.1 bertrand 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
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