1: SUBROUTINE DGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM, WR, WI,
2: $ VS, LDVS, WORK, LWORK, BWORK, 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 JOBVS, SORT
11: INTEGER INFO, LDA, LDVS, LWORK, N, SDIM
12: * ..
13: * .. Array Arguments ..
14: LOGICAL BWORK( * )
15: DOUBLE PRECISION A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ),
16: $ WR( * )
17: * ..
18: * .. Function Arguments ..
19: LOGICAL SELECT
20: EXTERNAL SELECT
21: * ..
22: *
23: * Purpose
24: * =======
25: *
26: * DGEES computes for an N-by-N real nonsymmetric matrix A, the
27: * eigenvalues, the real Schur form T, and, optionally, the matrix of
28: * Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**T).
29: *
30: * Optionally, it also orders the eigenvalues on the diagonal of the
31: * real Schur form so that selected eigenvalues are at the top left.
32: * The leading columns of Z then form an orthonormal basis for the
33: * invariant subspace corresponding to the selected eigenvalues.
34: *
35: * A matrix is in real Schur form if it is upper quasi-triangular with
36: * 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in the
37: * form
38: * [ a b ]
39: * [ c a ]
40: *
41: * where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).
42: *
43: * Arguments
44: * =========
45: *
46: * JOBVS (input) CHARACTER*1
47: * = 'N': Schur vectors are not computed;
48: * = 'V': Schur vectors are computed.
49: *
50: * SORT (input) CHARACTER*1
51: * Specifies whether or not to order the eigenvalues on the
52: * diagonal of the Schur form.
53: * = 'N': Eigenvalues are not ordered;
54: * = 'S': Eigenvalues are ordered (see SELECT).
55: *
56: * SELECT (external procedure) LOGICAL FUNCTION of two DOUBLE PRECISION arguments
57: * SELECT must be declared EXTERNAL in the calling subroutine.
58: * If SORT = 'S', SELECT is used to select eigenvalues to sort
59: * to the top left of the Schur form.
60: * If SORT = 'N', SELECT is not referenced.
61: * An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
62: * SELECT(WR(j),WI(j)) is true; i.e., if either one of a complex
63: * conjugate pair of eigenvalues is selected, then both complex
64: * eigenvalues are selected.
65: * Note that a selected complex eigenvalue may no longer
66: * satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
67: * ordering may change the value of complex eigenvalues
68: * (especially if the eigenvalue is ill-conditioned); in this
69: * case INFO is set to N+2 (see INFO below).
70: *
71: * N (input) INTEGER
72: * The order of the matrix A. N >= 0.
73: *
74: * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
75: * On entry, the N-by-N matrix A.
76: * On exit, A has been overwritten by its real Schur form T.
77: *
78: * LDA (input) INTEGER
79: * The leading dimension of the array A. LDA >= max(1,N).
80: *
81: * SDIM (output) INTEGER
82: * If SORT = 'N', SDIM = 0.
83: * If SORT = 'S', SDIM = number of eigenvalues (after sorting)
84: * for which SELECT is true. (Complex conjugate
85: * pairs for which SELECT is true for either
86: * eigenvalue count as 2.)
87: *
88: * WR (output) DOUBLE PRECISION array, dimension (N)
89: * WI (output) DOUBLE PRECISION array, dimension (N)
90: * WR and WI contain the real and imaginary parts,
91: * respectively, of the computed eigenvalues in the same order
92: * that they appear on the diagonal of the output Schur form T.
93: * Complex conjugate pairs of eigenvalues will appear
94: * consecutively with the eigenvalue having the positive
95: * imaginary part first.
96: *
97: * VS (output) DOUBLE PRECISION array, dimension (LDVS,N)
98: * If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
99: * vectors.
100: * If JOBVS = 'N', VS is not referenced.
101: *
102: * LDVS (input) INTEGER
103: * The leading dimension of the array VS. LDVS >= 1; if
104: * JOBVS = 'V', LDVS >= N.
105: *
106: * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
107: * On exit, if INFO = 0, WORK(1) contains the optimal LWORK.
108: *
109: * LWORK (input) INTEGER
110: * The dimension of the array WORK. LWORK >= max(1,3*N).
111: * For good performance, LWORK must generally be larger.
112: *
113: * If LWORK = -1, then a workspace query is assumed; the routine
114: * only calculates the optimal size of the WORK array, returns
115: * this value as the first entry of the WORK array, and no error
116: * message related to LWORK is issued by XERBLA.
117: *
118: * BWORK (workspace) LOGICAL array, dimension (N)
119: * Not referenced if SORT = 'N'.
120: *
121: * INFO (output) INTEGER
122: * = 0: successful exit
123: * < 0: if INFO = -i, the i-th argument had an illegal value.
124: * > 0: if INFO = i, and i is
125: * <= N: the QR algorithm failed to compute all the
126: * eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI
127: * contain those eigenvalues which have converged; if
128: * JOBVS = 'V', VS contains the matrix which reduces A
129: * to its partially converged Schur form.
130: * = N+1: the eigenvalues could not be reordered because some
131: * eigenvalues were too close to separate (the problem
132: * is very ill-conditioned);
133: * = N+2: after reordering, roundoff changed values of some
134: * complex eigenvalues so that leading eigenvalues in
135: * the Schur form no longer satisfy SELECT=.TRUE. This
136: * could also be caused by underflow due to scaling.
137: *
138: * =====================================================================
139: *
140: * .. Parameters ..
141: DOUBLE PRECISION ZERO, ONE
142: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
143: * ..
144: * .. Local Scalars ..
145: LOGICAL CURSL, LASTSL, LQUERY, LST2SL, SCALEA, WANTST,
146: $ WANTVS
147: INTEGER HSWORK, I, I1, I2, IBAL, ICOND, IERR, IEVAL,
148: $ IHI, ILO, INXT, IP, ITAU, IWRK, MAXWRK, MINWRK
149: DOUBLE PRECISION ANRM, BIGNUM, CSCALE, EPS, S, SEP, SMLNUM
150: * ..
151: * .. Local Arrays ..
152: INTEGER IDUM( 1 )
153: DOUBLE PRECISION DUM( 1 )
154: * ..
155: * .. External Subroutines ..
156: EXTERNAL DCOPY, DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLACPY,
157: $ DLABAD, DLASCL, DORGHR, DSWAP, DTRSEN, XERBLA
158: * ..
159: * .. External Functions ..
160: LOGICAL LSAME
161: INTEGER ILAENV
162: DOUBLE PRECISION DLAMCH, DLANGE
163: EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
164: * ..
165: * .. Intrinsic Functions ..
166: INTRINSIC MAX, SQRT
167: * ..
168: * .. Executable Statements ..
169: *
170: * Test the input arguments
171: *
172: INFO = 0
173: LQUERY = ( LWORK.EQ.-1 )
174: WANTVS = LSAME( JOBVS, 'V' )
175: WANTST = LSAME( SORT, 'S' )
176: IF( ( .NOT.WANTVS ) .AND. ( .NOT.LSAME( JOBVS, 'N' ) ) ) THEN
177: INFO = -1
178: ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
179: INFO = -2
180: ELSE IF( N.LT.0 ) THEN
181: INFO = -4
182: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
183: INFO = -6
184: ELSE IF( LDVS.LT.1 .OR. ( WANTVS .AND. LDVS.LT.N ) ) THEN
185: INFO = -11
186: END IF
187: *
188: * Compute workspace
189: * (Note: Comments in the code beginning "Workspace:" describe the
190: * minimal amount of workspace needed at that point in the code,
191: * as well as the preferred amount for good performance.
192: * NB refers to the optimal block size for the immediately
193: * following subroutine, as returned by ILAENV.
194: * HSWORK refers to the workspace preferred by DHSEQR, as
195: * calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
196: * the worst case.)
197: *
198: IF( INFO.EQ.0 ) THEN
199: IF( N.EQ.0 ) THEN
200: MINWRK = 1
201: MAXWRK = 1
202: ELSE
203: MAXWRK = 2*N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
204: MINWRK = 3*N
205: *
206: CALL DHSEQR( 'S', JOBVS, N, 1, N, A, LDA, WR, WI, VS, LDVS,
207: $ WORK, -1, IEVAL )
208: HSWORK = WORK( 1 )
209: *
210: IF( .NOT.WANTVS ) THEN
211: MAXWRK = MAX( MAXWRK, N + HSWORK )
212: ELSE
213: MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
214: $ 'DORGHR', ' ', N, 1, N, -1 ) )
215: MAXWRK = MAX( MAXWRK, N + HSWORK )
216: END IF
217: END IF
218: WORK( 1 ) = MAXWRK
219: *
220: IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
221: INFO = -13
222: END IF
223: END IF
224: *
225: IF( INFO.NE.0 ) THEN
226: CALL XERBLA( 'DGEES ', -INFO )
227: RETURN
228: ELSE IF( LQUERY ) THEN
229: RETURN
230: END IF
231: *
232: * Quick return if possible
233: *
234: IF( N.EQ.0 ) THEN
235: SDIM = 0
236: RETURN
237: END IF
238: *
239: * Get machine constants
240: *
241: EPS = DLAMCH( 'P' )
242: SMLNUM = DLAMCH( 'S' )
243: BIGNUM = ONE / SMLNUM
244: CALL DLABAD( SMLNUM, BIGNUM )
245: SMLNUM = SQRT( SMLNUM ) / EPS
246: BIGNUM = ONE / SMLNUM
247: *
248: * Scale A if max element outside range [SMLNUM,BIGNUM]
249: *
250: ANRM = DLANGE( 'M', N, N, A, LDA, DUM )
251: SCALEA = .FALSE.
252: IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
253: SCALEA = .TRUE.
254: CSCALE = SMLNUM
255: ELSE IF( ANRM.GT.BIGNUM ) THEN
256: SCALEA = .TRUE.
257: CSCALE = BIGNUM
258: END IF
259: IF( SCALEA )
260: $ CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
261: *
262: * Permute the matrix to make it more nearly triangular
263: * (Workspace: need N)
264: *
265: IBAL = 1
266: CALL DGEBAL( 'P', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR )
267: *
268: * Reduce to upper Hessenberg form
269: * (Workspace: need 3*N, prefer 2*N+N*NB)
270: *
271: ITAU = N + IBAL
272: IWRK = N + ITAU
273: CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
274: $ LWORK-IWRK+1, IERR )
275: *
276: IF( WANTVS ) THEN
277: *
278: * Copy Householder vectors to VS
279: *
280: CALL DLACPY( 'L', N, N, A, LDA, VS, LDVS )
281: *
282: * Generate orthogonal matrix in VS
283: * (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB)
284: *
285: CALL DORGHR( N, ILO, IHI, VS, LDVS, WORK( ITAU ), WORK( IWRK ),
286: $ LWORK-IWRK+1, IERR )
287: END IF
288: *
289: SDIM = 0
290: *
291: * Perform QR iteration, accumulating Schur vectors in VS if desired
292: * (Workspace: need N+1, prefer N+HSWORK (see comments) )
293: *
294: IWRK = ITAU
295: CALL DHSEQR( 'S', JOBVS, N, ILO, IHI, A, LDA, WR, WI, VS, LDVS,
296: $ WORK( IWRK ), LWORK-IWRK+1, IEVAL )
297: IF( IEVAL.GT.0 )
298: $ INFO = IEVAL
299: *
300: * Sort eigenvalues if desired
301: *
302: IF( WANTST .AND. INFO.EQ.0 ) THEN
303: IF( SCALEA ) THEN
304: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WR, N, IERR )
305: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WI, N, IERR )
306: END IF
307: DO 10 I = 1, N
308: BWORK( I ) = SELECT( WR( I ), WI( I ) )
309: 10 CONTINUE
310: *
311: * Reorder eigenvalues and transform Schur vectors
312: * (Workspace: none needed)
313: *
314: CALL DTRSEN( 'N', JOBVS, BWORK, N, A, LDA, VS, LDVS, WR, WI,
315: $ SDIM, S, SEP, WORK( IWRK ), LWORK-IWRK+1, IDUM, 1,
316: $ ICOND )
317: IF( ICOND.GT.0 )
318: $ INFO = N + ICOND
319: END IF
320: *
321: IF( WANTVS ) THEN
322: *
323: * Undo balancing
324: * (Workspace: need N)
325: *
326: CALL DGEBAK( 'P', 'R', N, ILO, IHI, WORK( IBAL ), N, VS, LDVS,
327: $ IERR )
328: END IF
329: *
330: IF( SCALEA ) THEN
331: *
332: * Undo scaling for the Schur form of A
333: *
334: CALL DLASCL( 'H', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR )
335: CALL DCOPY( N, A, LDA+1, WR, 1 )
336: IF( CSCALE.EQ.SMLNUM ) THEN
337: *
338: * If scaling back towards underflow, adjust WI if an
339: * offdiagonal element of a 2-by-2 block in the Schur form
340: * underflows.
341: *
342: IF( IEVAL.GT.0 ) THEN
343: I1 = IEVAL + 1
344: I2 = IHI - 1
345: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI,
346: $ MAX( ILO-1, 1 ), IERR )
347: ELSE IF( WANTST ) THEN
348: I1 = 1
349: I2 = N - 1
350: ELSE
351: I1 = ILO
352: I2 = IHI - 1
353: END IF
354: INXT = I1 - 1
355: DO 20 I = I1, I2
356: IF( I.LT.INXT )
357: $ GO TO 20
358: IF( WI( I ).EQ.ZERO ) THEN
359: INXT = I + 1
360: ELSE
361: IF( A( I+1, I ).EQ.ZERO ) THEN
362: WI( I ) = ZERO
363: WI( I+1 ) = ZERO
364: ELSE IF( A( I+1, I ).NE.ZERO .AND. A( I, I+1 ).EQ.
365: $ ZERO ) THEN
366: WI( I ) = ZERO
367: WI( I+1 ) = ZERO
368: IF( I.GT.1 )
369: $ CALL DSWAP( I-1, A( 1, I ), 1, A( 1, I+1 ), 1 )
370: IF( N.GT.I+1 )
371: $ CALL DSWAP( N-I-1, A( I, I+2 ), LDA,
372: $ A( I+1, I+2 ), LDA )
373: IF( WANTVS ) THEN
374: CALL DSWAP( N, VS( 1, I ), 1, VS( 1, I+1 ), 1 )
375: END IF
376: A( I, I+1 ) = A( I+1, I )
377: A( I+1, I ) = ZERO
378: END IF
379: INXT = I + 2
380: END IF
381: 20 CONTINUE
382: END IF
383: *
384: * Undo scaling for the imaginary part of the eigenvalues
385: *
386: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-IEVAL, 1,
387: $ WI( IEVAL+1 ), MAX( N-IEVAL, 1 ), IERR )
388: END IF
389: *
390: IF( WANTST .AND. INFO.EQ.0 ) THEN
391: *
392: * Check if reordering successful
393: *
394: LASTSL = .TRUE.
395: LST2SL = .TRUE.
396: SDIM = 0
397: IP = 0
398: DO 30 I = 1, N
399: CURSL = SELECT( WR( I ), WI( I ) )
400: IF( WI( I ).EQ.ZERO ) THEN
401: IF( CURSL )
402: $ SDIM = SDIM + 1
403: IP = 0
404: IF( CURSL .AND. .NOT.LASTSL )
405: $ INFO = N + 2
406: ELSE
407: IF( IP.EQ.1 ) THEN
408: *
409: * Last eigenvalue of conjugate pair
410: *
411: CURSL = CURSL .OR. LASTSL
412: LASTSL = CURSL
413: IF( CURSL )
414: $ SDIM = SDIM + 2
415: IP = -1
416: IF( CURSL .AND. .NOT.LST2SL )
417: $ INFO = N + 2
418: ELSE
419: *
420: * First eigenvalue of conjugate pair
421: *
422: IP = 1
423: END IF
424: END IF
425: LST2SL = LASTSL
426: LASTSL = CURSL
427: 30 CONTINUE
428: END IF
429: *
430: WORK( 1 ) = MAXWRK
431: RETURN
432: *
433: * End of DGEES
434: *
435: END
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