Annotation of rpl/lapack/lapack/zlaqr3.f, revision 1.3
1.1 bertrand 1: SUBROUTINE ZLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
2: $ IHIZ, Z, LDZ, NS, ND, SH, V, LDV, NH, T, LDT,
3: $ NV, WV, LDWV, WORK, LWORK )
4: *
5: * -- LAPACK auxiliary routine (version 3.2.1) --
6: * Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..
7: * -- April 2009 --
8: *
9: * .. Scalar Arguments ..
10: INTEGER IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
11: $ LDZ, LWORK, N, ND, NH, NS, NV, NW
12: LOGICAL WANTT, WANTZ
13: * ..
14: * .. Array Arguments ..
15: COMPLEX*16 H( LDH, * ), SH( * ), T( LDT, * ), V( LDV, * ),
16: $ WORK( * ), WV( LDWV, * ), Z( LDZ, * )
17: * ..
18: *
19: * ******************************************************************
20: * Aggressive early deflation:
21: *
22: * This subroutine accepts as input an upper Hessenberg matrix
23: * H and performs an unitary similarity transformation
24: * designed to detect and deflate fully converged eigenvalues from
25: * a trailing principal submatrix. On output H has been over-
26: * written by a new Hessenberg matrix that is a perturbation of
27: * an unitary similarity transformation of H. It is to be
28: * hoped that the final version of H has many zero subdiagonal
29: * entries.
30: *
31: * ******************************************************************
32: * WANTT (input) LOGICAL
33: * If .TRUE., then the Hessenberg matrix H is fully updated
34: * so that the triangular Schur factor may be
35: * computed (in cooperation with the calling subroutine).
36: * If .FALSE., then only enough of H is updated to preserve
37: * the eigenvalues.
38: *
39: * WANTZ (input) LOGICAL
40: * If .TRUE., then the unitary matrix Z is updated so
41: * so that the unitary Schur factor may be computed
42: * (in cooperation with the calling subroutine).
43: * If .FALSE., then Z is not referenced.
44: *
45: * N (input) INTEGER
46: * The order of the matrix H and (if WANTZ is .TRUE.) the
47: * order of the unitary matrix Z.
48: *
49: * KTOP (input) INTEGER
50: * It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
51: * KBOT and KTOP together determine an isolated block
52: * along the diagonal of the Hessenberg matrix.
53: *
54: * KBOT (input) INTEGER
55: * It is assumed without a check that either
56: * KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together
57: * determine an isolated block along the diagonal of the
58: * Hessenberg matrix.
59: *
60: * NW (input) INTEGER
61: * Deflation window size. 1 .LE. NW .LE. (KBOT-KTOP+1).
62: *
63: * H (input/output) COMPLEX*16 array, dimension (LDH,N)
64: * On input the initial N-by-N section of H stores the
65: * Hessenberg matrix undergoing aggressive early deflation.
66: * On output H has been transformed by a unitary
67: * similarity transformation, perturbed, and the returned
68: * to Hessenberg form that (it is to be hoped) has some
69: * zero subdiagonal entries.
70: *
71: * LDH (input) integer
72: * Leading dimension of H just as declared in the calling
73: * subroutine. N .LE. LDH
74: *
75: * ILOZ (input) INTEGER
76: * IHIZ (input) INTEGER
77: * Specify the rows of Z to which transformations must be
78: * applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.
79: *
80: * Z (input/output) COMPLEX*16 array, dimension (LDZ,N)
81: * IF WANTZ is .TRUE., then on output, the unitary
82: * similarity transformation mentioned above has been
83: * accumulated into Z(ILOZ:IHIZ,ILO:IHI) from the right.
84: * If WANTZ is .FALSE., then Z is unreferenced.
85: *
86: * LDZ (input) integer
87: * The leading dimension of Z just as declared in the
88: * calling subroutine. 1 .LE. LDZ.
89: *
90: * NS (output) integer
91: * The number of unconverged (ie approximate) eigenvalues
92: * returned in SR and SI that may be used as shifts by the
93: * calling subroutine.
94: *
95: * ND (output) integer
96: * The number of converged eigenvalues uncovered by this
97: * subroutine.
98: *
99: * SH (output) COMPLEX*16 array, dimension KBOT
100: * On output, approximate eigenvalues that may
101: * be used for shifts are stored in SH(KBOT-ND-NS+1)
102: * through SR(KBOT-ND). Converged eigenvalues are
103: * stored in SH(KBOT-ND+1) through SH(KBOT).
104: *
105: * V (workspace) COMPLEX*16 array, dimension (LDV,NW)
106: * An NW-by-NW work array.
107: *
108: * LDV (input) integer scalar
109: * The leading dimension of V just as declared in the
110: * calling subroutine. NW .LE. LDV
111: *
112: * NH (input) integer scalar
113: * The number of columns of T. NH.GE.NW.
114: *
115: * T (workspace) COMPLEX*16 array, dimension (LDT,NW)
116: *
117: * LDT (input) integer
118: * The leading dimension of T just as declared in the
119: * calling subroutine. NW .LE. LDT
120: *
121: * NV (input) integer
122: * The number of rows of work array WV available for
123: * workspace. NV.GE.NW.
124: *
125: * WV (workspace) COMPLEX*16 array, dimension (LDWV,NW)
126: *
127: * LDWV (input) integer
128: * The leading dimension of W just as declared in the
129: * calling subroutine. NW .LE. LDV
130: *
131: * WORK (workspace) COMPLEX*16 array, dimension LWORK.
132: * On exit, WORK(1) is set to an estimate of the optimal value
133: * of LWORK for the given values of N, NW, KTOP and KBOT.
134: *
135: * LWORK (input) integer
136: * The dimension of the work array WORK. LWORK = 2*NW
137: * suffices, but greater efficiency may result from larger
138: * values of LWORK.
139: *
140: * If LWORK = -1, then a workspace query is assumed; ZLAQR3
141: * only estimates the optimal workspace size for the given
142: * values of N, NW, KTOP and KBOT. The estimate is returned
143: * in WORK(1). No error message related to LWORK is issued
144: * by XERBLA. Neither H nor Z are accessed.
145: *
146: * ================================================================
147: * Based on contributions by
148: * Karen Braman and Ralph Byers, Department of Mathematics,
149: * University of Kansas, USA
150: *
151: * ================================================================
152: * .. Parameters ..
153: COMPLEX*16 ZERO, ONE
154: PARAMETER ( ZERO = ( 0.0d0, 0.0d0 ),
155: $ ONE = ( 1.0d0, 0.0d0 ) )
156: DOUBLE PRECISION RZERO, RONE
157: PARAMETER ( RZERO = 0.0d0, RONE = 1.0d0 )
158: * ..
159: * .. Local Scalars ..
160: COMPLEX*16 BETA, CDUM, S, TAU
161: DOUBLE PRECISION FOO, SAFMAX, SAFMIN, SMLNUM, ULP
162: INTEGER I, IFST, ILST, INFO, INFQR, J, JW, KCOL, KLN,
163: $ KNT, KROW, KWTOP, LTOP, LWK1, LWK2, LWK3,
164: $ LWKOPT, NMIN
165: * ..
166: * .. External Functions ..
167: DOUBLE PRECISION DLAMCH
168: INTEGER ILAENV
169: EXTERNAL DLAMCH, ILAENV
170: * ..
171: * .. External Subroutines ..
172: EXTERNAL DLABAD, ZCOPY, ZGEHRD, ZGEMM, ZLACPY, ZLAHQR,
173: $ ZLAQR4, ZLARF, ZLARFG, ZLASET, ZTREXC, ZUNMHR
174: * ..
175: * .. Intrinsic Functions ..
176: INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, INT, MAX, MIN
177: * ..
178: * .. Statement Functions ..
179: DOUBLE PRECISION CABS1
180: * ..
181: * .. Statement Function definitions ..
182: CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
183: * ..
184: * .. Executable Statements ..
185: *
186: * ==== Estimate optimal workspace. ====
187: *
188: JW = MIN( NW, KBOT-KTOP+1 )
189: IF( JW.LE.2 ) THEN
190: LWKOPT = 1
191: ELSE
192: *
193: * ==== Workspace query call to ZGEHRD ====
194: *
195: CALL ZGEHRD( JW, 1, JW-1, T, LDT, WORK, WORK, -1, INFO )
196: LWK1 = INT( WORK( 1 ) )
197: *
198: * ==== Workspace query call to ZUNMHR ====
199: *
200: CALL ZUNMHR( 'R', 'N', JW, JW, 1, JW-1, T, LDT, WORK, V, LDV,
201: $ WORK, -1, INFO )
202: LWK2 = INT( WORK( 1 ) )
203: *
204: * ==== Workspace query call to ZLAQR4 ====
205: *
206: CALL ZLAQR4( .true., .true., JW, 1, JW, T, LDT, SH, 1, JW, V,
207: $ LDV, WORK, -1, INFQR )
208: LWK3 = INT( WORK( 1 ) )
209: *
210: * ==== Optimal workspace ====
211: *
212: LWKOPT = MAX( JW+MAX( LWK1, LWK2 ), LWK3 )
213: END IF
214: *
215: * ==== Quick return in case of workspace query. ====
216: *
217: IF( LWORK.EQ.-1 ) THEN
218: WORK( 1 ) = DCMPLX( LWKOPT, 0 )
219: RETURN
220: END IF
221: *
222: * ==== Nothing to do ...
223: * ... for an empty active block ... ====
224: NS = 0
225: ND = 0
226: WORK( 1 ) = ONE
227: IF( KTOP.GT.KBOT )
228: $ RETURN
229: * ... nor for an empty deflation window. ====
230: IF( NW.LT.1 )
231: $ RETURN
232: *
233: * ==== Machine constants ====
234: *
235: SAFMIN = DLAMCH( 'SAFE MINIMUM' )
236: SAFMAX = RONE / SAFMIN
237: CALL DLABAD( SAFMIN, SAFMAX )
238: ULP = DLAMCH( 'PRECISION' )
239: SMLNUM = SAFMIN*( DBLE( N ) / ULP )
240: *
241: * ==== Setup deflation window ====
242: *
243: JW = MIN( NW, KBOT-KTOP+1 )
244: KWTOP = KBOT - JW + 1
245: IF( KWTOP.EQ.KTOP ) THEN
246: S = ZERO
247: ELSE
248: S = H( KWTOP, KWTOP-1 )
249: END IF
250: *
251: IF( KBOT.EQ.KWTOP ) THEN
252: *
253: * ==== 1-by-1 deflation window: not much to do ====
254: *
255: SH( KWTOP ) = H( KWTOP, KWTOP )
256: NS = 1
257: ND = 0
258: IF( CABS1( S ).LE.MAX( SMLNUM, ULP*CABS1( H( KWTOP,
259: $ KWTOP ) ) ) ) THEN
260: NS = 0
261: ND = 1
262: IF( KWTOP.GT.KTOP )
263: $ H( KWTOP, KWTOP-1 ) = ZERO
264: END IF
265: WORK( 1 ) = ONE
266: RETURN
267: END IF
268: *
269: * ==== Convert to spike-triangular form. (In case of a
270: * . rare QR failure, this routine continues to do
271: * . aggressive early deflation using that part of
272: * . the deflation window that converged using INFQR
273: * . here and there to keep track.) ====
274: *
275: CALL ZLACPY( 'U', JW, JW, H( KWTOP, KWTOP ), LDH, T, LDT )
276: CALL ZCOPY( JW-1, H( KWTOP+1, KWTOP ), LDH+1, T( 2, 1 ), LDT+1 )
277: *
278: CALL ZLASET( 'A', JW, JW, ZERO, ONE, V, LDV )
279: NMIN = ILAENV( 12, 'ZLAQR3', 'SV', JW, 1, JW, LWORK )
280: IF( JW.GT.NMIN ) THEN
281: CALL ZLAQR4( .true., .true., JW, 1, JW, T, LDT, SH( KWTOP ), 1,
282: $ JW, V, LDV, WORK, LWORK, INFQR )
283: ELSE
284: CALL ZLAHQR( .true., .true., JW, 1, JW, T, LDT, SH( KWTOP ), 1,
285: $ JW, V, LDV, INFQR )
286: END IF
287: *
288: * ==== Deflation detection loop ====
289: *
290: NS = JW
291: ILST = INFQR + 1
292: DO 10 KNT = INFQR + 1, JW
293: *
294: * ==== Small spike tip deflation test ====
295: *
296: FOO = CABS1( T( NS, NS ) )
297: IF( FOO.EQ.RZERO )
298: $ FOO = CABS1( S )
299: IF( CABS1( S )*CABS1( V( 1, NS ) ).LE.MAX( SMLNUM, ULP*FOO ) )
300: $ THEN
301: *
302: * ==== One more converged eigenvalue ====
303: *
304: NS = NS - 1
305: ELSE
306: *
307: * ==== One undeflatable eigenvalue. Move it up out of the
308: * . way. (ZTREXC can not fail in this case.) ====
309: *
310: IFST = NS
311: CALL ZTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, INFO )
312: ILST = ILST + 1
313: END IF
314: 10 CONTINUE
315: *
316: * ==== Return to Hessenberg form ====
317: *
318: IF( NS.EQ.0 )
319: $ S = ZERO
320: *
321: IF( NS.LT.JW ) THEN
322: *
323: * ==== sorting the diagonal of T improves accuracy for
324: * . graded matrices. ====
325: *
326: DO 30 I = INFQR + 1, NS
327: IFST = I
328: DO 20 J = I + 1, NS
329: IF( CABS1( T( J, J ) ).GT.CABS1( T( IFST, IFST ) ) )
330: $ IFST = J
331: 20 CONTINUE
332: ILST = I
333: IF( IFST.NE.ILST )
334: $ CALL ZTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, INFO )
335: 30 CONTINUE
336: END IF
337: *
338: * ==== Restore shift/eigenvalue array from T ====
339: *
340: DO 40 I = INFQR + 1, JW
341: SH( KWTOP+I-1 ) = T( I, I )
342: 40 CONTINUE
343: *
344: *
345: IF( NS.LT.JW .OR. S.EQ.ZERO ) THEN
346: IF( NS.GT.1 .AND. S.NE.ZERO ) THEN
347: *
348: * ==== Reflect spike back into lower triangle ====
349: *
350: CALL ZCOPY( NS, V, LDV, WORK, 1 )
351: DO 50 I = 1, NS
352: WORK( I ) = DCONJG( WORK( I ) )
353: 50 CONTINUE
354: BETA = WORK( 1 )
355: CALL ZLARFG( NS, BETA, WORK( 2 ), 1, TAU )
356: WORK( 1 ) = ONE
357: *
358: CALL ZLASET( 'L', JW-2, JW-2, ZERO, ZERO, T( 3, 1 ), LDT )
359: *
360: CALL ZLARF( 'L', NS, JW, WORK, 1, DCONJG( TAU ), T, LDT,
361: $ WORK( JW+1 ) )
362: CALL ZLARF( 'R', NS, NS, WORK, 1, TAU, T, LDT,
363: $ WORK( JW+1 ) )
364: CALL ZLARF( 'R', JW, NS, WORK, 1, TAU, V, LDV,
365: $ WORK( JW+1 ) )
366: *
367: CALL ZGEHRD( JW, 1, NS, T, LDT, WORK, WORK( JW+1 ),
368: $ LWORK-JW, INFO )
369: END IF
370: *
371: * ==== Copy updated reduced window into place ====
372: *
373: IF( KWTOP.GT.1 )
374: $ H( KWTOP, KWTOP-1 ) = S*DCONJG( V( 1, 1 ) )
375: CALL ZLACPY( 'U', JW, JW, T, LDT, H( KWTOP, KWTOP ), LDH )
376: CALL ZCOPY( JW-1, T( 2, 1 ), LDT+1, H( KWTOP+1, KWTOP ),
377: $ LDH+1 )
378: *
379: * ==== Accumulate orthogonal matrix in order update
380: * . H and Z, if requested. ====
381: *
382: IF( NS.GT.1 .AND. S.NE.ZERO )
383: $ CALL ZUNMHR( 'R', 'N', JW, NS, 1, NS, T, LDT, WORK, V, LDV,
384: $ WORK( JW+1 ), LWORK-JW, INFO )
385: *
386: * ==== Update vertical slab in H ====
387: *
388: IF( WANTT ) THEN
389: LTOP = 1
390: ELSE
391: LTOP = KTOP
392: END IF
393: DO 60 KROW = LTOP, KWTOP - 1, NV
394: KLN = MIN( NV, KWTOP-KROW )
395: CALL ZGEMM( 'N', 'N', KLN, JW, JW, ONE, H( KROW, KWTOP ),
396: $ LDH, V, LDV, ZERO, WV, LDWV )
397: CALL ZLACPY( 'A', KLN, JW, WV, LDWV, H( KROW, KWTOP ), LDH )
398: 60 CONTINUE
399: *
400: * ==== Update horizontal slab in H ====
401: *
402: IF( WANTT ) THEN
403: DO 70 KCOL = KBOT + 1, N, NH
404: KLN = MIN( NH, N-KCOL+1 )
405: CALL ZGEMM( 'C', 'N', JW, KLN, JW, ONE, V, LDV,
406: $ H( KWTOP, KCOL ), LDH, ZERO, T, LDT )
407: CALL ZLACPY( 'A', JW, KLN, T, LDT, H( KWTOP, KCOL ),
408: $ LDH )
409: 70 CONTINUE
410: END IF
411: *
412: * ==== Update vertical slab in Z ====
413: *
414: IF( WANTZ ) THEN
415: DO 80 KROW = ILOZ, IHIZ, NV
416: KLN = MIN( NV, IHIZ-KROW+1 )
417: CALL ZGEMM( 'N', 'N', KLN, JW, JW, ONE, Z( KROW, KWTOP ),
418: $ LDZ, V, LDV, ZERO, WV, LDWV )
419: CALL ZLACPY( 'A', KLN, JW, WV, LDWV, Z( KROW, KWTOP ),
420: $ LDZ )
421: 80 CONTINUE
422: END IF
423: END IF
424: *
425: * ==== Return the number of deflations ... ====
426: *
427: ND = JW - NS
428: *
429: * ==== ... and the number of shifts. (Subtracting
430: * . INFQR from the spike length takes care
431: * . of the case of a rare QR failure while
432: * . calculating eigenvalues of the deflation
433: * . window.) ====
434: *
435: NS = NS - INFQR
436: *
437: * ==== Return optimal workspace. ====
438: *
439: WORK( 1 ) = DCMPLX( LWKOPT, 0 )
440: *
441: * ==== End of ZLAQR3 ====
442: *
443: END
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