Annotation of rpl/lapack/lapack/zhseqr.f, revision 1.7
1.1 bertrand 1: SUBROUTINE ZHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ,
2: $ WORK, LWORK, INFO )
3: *
1.5 bertrand 4: * -- LAPACK computational routine (version 3.2.2) --
1.1 bertrand 5: * Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..
1.5 bertrand 6: * June 2010
1.1 bertrand 7: *
8: * .. Scalar Arguments ..
9: INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
10: CHARACTER COMPZ, JOB
11: * ..
12: * .. Array Arguments ..
13: COMPLEX*16 H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
14: * ..
15: * Purpose
16: * =======
17: *
18: * ZHSEQR computes the eigenvalues of a Hessenberg matrix H
19: * and, optionally, the matrices T and Z from the Schur decomposition
20: * H = Z T Z**H, where T is an upper triangular matrix (the
21: * Schur form), and Z is the unitary matrix of Schur vectors.
22: *
23: * Optionally Z may be postmultiplied into an input unitary
24: * matrix Q so that this routine can give the Schur factorization
25: * of a matrix A which has been reduced to the Hessenberg form H
26: * by the unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H.
27: *
28: * Arguments
29: * =========
30: *
31: * JOB (input) CHARACTER*1
32: * = 'E': compute eigenvalues only;
33: * = 'S': compute eigenvalues and the Schur form T.
34: *
35: * COMPZ (input) CHARACTER*1
36: * = 'N': no Schur vectors are computed;
37: * = 'I': Z is initialized to the unit matrix and the matrix Z
38: * of Schur vectors of H is returned;
39: * = 'V': Z must contain an unitary matrix Q on entry, and
40: * the product Q*Z is returned.
41: *
42: * N (input) INTEGER
43: * The order of the matrix H. N .GE. 0.
44: *
45: * ILO (input) INTEGER
46: * IHI (input) INTEGER
47: * It is assumed that H is already upper triangular in rows
48: * and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
49: * set by a previous call to ZGEBAL, and then passed to ZGEHRD
50: * when the matrix output by ZGEBAL is reduced to Hessenberg
51: * form. Otherwise ILO and IHI should be set to 1 and N
52: * respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
53: * If N = 0, then ILO = 1 and IHI = 0.
54: *
55: * H (input/output) COMPLEX*16 array, dimension (LDH,N)
56: * On entry, the upper Hessenberg matrix H.
57: * On exit, if INFO = 0 and JOB = 'S', H contains the upper
58: * triangular matrix T from the Schur decomposition (the
59: * Schur form). If INFO = 0 and JOB = 'E', the contents of
60: * H are unspecified on exit. (The output value of H when
61: * INFO.GT.0 is given under the description of INFO below.)
62: *
63: * Unlike earlier versions of ZHSEQR, this subroutine may
64: * explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1
65: * or j = IHI+1, IHI+2, ... N.
66: *
67: * LDH (input) INTEGER
68: * The leading dimension of the array H. LDH .GE. max(1,N).
69: *
70: * W (output) COMPLEX*16 array, dimension (N)
71: * The computed eigenvalues. If JOB = 'S', the eigenvalues are
72: * stored in the same order as on the diagonal of the Schur
73: * form returned in H, with W(i) = H(i,i).
74: *
75: * Z (input/output) COMPLEX*16 array, dimension (LDZ,N)
76: * If COMPZ = 'N', Z is not referenced.
77: * If COMPZ = 'I', on entry Z need not be set and on exit,
78: * if INFO = 0, Z contains the unitary matrix Z of the Schur
79: * vectors of H. If COMPZ = 'V', on entry Z must contain an
80: * N-by-N matrix Q, which is assumed to be equal to the unit
81: * matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
82: * if INFO = 0, Z contains Q*Z.
83: * Normally Q is the unitary matrix generated by ZUNGHR
84: * after the call to ZGEHRD which formed the Hessenberg matrix
85: * H. (The output value of Z when INFO.GT.0 is given under
86: * the description of INFO below.)
87: *
88: * LDZ (input) INTEGER
89: * The leading dimension of the array Z. if COMPZ = 'I' or
90: * COMPZ = 'V', then LDZ.GE.MAX(1,N). Otherwize, LDZ.GE.1.
91: *
92: * WORK (workspace/output) COMPLEX*16 array, dimension (LWORK)
93: * On exit, if INFO = 0, WORK(1) returns an estimate of
94: * the optimal value for LWORK.
95: *
96: * LWORK (input) INTEGER
97: * The dimension of the array WORK. LWORK .GE. max(1,N)
98: * is sufficient and delivers very good and sometimes
99: * optimal performance. However, LWORK as large as 11*N
100: * may be required for optimal performance. A workspace
101: * query is recommended to determine the optimal workspace
102: * size.
103: *
104: * If LWORK = -1, then ZHSEQR does a workspace query.
105: * In this case, ZHSEQR checks the input parameters and
106: * estimates the optimal workspace size for the given
107: * values of N, ILO and IHI. The estimate is returned
108: * in WORK(1). No error message related to LWORK is
109: * issued by XERBLA. Neither H nor Z are accessed.
110: *
111: *
112: * INFO (output) INTEGER
113: * = 0: successful exit
114: * .LT. 0: if INFO = -i, the i-th argument had an illegal
115: * value
116: * .GT. 0: if INFO = i, ZHSEQR failed to compute all of
117: * the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
118: * and WI contain those eigenvalues which have been
119: * successfully computed. (Failures are rare.)
120: *
121: * If INFO .GT. 0 and JOB = 'E', then on exit, the
122: * remaining unconverged eigenvalues are the eigen-
123: * values of the upper Hessenberg matrix rows and
124: * columns ILO through INFO of the final, output
125: * value of H.
126: *
127: * If INFO .GT. 0 and JOB = 'S', then on exit
128: *
129: * (*) (initial value of H)*U = U*(final value of H)
130: *
131: * where U is a unitary matrix. The final
132: * value of H is upper Hessenberg and triangular in
133: * rows and columns INFO+1 through IHI.
134: *
135: * If INFO .GT. 0 and COMPZ = 'V', then on exit
136: *
137: * (final value of Z) = (initial value of Z)*U
138: *
139: * where U is the unitary matrix in (*) (regard-
140: * less of the value of JOB.)
141: *
142: * If INFO .GT. 0 and COMPZ = 'I', then on exit
143: * (final value of Z) = U
144: * where U is the unitary matrix in (*) (regard-
145: * less of the value of JOB.)
146: *
147: * If INFO .GT. 0 and COMPZ = 'N', then Z is not
148: * accessed.
149: *
150: * ================================================================
151: * Default values supplied by
152: * ILAENV(ISPEC,'ZHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
153: * It is suggested that these defaults be adjusted in order
154: * to attain best performance in each particular
155: * computational environment.
156: *
157: * ISPEC=12: The ZLAHQR vs ZLAQR0 crossover point.
158: * Default: 75. (Must be at least 11.)
159: *
160: * ISPEC=13: Recommended deflation window size.
161: * This depends on ILO, IHI and NS. NS is the
162: * number of simultaneous shifts returned
163: * by ILAENV(ISPEC=15). (See ISPEC=15 below.)
164: * The default for (IHI-ILO+1).LE.500 is NS.
165: * The default for (IHI-ILO+1).GT.500 is 3*NS/2.
166: *
167: * ISPEC=14: Nibble crossover point. (See IPARMQ for
168: * details.) Default: 14% of deflation window
169: * size.
170: *
171: * ISPEC=15: Number of simultaneous shifts in a multishift
172: * QR iteration.
173: *
174: * If IHI-ILO+1 is ...
175: *
176: * greater than ...but less ... the
177: * or equal to ... than default is
178: *
179: * 1 30 NS = 2(+)
180: * 30 60 NS = 4(+)
181: * 60 150 NS = 10(+)
182: * 150 590 NS = **
183: * 590 3000 NS = 64
184: * 3000 6000 NS = 128
185: * 6000 infinity NS = 256
186: *
187: * (+) By default some or all matrices of this order
188: * are passed to the implicit double shift routine
189: * ZLAHQR and this parameter is ignored. See
190: * ISPEC=12 above and comments in IPARMQ for
191: * details.
192: *
193: * (**) The asterisks (**) indicate an ad-hoc
194: * function of N increasing from 10 to 64.
195: *
196: * ISPEC=16: Select structured matrix multiply.
197: * If the number of simultaneous shifts (specified
198: * by ISPEC=15) is less than 14, then the default
199: * for ISPEC=16 is 0. Otherwise the default for
200: * ISPEC=16 is 2.
201: *
202: * ================================================================
203: * Based on contributions by
204: * Karen Braman and Ralph Byers, Department of Mathematics,
205: * University of Kansas, USA
206: *
207: * ================================================================
208: * References:
209: * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
210: * Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
211: * Performance, SIAM Journal of Matrix Analysis, volume 23, pages
212: * 929--947, 2002.
213: *
214: * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
215: * Algorithm Part II: Aggressive Early Deflation, SIAM Journal
216: * of Matrix Analysis, volume 23, pages 948--973, 2002.
217: *
218: * ================================================================
219: * .. Parameters ..
220: *
221: * ==== Matrices of order NTINY or smaller must be processed by
222: * . ZLAHQR because of insufficient subdiagonal scratch space.
223: * . (This is a hard limit.) ====
224: INTEGER NTINY
225: PARAMETER ( NTINY = 11 )
226: *
227: * ==== NL allocates some local workspace to help small matrices
228: * . through a rare ZLAHQR failure. NL .GT. NTINY = 11 is
229: * . required and NL .LE. NMIN = ILAENV(ISPEC=12,...) is recom-
230: * . mended. (The default value of NMIN is 75.) Using NL = 49
231: * . allows up to six simultaneous shifts and a 16-by-16
232: * . deflation window. ====
233: INTEGER NL
234: PARAMETER ( NL = 49 )
235: COMPLEX*16 ZERO, ONE
236: PARAMETER ( ZERO = ( 0.0d0, 0.0d0 ),
237: $ ONE = ( 1.0d0, 0.0d0 ) )
238: DOUBLE PRECISION RZERO
239: PARAMETER ( RZERO = 0.0d0 )
240: * ..
241: * .. Local Arrays ..
242: COMPLEX*16 HL( NL, NL ), WORKL( NL )
243: * ..
244: * .. Local Scalars ..
245: INTEGER KBOT, NMIN
246: LOGICAL INITZ, LQUERY, WANTT, WANTZ
247: * ..
248: * .. External Functions ..
249: INTEGER ILAENV
250: LOGICAL LSAME
251: EXTERNAL ILAENV, LSAME
252: * ..
253: * .. External Subroutines ..
254: EXTERNAL XERBLA, ZCOPY, ZLACPY, ZLAHQR, ZLAQR0, ZLASET
255: * ..
256: * .. Intrinsic Functions ..
257: INTRINSIC DBLE, DCMPLX, MAX, MIN
258: * ..
259: * .. Executable Statements ..
260: *
261: * ==== Decode and check the input parameters. ====
262: *
263: WANTT = LSAME( JOB, 'S' )
264: INITZ = LSAME( COMPZ, 'I' )
265: WANTZ = INITZ .OR. LSAME( COMPZ, 'V' )
266: WORK( 1 ) = DCMPLX( DBLE( MAX( 1, N ) ), RZERO )
267: LQUERY = LWORK.EQ.-1
268: *
269: INFO = 0
270: IF( .NOT.LSAME( JOB, 'E' ) .AND. .NOT.WANTT ) THEN
271: INFO = -1
272: ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN
273: INFO = -2
274: ELSE IF( N.LT.0 ) THEN
275: INFO = -3
276: ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
277: INFO = -4
278: ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
279: INFO = -5
280: ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
281: INFO = -7
282: ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.MAX( 1, N ) ) ) THEN
283: INFO = -10
284: ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
285: INFO = -12
286: END IF
287: *
288: IF( INFO.NE.0 ) THEN
289: *
290: * ==== Quick return in case of invalid argument. ====
291: *
292: CALL XERBLA( 'ZHSEQR', -INFO )
293: RETURN
294: *
295: ELSE IF( N.EQ.0 ) THEN
296: *
297: * ==== Quick return in case N = 0; nothing to do. ====
298: *
299: RETURN
300: *
301: ELSE IF( LQUERY ) THEN
302: *
303: * ==== Quick return in case of a workspace query ====
304: *
305: CALL ZLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILO, IHI, Z,
306: $ LDZ, WORK, LWORK, INFO )
307: * ==== Ensure reported workspace size is backward-compatible with
308: * . previous LAPACK versions. ====
309: WORK( 1 ) = DCMPLX( MAX( DBLE( WORK( 1 ) ), DBLE( MAX( 1,
310: $ N ) ) ), RZERO )
311: RETURN
312: *
313: ELSE
314: *
315: * ==== copy eigenvalues isolated by ZGEBAL ====
316: *
317: IF( ILO.GT.1 )
318: $ CALL ZCOPY( ILO-1, H, LDH+1, W, 1 )
319: IF( IHI.LT.N )
320: $ CALL ZCOPY( N-IHI, H( IHI+1, IHI+1 ), LDH+1, W( IHI+1 ), 1 )
321: *
322: * ==== Initialize Z, if requested ====
323: *
324: IF( INITZ )
325: $ CALL ZLASET( 'A', N, N, ZERO, ONE, Z, LDZ )
326: *
327: * ==== Quick return if possible ====
328: *
329: IF( ILO.EQ.IHI ) THEN
330: W( ILO ) = H( ILO, ILO )
331: RETURN
332: END IF
333: *
334: * ==== ZLAHQR/ZLAQR0 crossover point ====
335: *
336: NMIN = ILAENV( 12, 'ZHSEQR', JOB( : 1 ) // COMPZ( : 1 ), N,
337: $ ILO, IHI, LWORK )
338: NMIN = MAX( NTINY, NMIN )
339: *
340: * ==== ZLAQR0 for big matrices; ZLAHQR for small ones ====
341: *
342: IF( N.GT.NMIN ) THEN
343: CALL ZLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILO, IHI,
344: $ Z, LDZ, WORK, LWORK, INFO )
345: ELSE
346: *
347: * ==== Small matrix ====
348: *
349: CALL ZLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILO, IHI,
350: $ Z, LDZ, INFO )
351: *
352: IF( INFO.GT.0 ) THEN
353: *
354: * ==== A rare ZLAHQR failure! ZLAQR0 sometimes succeeds
355: * . when ZLAHQR fails. ====
356: *
357: KBOT = INFO
358: *
359: IF( N.GE.NL ) THEN
360: *
361: * ==== Larger matrices have enough subdiagonal scratch
362: * . space to call ZLAQR0 directly. ====
363: *
364: CALL ZLAQR0( WANTT, WANTZ, N, ILO, KBOT, H, LDH, W,
365: $ ILO, IHI, Z, LDZ, WORK, LWORK, INFO )
366: *
367: ELSE
368: *
369: * ==== Tiny matrices don't have enough subdiagonal
370: * . scratch space to benefit from ZLAQR0. Hence,
371: * . tiny matrices must be copied into a larger
372: * . array before calling ZLAQR0. ====
373: *
374: CALL ZLACPY( 'A', N, N, H, LDH, HL, NL )
375: HL( N+1, N ) = ZERO
376: CALL ZLASET( 'A', NL, NL-N, ZERO, ZERO, HL( 1, N+1 ),
377: $ NL )
378: CALL ZLAQR0( WANTT, WANTZ, NL, ILO, KBOT, HL, NL, W,
379: $ ILO, IHI, Z, LDZ, WORKL, NL, INFO )
380: IF( WANTT .OR. INFO.NE.0 )
381: $ CALL ZLACPY( 'A', N, N, HL, NL, H, LDH )
382: END IF
383: END IF
384: END IF
385: *
386: * ==== Clear out the trash, if necessary. ====
387: *
388: IF( ( WANTT .OR. INFO.NE.0 ) .AND. N.GT.2 )
389: $ CALL ZLASET( 'L', N-2, N-2, ZERO, ZERO, H( 3, 1 ), LDH )
390: *
391: * ==== Ensure reported workspace size is backward-compatible with
392: * . previous LAPACK versions. ====
393: *
394: WORK( 1 ) = DCMPLX( MAX( DBLE( MAX( 1, N ) ),
395: $ DBLE( WORK( 1 ) ) ), RZERO )
396: END IF
397: *
398: * ==== End of ZHSEQR ====
399: *
400: END
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