1: SUBROUTINE ZTREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
2: $ LDVR, MM, M, WORK, RWORK, INFO )
3: *
4: * -- LAPACK 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 HOWMNY, SIDE
11: INTEGER INFO, LDT, LDVL, LDVR, M, MM, N
12: * ..
13: * .. Array Arguments ..
14: LOGICAL SELECT( * )
15: DOUBLE PRECISION RWORK( * )
16: COMPLEX*16 T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
17: $ WORK( * )
18: * ..
19: *
20: * Purpose
21: * =======
22: *
23: * ZTREVC computes some or all of the right and/or left eigenvectors of
24: * a complex upper triangular matrix T.
25: * Matrices of this type are produced by the Schur factorization of
26: * a complex general matrix: A = Q*T*Q**H, as computed by ZHSEQR.
27: *
28: * The right eigenvector x and the left eigenvector y of T corresponding
29: * to an eigenvalue w are defined by:
30: *
31: * T*x = w*x, (y**H)*T = w*(y**H)
32: *
33: * where y**H denotes the conjugate transpose of the vector y.
34: * The eigenvalues are not input to this routine, but are read directly
35: * from the diagonal of T.
36: *
37: * This routine returns the matrices X and/or Y of right and left
38: * eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
39: * input matrix. If Q is the unitary factor that reduces a matrix A to
40: * Schur form T, then Q*X and Q*Y are the matrices of right and left
41: * eigenvectors of A.
42: *
43: * Arguments
44: * =========
45: *
46: * SIDE (input) CHARACTER*1
47: * = 'R': compute right eigenvectors only;
48: * = 'L': compute left eigenvectors only;
49: * = 'B': compute both right and left eigenvectors.
50: *
51: * HOWMNY (input) CHARACTER*1
52: * = 'A': compute all right and/or left eigenvectors;
53: * = 'B': compute all right and/or left eigenvectors,
54: * backtransformed using the matrices supplied in
55: * VR and/or VL;
56: * = 'S': compute selected right and/or left eigenvectors,
57: * as indicated by the logical array SELECT.
58: *
59: * SELECT (input) LOGICAL array, dimension (N)
60: * If HOWMNY = 'S', SELECT specifies the eigenvectors to be
61: * computed.
62: * The eigenvector corresponding to the j-th eigenvalue is
63: * computed if SELECT(j) = .TRUE..
64: * Not referenced if HOWMNY = 'A' or 'B'.
65: *
66: * N (input) INTEGER
67: * The order of the matrix T. N >= 0.
68: *
69: * T (input/output) COMPLEX*16 array, dimension (LDT,N)
70: * The upper triangular matrix T. T is modified, but restored
71: * on exit.
72: *
73: * LDT (input) INTEGER
74: * The leading dimension of the array T. LDT >= max(1,N).
75: *
76: * VL (input/output) COMPLEX*16 array, dimension (LDVL,MM)
77: * On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
78: * contain an N-by-N matrix Q (usually the unitary matrix Q of
79: * Schur vectors returned by ZHSEQR).
80: * On exit, if SIDE = 'L' or 'B', VL contains:
81: * if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
82: * if HOWMNY = 'B', the matrix Q*Y;
83: * if HOWMNY = 'S', the left eigenvectors of T specified by
84: * SELECT, stored consecutively in the columns
85: * of VL, in the same order as their
86: * eigenvalues.
87: * Not referenced if SIDE = 'R'.
88: *
89: * LDVL (input) INTEGER
90: * The leading dimension of the array VL. LDVL >= 1, and if
91: * SIDE = 'L' or 'B', LDVL >= N.
92: *
93: * VR (input/output) COMPLEX*16 array, dimension (LDVR,MM)
94: * On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
95: * contain an N-by-N matrix Q (usually the unitary matrix Q of
96: * Schur vectors returned by ZHSEQR).
97: * On exit, if SIDE = 'R' or 'B', VR contains:
98: * if HOWMNY = 'A', the matrix X of right eigenvectors of T;
99: * if HOWMNY = 'B', the matrix Q*X;
100: * if HOWMNY = 'S', the right eigenvectors of T specified by
101: * SELECT, stored consecutively in the columns
102: * of VR, in the same order as their
103: * eigenvalues.
104: * Not referenced if SIDE = 'L'.
105: *
106: * LDVR (input) INTEGER
107: * The leading dimension of the array VR. LDVR >= 1, and if
108: * SIDE = 'R' or 'B'; LDVR >= N.
109: *
110: * MM (input) INTEGER
111: * The number of columns in the arrays VL and/or VR. MM >= M.
112: *
113: * M (output) INTEGER
114: * The number of columns in the arrays VL and/or VR actually
115: * used to store the eigenvectors. If HOWMNY = 'A' or 'B', M
116: * is set to N. Each selected eigenvector occupies one
117: * column.
118: *
119: * WORK (workspace) COMPLEX*16 array, dimension (2*N)
120: *
121: * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
122: *
123: * INFO (output) INTEGER
124: * = 0: successful exit
125: * < 0: if INFO = -i, the i-th argument had an illegal value
126: *
127: * Further Details
128: * ===============
129: *
130: * The algorithm used in this program is basically backward (forward)
131: * substitution, with scaling to make the the code robust against
132: * possible overflow.
133: *
134: * Each eigenvector is normalized so that the element of largest
135: * magnitude has magnitude 1; here the magnitude of a complex number
136: * (x,y) is taken to be |x| + |y|.
137: *
138: * =====================================================================
139: *
140: * .. Parameters ..
141: DOUBLE PRECISION ZERO, ONE
142: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
143: COMPLEX*16 CMZERO, CMONE
144: PARAMETER ( CMZERO = ( 0.0D+0, 0.0D+0 ),
145: $ CMONE = ( 1.0D+0, 0.0D+0 ) )
146: * ..
147: * .. Local Scalars ..
148: LOGICAL ALLV, BOTHV, LEFTV, OVER, RIGHTV, SOMEV
149: INTEGER I, II, IS, J, K, KI
150: DOUBLE PRECISION OVFL, REMAX, SCALE, SMIN, SMLNUM, ULP, UNFL
151: COMPLEX*16 CDUM
152: * ..
153: * .. External Functions ..
154: LOGICAL LSAME
155: INTEGER IZAMAX
156: DOUBLE PRECISION DLAMCH, DZASUM
157: EXTERNAL LSAME, IZAMAX, DLAMCH, DZASUM
158: * ..
159: * .. External Subroutines ..
160: EXTERNAL XERBLA, ZCOPY, ZDSCAL, ZGEMV, ZLATRS
161: * ..
162: * .. Intrinsic Functions ..
163: INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX
164: * ..
165: * .. Statement Functions ..
166: DOUBLE PRECISION CABS1
167: * ..
168: * .. Statement Function definitions ..
169: CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
170: * ..
171: * .. Executable Statements ..
172: *
173: * Decode and test the input parameters
174: *
175: BOTHV = LSAME( SIDE, 'B' )
176: RIGHTV = LSAME( SIDE, 'R' ) .OR. BOTHV
177: LEFTV = LSAME( SIDE, 'L' ) .OR. BOTHV
178: *
179: ALLV = LSAME( HOWMNY, 'A' )
180: OVER = LSAME( HOWMNY, 'B' )
181: SOMEV = LSAME( HOWMNY, 'S' )
182: *
183: * Set M to the number of columns required to store the selected
184: * eigenvectors.
185: *
186: IF( SOMEV ) THEN
187: M = 0
188: DO 10 J = 1, N
189: IF( SELECT( J ) )
190: $ M = M + 1
191: 10 CONTINUE
192: ELSE
193: M = N
194: END IF
195: *
196: INFO = 0
197: IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN
198: INFO = -1
199: ELSE IF( .NOT.ALLV .AND. .NOT.OVER .AND. .NOT.SOMEV ) THEN
200: INFO = -2
201: ELSE IF( N.LT.0 ) THEN
202: INFO = -4
203: ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
204: INFO = -6
205: ELSE IF( LDVL.LT.1 .OR. ( LEFTV .AND. LDVL.LT.N ) ) THEN
206: INFO = -8
207: ELSE IF( LDVR.LT.1 .OR. ( RIGHTV .AND. LDVR.LT.N ) ) THEN
208: INFO = -10
209: ELSE IF( MM.LT.M ) THEN
210: INFO = -11
211: END IF
212: IF( INFO.NE.0 ) THEN
213: CALL XERBLA( 'ZTREVC', -INFO )
214: RETURN
215: END IF
216: *
217: * Quick return if possible.
218: *
219: IF( N.EQ.0 )
220: $ RETURN
221: *
222: * Set the constants to control overflow.
223: *
224: UNFL = DLAMCH( 'Safe minimum' )
225: OVFL = ONE / UNFL
226: CALL DLABAD( UNFL, OVFL )
227: ULP = DLAMCH( 'Precision' )
228: SMLNUM = UNFL*( N / ULP )
229: *
230: * Store the diagonal elements of T in working array WORK.
231: *
232: DO 20 I = 1, N
233: WORK( I+N ) = T( I, I )
234: 20 CONTINUE
235: *
236: * Compute 1-norm of each column of strictly upper triangular
237: * part of T to control overflow in triangular solver.
238: *
239: RWORK( 1 ) = ZERO
240: DO 30 J = 2, N
241: RWORK( J ) = DZASUM( J-1, T( 1, J ), 1 )
242: 30 CONTINUE
243: *
244: IF( RIGHTV ) THEN
245: *
246: * Compute right eigenvectors.
247: *
248: IS = M
249: DO 80 KI = N, 1, -1
250: *
251: IF( SOMEV ) THEN
252: IF( .NOT.SELECT( KI ) )
253: $ GO TO 80
254: END IF
255: SMIN = MAX( ULP*( CABS1( T( KI, KI ) ) ), SMLNUM )
256: *
257: WORK( 1 ) = CMONE
258: *
259: * Form right-hand side.
260: *
261: DO 40 K = 1, KI - 1
262: WORK( K ) = -T( K, KI )
263: 40 CONTINUE
264: *
265: * Solve the triangular system:
266: * (T(1:KI-1,1:KI-1) - T(KI,KI))*X = SCALE*WORK.
267: *
268: DO 50 K = 1, KI - 1
269: T( K, K ) = T( K, K ) - T( KI, KI )
270: IF( CABS1( T( K, K ) ).LT.SMIN )
271: $ T( K, K ) = SMIN
272: 50 CONTINUE
273: *
274: IF( KI.GT.1 ) THEN
275: CALL ZLATRS( 'Upper', 'No transpose', 'Non-unit', 'Y',
276: $ KI-1, T, LDT, WORK( 1 ), SCALE, RWORK,
277: $ INFO )
278: WORK( KI ) = SCALE
279: END IF
280: *
281: * Copy the vector x or Q*x to VR and normalize.
282: *
283: IF( .NOT.OVER ) THEN
284: CALL ZCOPY( KI, WORK( 1 ), 1, VR( 1, IS ), 1 )
285: *
286: II = IZAMAX( KI, VR( 1, IS ), 1 )
287: REMAX = ONE / CABS1( VR( II, IS ) )
288: CALL ZDSCAL( KI, REMAX, VR( 1, IS ), 1 )
289: *
290: DO 60 K = KI + 1, N
291: VR( K, IS ) = CMZERO
292: 60 CONTINUE
293: ELSE
294: IF( KI.GT.1 )
295: $ CALL ZGEMV( 'N', N, KI-1, CMONE, VR, LDVR, WORK( 1 ),
296: $ 1, DCMPLX( SCALE ), VR( 1, KI ), 1 )
297: *
298: II = IZAMAX( N, VR( 1, KI ), 1 )
299: REMAX = ONE / CABS1( VR( II, KI ) )
300: CALL ZDSCAL( N, REMAX, VR( 1, KI ), 1 )
301: END IF
302: *
303: * Set back the original diagonal elements of T.
304: *
305: DO 70 K = 1, KI - 1
306: T( K, K ) = WORK( K+N )
307: 70 CONTINUE
308: *
309: IS = IS - 1
310: 80 CONTINUE
311: END IF
312: *
313: IF( LEFTV ) THEN
314: *
315: * Compute left eigenvectors.
316: *
317: IS = 1
318: DO 130 KI = 1, N
319: *
320: IF( SOMEV ) THEN
321: IF( .NOT.SELECT( KI ) )
322: $ GO TO 130
323: END IF
324: SMIN = MAX( ULP*( CABS1( T( KI, KI ) ) ), SMLNUM )
325: *
326: WORK( N ) = CMONE
327: *
328: * Form right-hand side.
329: *
330: DO 90 K = KI + 1, N
331: WORK( K ) = -DCONJG( T( KI, K ) )
332: 90 CONTINUE
333: *
334: * Solve the triangular system:
335: * (T(KI+1:N,KI+1:N) - T(KI,KI))'*X = SCALE*WORK.
336: *
337: DO 100 K = KI + 1, N
338: T( K, K ) = T( K, K ) - T( KI, KI )
339: IF( CABS1( T( K, K ) ).LT.SMIN )
340: $ T( K, K ) = SMIN
341: 100 CONTINUE
342: *
343: IF( KI.LT.N ) THEN
344: CALL ZLATRS( 'Upper', 'Conjugate transpose', 'Non-unit',
345: $ 'Y', N-KI, T( KI+1, KI+1 ), LDT,
346: $ WORK( KI+1 ), SCALE, RWORK, INFO )
347: WORK( KI ) = SCALE
348: END IF
349: *
350: * Copy the vector x or Q*x to VL and normalize.
351: *
352: IF( .NOT.OVER ) THEN
353: CALL ZCOPY( N-KI+1, WORK( KI ), 1, VL( KI, IS ), 1 )
354: *
355: II = IZAMAX( N-KI+1, VL( KI, IS ), 1 ) + KI - 1
356: REMAX = ONE / CABS1( VL( II, IS ) )
357: CALL ZDSCAL( N-KI+1, REMAX, VL( KI, IS ), 1 )
358: *
359: DO 110 K = 1, KI - 1
360: VL( K, IS ) = CMZERO
361: 110 CONTINUE
362: ELSE
363: IF( KI.LT.N )
364: $ CALL ZGEMV( 'N', N, N-KI, CMONE, VL( 1, KI+1 ), LDVL,
365: $ WORK( KI+1 ), 1, DCMPLX( SCALE ),
366: $ VL( 1, KI ), 1 )
367: *
368: II = IZAMAX( N, VL( 1, KI ), 1 )
369: REMAX = ONE / CABS1( VL( II, KI ) )
370: CALL ZDSCAL( N, REMAX, VL( 1, KI ), 1 )
371: END IF
372: *
373: * Set back the original diagonal elements of T.
374: *
375: DO 120 K = KI + 1, N
376: T( K, K ) = WORK( K+N )
377: 120 CONTINUE
378: *
379: IS = IS + 1
380: 130 CONTINUE
381: END IF
382: *
383: RETURN
384: *
385: * End of ZTREVC
386: *
387: END
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