1: SUBROUTINE DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,
2: $ ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
3: $ LWORK, INFO )
4: *
5: * -- LAPACK driver routine (version 3.2) --
6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8: * November 2006
9: *
10: * .. Scalar Arguments ..
11: CHARACTER JOBVSL, JOBVSR
12: INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
13: * ..
14: * .. Array Arguments ..
15: DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
16: $ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
17: $ VSR( LDVSR, * ), WORK( * )
18: * ..
19: *
20: * Purpose
21: * =======
22: *
23: * This routine is deprecated and has been replaced by routine DGGES.
24: *
25: * DGEGS computes the eigenvalues, real Schur form, and, optionally,
26: * left and or/right Schur vectors of a real matrix pair (A,B).
27: * Given two square matrices A and B, the generalized real Schur
28: * factorization has the form
29: *
30: * A = Q*S*Z**T, B = Q*T*Z**T
31: *
32: * where Q and Z are orthogonal matrices, T is upper triangular, and S
33: * is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal
34: * blocks, the 2-by-2 blocks corresponding to complex conjugate pairs
35: * of eigenvalues of (A,B). The columns of Q are the left Schur vectors
36: * and the columns of Z are the right Schur vectors.
37: *
38: * If only the eigenvalues of (A,B) are needed, the driver routine
39: * DGEGV should be used instead. See DGEGV for a description of the
40: * eigenvalues of the generalized nonsymmetric eigenvalue problem
41: * (GNEP).
42: *
43: * Arguments
44: * =========
45: *
46: * JOBVSL (input) CHARACTER*1
47: * = 'N': do not compute the left Schur vectors;
48: * = 'V': compute the left Schur vectors (returned in VSL).
49: *
50: * JOBVSR (input) CHARACTER*1
51: * = 'N': do not compute the right Schur vectors;
52: * = 'V': compute the right Schur vectors (returned in VSR).
53: *
54: * N (input) INTEGER
55: * The order of the matrices A, B, VSL, and VSR. N >= 0.
56: *
57: * A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
58: * On entry, the matrix A.
59: * On exit, the upper quasi-triangular matrix S from the
60: * generalized real Schur factorization.
61: *
62: * LDA (input) INTEGER
63: * The leading dimension of A. LDA >= max(1,N).
64: *
65: * B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
66: * On entry, the matrix B.
67: * On exit, the upper triangular matrix T from the generalized
68: * real Schur factorization.
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: * The real parts of each scalar alpha defining an eigenvalue
75: * of GNEP.
76: *
77: * ALPHAI (output) DOUBLE PRECISION array, dimension (N)
78: * The imaginary parts of each scalar alpha defining an
79: * eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th
80: * eigenvalue is real; if positive, then the j-th and (j+1)-st
81: * eigenvalues are a complex conjugate pair, with
82: * ALPHAI(j+1) = -ALPHAI(j).
83: *
84: * BETA (output) DOUBLE PRECISION array, dimension (N)
85: * The scalars beta that define the eigenvalues of GNEP.
86: * Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
87: * beta = BETA(j) represent the j-th eigenvalue of the matrix
88: * pair (A,B), in one of the forms lambda = alpha/beta or
89: * mu = beta/alpha. Since either lambda or mu may overflow,
90: * they should not, in general, be computed.
91: *
92: * VSL (output) DOUBLE PRECISION array, dimension (LDVSL,N)
93: * If JOBVSL = 'V', the matrix of left Schur vectors Q.
94: * Not referenced if JOBVSL = 'N'.
95: *
96: * LDVSL (input) INTEGER
97: * The leading dimension of the matrix VSL. LDVSL >=1, and
98: * if JOBVSL = 'V', LDVSL >= N.
99: *
100: * VSR (output) DOUBLE PRECISION array, dimension (LDVSR,N)
101: * If JOBVSR = 'V', the matrix of right Schur vectors Z.
102: * Not referenced if JOBVSR = 'N'.
103: *
104: * LDVSR (input) INTEGER
105: * The leading dimension of the matrix VSR. LDVSR >= 1, and
106: * if JOBVSR = 'V', LDVSR >= N.
107: *
108: * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
109: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
110: *
111: * LWORK (input) INTEGER
112: * The dimension of the array WORK. LWORK >= max(1,4*N).
113: * For good performance, LWORK must generally be larger.
114: * To compute the optimal value of LWORK, call ILAENV to get
115: * blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute:
116: * NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR
117: * The optimal LWORK is 2*N + N*(NB+1).
118: *
119: * If LWORK = -1, then a workspace query is assumed; the routine
120: * only calculates the optimal size of the WORK array, returns
121: * this value as the first entry of the WORK array, and no error
122: * message related to LWORK is issued by XERBLA.
123: *
124: * INFO (output) INTEGER
125: * = 0: successful exit
126: * < 0: if INFO = -i, the i-th argument had an illegal value.
127: * = 1,...,N:
128: * The QZ iteration failed. (A,B) are not in Schur
129: * form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
130: * be correct for j=INFO+1,...,N.
131: * > N: errors that usually indicate LAPACK problems:
132: * =N+1: error return from DGGBAL
133: * =N+2: error return from DGEQRF
134: * =N+3: error return from DORMQR
135: * =N+4: error return from DORGQR
136: * =N+5: error return from DGGHRD
137: * =N+6: error return from DHGEQZ (other than failed
138: * iteration)
139: * =N+7: error return from DGGBAK (computing VSL)
140: * =N+8: error return from DGGBAK (computing VSR)
141: * =N+9: error return from DLASCL (various places)
142: *
143: * =====================================================================
144: *
145: * .. Parameters ..
146: DOUBLE PRECISION ZERO, ONE
147: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
148: * ..
149: * .. Local Scalars ..
150: LOGICAL ILASCL, ILBSCL, ILVSL, ILVSR, LQUERY
151: INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
152: $ IRIGHT, IROWS, ITAU, IWORK, LOPT, LWKMIN,
153: $ LWKOPT, NB, NB1, NB2, NB3
154: DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
155: $ SAFMIN, SMLNUM
156: * ..
157: * .. External Subroutines ..
158: EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLACPY,
159: $ DLASCL, DLASET, DORGQR, DORMQR, XERBLA
160: * ..
161: * .. External Functions ..
162: LOGICAL LSAME
163: INTEGER ILAENV
164: DOUBLE PRECISION DLAMCH, DLANGE
165: EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
166: * ..
167: * .. Intrinsic Functions ..
168: INTRINSIC INT, MAX
169: * ..
170: * .. Executable Statements ..
171: *
172: * Decode the input arguments
173: *
174: IF( LSAME( JOBVSL, 'N' ) ) THEN
175: IJOBVL = 1
176: ILVSL = .FALSE.
177: ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN
178: IJOBVL = 2
179: ILVSL = .TRUE.
180: ELSE
181: IJOBVL = -1
182: ILVSL = .FALSE.
183: END IF
184: *
185: IF( LSAME( JOBVSR, 'N' ) ) THEN
186: IJOBVR = 1
187: ILVSR = .FALSE.
188: ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN
189: IJOBVR = 2
190: ILVSR = .TRUE.
191: ELSE
192: IJOBVR = -1
193: ILVSR = .FALSE.
194: END IF
195: *
196: * Test the input arguments
197: *
198: LWKMIN = MAX( 4*N, 1 )
199: LWKOPT = LWKMIN
200: WORK( 1 ) = LWKOPT
201: LQUERY = ( LWORK.EQ.-1 )
202: INFO = 0
203: IF( IJOBVL.LE.0 ) THEN
204: INFO = -1
205: ELSE IF( IJOBVR.LE.0 ) THEN
206: INFO = -2
207: ELSE IF( N.LT.0 ) THEN
208: INFO = -3
209: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
210: INFO = -5
211: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
212: INFO = -7
213: ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
214: INFO = -12
215: ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
216: INFO = -14
217: ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
218: INFO = -16
219: END IF
220: *
221: IF( INFO.EQ.0 ) THEN
222: NB1 = ILAENV( 1, 'DGEQRF', ' ', N, N, -1, -1 )
223: NB2 = ILAENV( 1, 'DORMQR', ' ', N, N, N, -1 )
224: NB3 = ILAENV( 1, 'DORGQR', ' ', N, N, N, -1 )
225: NB = MAX( NB1, NB2, NB3 )
226: LOPT = 2*N + N*( NB+1 )
227: WORK( 1 ) = LOPT
228: END IF
229: *
230: IF( INFO.NE.0 ) THEN
231: CALL XERBLA( 'DGEGS ', -INFO )
232: RETURN
233: ELSE IF( LQUERY ) THEN
234: RETURN
235: END IF
236: *
237: * Quick return if possible
238: *
239: IF( N.EQ.0 )
240: $ RETURN
241: *
242: * Get machine constants
243: *
244: EPS = DLAMCH( 'E' )*DLAMCH( 'B' )
245: SAFMIN = DLAMCH( 'S' )
246: SMLNUM = N*SAFMIN / EPS
247: BIGNUM = ONE / SMLNUM
248: *
249: * Scale A if max element outside range [SMLNUM,BIGNUM]
250: *
251: ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
252: ILASCL = .FALSE.
253: IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
254: ANRMTO = SMLNUM
255: ILASCL = .TRUE.
256: ELSE IF( ANRM.GT.BIGNUM ) THEN
257: ANRMTO = BIGNUM
258: ILASCL = .TRUE.
259: END IF
260: *
261: IF( ILASCL ) THEN
262: CALL DLASCL( 'G', -1, -1, ANRM, ANRMTO, N, N, A, LDA, IINFO )
263: IF( IINFO.NE.0 ) THEN
264: INFO = N + 9
265: RETURN
266: END IF
267: END IF
268: *
269: * Scale B if max element outside range [SMLNUM,BIGNUM]
270: *
271: BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
272: ILBSCL = .FALSE.
273: IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
274: BNRMTO = SMLNUM
275: ILBSCL = .TRUE.
276: ELSE IF( BNRM.GT.BIGNUM ) THEN
277: BNRMTO = BIGNUM
278: ILBSCL = .TRUE.
279: END IF
280: *
281: IF( ILBSCL ) THEN
282: CALL DLASCL( 'G', -1, -1, BNRM, BNRMTO, N, N, B, LDB, IINFO )
283: IF( IINFO.NE.0 ) THEN
284: INFO = N + 9
285: RETURN
286: END IF
287: END IF
288: *
289: * Permute the matrix to make it more nearly triangular
290: * Workspace layout: (2*N words -- "work..." not actually used)
291: * left_permutation, right_permutation, work...
292: *
293: ILEFT = 1
294: IRIGHT = N + 1
295: IWORK = IRIGHT + N
296: CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
297: $ WORK( IRIGHT ), WORK( IWORK ), IINFO )
298: IF( IINFO.NE.0 ) THEN
299: INFO = N + 1
300: GO TO 10
301: END IF
302: *
303: * Reduce B to triangular form, and initialize VSL and/or VSR
304: * Workspace layout: ("work..." must have at least N words)
305: * left_permutation, right_permutation, tau, work...
306: *
307: IROWS = IHI + 1 - ILO
308: ICOLS = N + 1 - ILO
309: ITAU = IWORK
310: IWORK = ITAU + IROWS
311: CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
312: $ WORK( IWORK ), LWORK+1-IWORK, IINFO )
313: IF( IINFO.GE.0 )
314: $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
315: IF( IINFO.NE.0 ) THEN
316: INFO = N + 2
317: GO TO 10
318: END IF
319: *
320: CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
321: $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
322: $ LWORK+1-IWORK, IINFO )
323: IF( IINFO.GE.0 )
324: $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
325: IF( IINFO.NE.0 ) THEN
326: INFO = N + 3
327: GO TO 10
328: END IF
329: *
330: IF( ILVSL ) THEN
331: CALL DLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL )
332: CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
333: $ VSL( ILO+1, ILO ), LDVSL )
334: CALL DORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
335: $ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
336: $ IINFO )
337: IF( IINFO.GE.0 )
338: $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
339: IF( IINFO.NE.0 ) THEN
340: INFO = N + 4
341: GO TO 10
342: END IF
343: END IF
344: *
345: IF( ILVSR )
346: $ CALL DLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR )
347: *
348: * Reduce to generalized Hessenberg form
349: *
350: CALL DGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
351: $ LDVSL, VSR, LDVSR, IINFO )
352: IF( IINFO.NE.0 ) THEN
353: INFO = N + 5
354: GO TO 10
355: END IF
356: *
357: * Perform QZ algorithm, computing Schur vectors if desired
358: * Workspace layout: ("work..." must have at least 1 word)
359: * left_permutation, right_permutation, work...
360: *
361: IWORK = ITAU
362: CALL DHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
363: $ ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
364: $ WORK( IWORK ), LWORK+1-IWORK, IINFO )
365: IF( IINFO.GE.0 )
366: $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
367: IF( IINFO.NE.0 ) THEN
368: IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
369: INFO = IINFO
370: ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
371: INFO = IINFO - N
372: ELSE
373: INFO = N + 6
374: END IF
375: GO TO 10
376: END IF
377: *
378: * Apply permutation to VSL and VSR
379: *
380: IF( ILVSL ) THEN
381: CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
382: $ WORK( IRIGHT ), N, VSL, LDVSL, IINFO )
383: IF( IINFO.NE.0 ) THEN
384: INFO = N + 7
385: GO TO 10
386: END IF
387: END IF
388: IF( ILVSR ) THEN
389: CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
390: $ WORK( IRIGHT ), N, VSR, LDVSR, IINFO )
391: IF( IINFO.NE.0 ) THEN
392: INFO = N + 8
393: GO TO 10
394: END IF
395: END IF
396: *
397: * Undo scaling
398: *
399: IF( ILASCL ) THEN
400: CALL DLASCL( 'H', -1, -1, ANRMTO, ANRM, N, N, A, LDA, IINFO )
401: IF( IINFO.NE.0 ) THEN
402: INFO = N + 9
403: RETURN
404: END IF
405: CALL DLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAR, N,
406: $ IINFO )
407: IF( IINFO.NE.0 ) THEN
408: INFO = N + 9
409: RETURN
410: END IF
411: CALL DLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAI, N,
412: $ IINFO )
413: IF( IINFO.NE.0 ) THEN
414: INFO = N + 9
415: RETURN
416: END IF
417: END IF
418: *
419: IF( ILBSCL ) THEN
420: CALL DLASCL( 'U', -1, -1, BNRMTO, BNRM, N, N, B, LDB, IINFO )
421: IF( IINFO.NE.0 ) THEN
422: INFO = N + 9
423: RETURN
424: END IF
425: CALL DLASCL( 'G', -1, -1, BNRMTO, BNRM, N, 1, BETA, N, IINFO )
426: IF( IINFO.NE.0 ) THEN
427: INFO = N + 9
428: RETURN
429: END IF
430: END IF
431: *
432: 10 CONTINUE
433: WORK( 1 ) = LWKOPT
434: *
435: RETURN
436: *
437: * End of DGEGS
438: *
439: END
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