Annotation of rpl/lapack/lapack/zgsvj1.f, revision 1.5
1.2 bertrand 1: *> \brief \b ZGSVJ1 pre-processor for the routine zgesvj, applies Jacobi rotations targeting only particular pivots.
1.1 bertrand 2: *
3: * =========== DOCUMENTATION ===========
4: *
1.4 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.1 bertrand 7: *
8: *> \htmlonly
1.4 bertrand 9: *> Download ZGSVJ1 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgsvj1.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgsvj1.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgsvj1.f">
1.1 bertrand 15: *> [TXT]</a>
1.4 bertrand 16: *> \endhtmlonly
1.1 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZGSVJ1( JOBV, M, N, N1, A, LDA, D, SVA, MV, V, LDV,
22: * EPS, SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
1.4 bertrand 23: *
1.1 bertrand 24: * .. Scalar Arguments ..
25: * DOUBLE PRECISION EPS, SFMIN, TOL
26: * INTEGER INFO, LDA, LDV, LWORK, M, MV, N, N1, NSWEEP
27: * CHARACTER*1 JOBV
28: * ..
29: * .. Array Arguments ..
1.4 bertrand 30: * COMPLEX*16 A( LDA, * ), D( N ), V( LDV, * ), WORK( LWORK )
31: * DOUBLE PRECISION SVA( N )
1.1 bertrand 32: * ..
1.4 bertrand 33: *
1.1 bertrand 34: *
35: *> \par Purpose:
36: * =============
37: *>
38: *> \verbatim
39: *>
40: *> ZGSVJ1 is called from ZGESVJ as a pre-processor and that is its main
41: *> purpose. It applies Jacobi rotations in the same way as ZGESVJ does, but
42: *> it targets only particular pivots and it does not check convergence
43: *> (stopping criterion). Few tunning parameters (marked by [TP]) are
44: *> available for the implementer.
45: *>
46: *> Further Details
47: *> ~~~~~~~~~~~~~~~
48: *> ZGSVJ1 applies few sweeps of Jacobi rotations in the column space of
49: *> the input M-by-N matrix A. The pivot pairs are taken from the (1,2)
50: *> off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The
51: *> block-entries (tiles) of the (1,2) off-diagonal block are marked by the
52: *> [x]'s in the following scheme:
53: *>
54: *> | * * * [x] [x] [x]|
55: *> | * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x] blocks.
56: *> | * * * [x] [x] [x]| Row-cyclic pivoting inside each [x] block.
57: *> |[x] [x] [x] * * * |
58: *> |[x] [x] [x] * * * |
59: *> |[x] [x] [x] * * * |
60: *>
61: *> In terms of the columns of A, the first N1 columns are rotated 'against'
62: *> the remaining N-N1 columns, trying to increase the angle between the
63: *> corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is
64: *> tiled using quadratic tiles of side KBL. Here, KBL is a tunning parmeter.
65: *> The number of sweeps is given in NSWEEP and the orthogonality threshold
66: *> is given in TOL.
67: *> \endverbatim
68: *
69: * Arguments:
70: * ==========
71: *
72: *> \param[in] JOBV
73: *> \verbatim
74: *> JOBV is CHARACTER*1
75: *> Specifies whether the output from this procedure is used
76: *> to compute the matrix V:
77: *> = 'V': the product of the Jacobi rotations is accumulated
78: *> by postmulyiplying the N-by-N array V.
79: *> (See the description of V.)
80: *> = 'A': the product of the Jacobi rotations is accumulated
81: *> by postmulyiplying the MV-by-N array V.
82: *> (See the descriptions of MV and V.)
83: *> = 'N': the Jacobi rotations are not accumulated.
84: *> \endverbatim
85: *>
86: *> \param[in] M
87: *> \verbatim
88: *> M is INTEGER
89: *> The number of rows of the input matrix A. M >= 0.
90: *> \endverbatim
91: *>
92: *> \param[in] N
93: *> \verbatim
94: *> N is INTEGER
95: *> The number of columns of the input matrix A.
96: *> M >= N >= 0.
97: *> \endverbatim
98: *>
99: *> \param[in] N1
100: *> \verbatim
101: *> N1 is INTEGER
102: *> N1 specifies the 2 x 2 block partition, the first N1 columns are
103: *> rotated 'against' the remaining N-N1 columns of A.
104: *> \endverbatim
105: *>
106: *> \param[in,out] A
107: *> \verbatim
1.2 bertrand 108: *> A is COMPLEX*16 array, dimension (LDA,N)
1.1 bertrand 109: *> On entry, M-by-N matrix A, such that A*diag(D) represents
110: *> the input matrix.
111: *> On exit,
112: *> A_onexit * D_onexit represents the input matrix A*diag(D)
113: *> post-multiplied by a sequence of Jacobi rotations, where the
114: *> rotation threshold and the total number of sweeps are given in
115: *> TOL and NSWEEP, respectively.
116: *> (See the descriptions of N1, D, TOL and NSWEEP.)
117: *> \endverbatim
118: *>
119: *> \param[in] LDA
120: *> \verbatim
121: *> LDA is INTEGER
122: *> The leading dimension of the array A. LDA >= max(1,M).
123: *> \endverbatim
124: *>
125: *> \param[in,out] D
126: *> \verbatim
1.2 bertrand 127: *> D is COMPLEX*16 array, dimension (N)
1.1 bertrand 128: *> The array D accumulates the scaling factors from the fast scaled
129: *> Jacobi rotations.
130: *> On entry, A*diag(D) represents the input matrix.
131: *> On exit, A_onexit*diag(D_onexit) represents the input matrix
132: *> post-multiplied by a sequence of Jacobi rotations, where the
133: *> rotation threshold and the total number of sweeps are given in
134: *> TOL and NSWEEP, respectively.
135: *> (See the descriptions of N1, A, TOL and NSWEEP.)
136: *> \endverbatim
137: *>
138: *> \param[in,out] SVA
139: *> \verbatim
140: *> SVA is DOUBLE PRECISION array, dimension (N)
141: *> On entry, SVA contains the Euclidean norms of the columns of
142: *> the matrix A*diag(D).
143: *> On exit, SVA contains the Euclidean norms of the columns of
144: *> the matrix onexit*diag(D_onexit).
145: *> \endverbatim
146: *>
147: *> \param[in] MV
148: *> \verbatim
149: *> MV is INTEGER
150: *> If JOBV .EQ. 'A', then MV rows of V are post-multipled by a
151: *> sequence of Jacobi rotations.
152: *> If JOBV = 'N', then MV is not referenced.
153: *> \endverbatim
154: *>
155: *> \param[in,out] V
156: *> \verbatim
1.2 bertrand 157: *> V is COMPLEX*16 array, dimension (LDV,N)
1.1 bertrand 158: *> If JOBV .EQ. 'V' then N rows of V are post-multipled by a
159: *> sequence of Jacobi rotations.
160: *> If JOBV .EQ. 'A' then MV rows of V are post-multipled by a
161: *> sequence of Jacobi rotations.
162: *> If JOBV = 'N', then V is not referenced.
163: *> \endverbatim
164: *>
165: *> \param[in] LDV
166: *> \verbatim
167: *> LDV is INTEGER
168: *> The leading dimension of the array V, LDV >= 1.
169: *> If JOBV = 'V', LDV .GE. N.
170: *> If JOBV = 'A', LDV .GE. MV.
171: *> \endverbatim
172: *>
173: *> \param[in] EPS
174: *> \verbatim
175: *> EPS is DOUBLE PRECISION
176: *> EPS = DLAMCH('Epsilon')
177: *> \endverbatim
178: *>
179: *> \param[in] SFMIN
180: *> \verbatim
181: *> SFMIN is DOUBLE PRECISION
182: *> SFMIN = DLAMCH('Safe Minimum')
183: *> \endverbatim
184: *>
185: *> \param[in] TOL
186: *> \verbatim
187: *> TOL is DOUBLE PRECISION
188: *> TOL is the threshold for Jacobi rotations. For a pair
189: *> A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
190: *> applied only if ABS(COS(angle(A(:,p),A(:,q)))) .GT. TOL.
191: *> \endverbatim
192: *>
193: *> \param[in] NSWEEP
194: *> \verbatim
195: *> NSWEEP is INTEGER
196: *> NSWEEP is the number of sweeps of Jacobi rotations to be
197: *> performed.
198: *> \endverbatim
199: *>
200: *> \param[out] WORK
201: *> \verbatim
1.2 bertrand 202: *> WORK is COMPLEX*16 array, dimension (LWORK)
1.1 bertrand 203: *> \endverbatim
204: *>
205: *> \param[in] LWORK
206: *> \verbatim
207: *> LWORK is INTEGER
208: *> LWORK is the dimension of WORK. LWORK .GE. M.
209: *> \endverbatim
210: *>
211: *> \param[out] INFO
212: *> \verbatim
213: *> INFO is INTEGER
214: *> = 0 : successful exit.
215: *> < 0 : if INFO = -i, then the i-th argument had an illegal value
216: *> \endverbatim
217: *
218: * Authors:
219: * ========
220: *
1.4 bertrand 221: *> \author Univ. of Tennessee
222: *> \author Univ. of California Berkeley
223: *> \author Univ. of Colorado Denver
224: *> \author NAG Ltd.
1.1 bertrand 225: *
1.2 bertrand 226: *> \date June 2016
1.1 bertrand 227: *
228: *> \ingroup complex16OTHERcomputational
229: *
1.4 bertrand 230: *> \par Contributor:
1.1 bertrand 231: * ==================
232: *>
1.4 bertrand 233: *> Zlatko Drmac (Zagreb, Croatia)
1.1 bertrand 234: *
235: * =====================================================================
236: SUBROUTINE ZGSVJ1( JOBV, M, N, N1, A, LDA, D, SVA, MV, V, LDV,
237: $ EPS, SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
238: *
1.4 bertrand 239: * -- LAPACK computational routine (version 3.7.0) --
1.1 bertrand 240: * -- LAPACK is a software package provided by Univ. of Tennessee, --
241: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.2 bertrand 242: * June 2016
1.1 bertrand 243: *
1.4 bertrand 244: IMPLICIT NONE
1.1 bertrand 245: * .. Scalar Arguments ..
246: DOUBLE PRECISION EPS, SFMIN, TOL
247: INTEGER INFO, LDA, LDV, LWORK, M, MV, N, N1, NSWEEP
248: CHARACTER*1 JOBV
249: * ..
250: * .. Array Arguments ..
251: COMPLEX*16 A( LDA, * ), D( N ), V( LDV, * ), WORK( LWORK )
1.4 bertrand 252: DOUBLE PRECISION SVA( N )
1.1 bertrand 253: * ..
254: *
255: * =====================================================================
256: *
257: * .. Local Parameters ..
258: DOUBLE PRECISION ZERO, HALF, ONE
259: PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0)
260: * ..
261: * .. Local Scalars ..
262: COMPLEX*16 AAPQ, OMPQ
263: DOUBLE PRECISION AAPP, AAPP0, AAPQ1, AAQQ, APOAQ, AQOAP, BIG,
1.4 bertrand 264: $ BIGTHETA, CS, MXAAPQ, MXSINJ, ROOTBIG,
1.1 bertrand 265: $ ROOTEPS, ROOTSFMIN, ROOTTOL, SMALL, SN, T,
266: $ TEMP1, THETA, THSIGN
267: INTEGER BLSKIP, EMPTSW, i, ibr, igl, IERR, IJBLSK,
268: $ ISWROT, jbc, jgl, KBL, MVL, NOTROT, nblc, nblr,
269: $ p, PSKIPPED, q, ROWSKIP, SWBAND
270: LOGICAL APPLV, ROTOK, RSVEC
271: * ..
272: * ..
273: * .. Intrinsic Functions ..
1.4 bertrand 274: INTRINSIC ABS, CONJG, MAX, DBLE, MIN, SIGN, SQRT
1.1 bertrand 275: * ..
276: * .. External Functions ..
277: DOUBLE PRECISION DZNRM2
278: COMPLEX*16 ZDOTC
279: INTEGER IDAMAX
280: LOGICAL LSAME
281: EXTERNAL IDAMAX, LSAME, ZDOTC, DZNRM2
282: * ..
283: * .. External Subroutines ..
1.4 bertrand 284: * .. from BLAS
1.1 bertrand 285: EXTERNAL ZCOPY, ZROT, ZSWAP
286: * .. from LAPACK
287: EXTERNAL ZLASCL, ZLASSQ, XERBLA
288: * ..
289: * .. Executable Statements ..
290: *
291: * Test the input parameters.
292: *
293: APPLV = LSAME( JOBV, 'A' )
294: RSVEC = LSAME( JOBV, 'V' )
295: IF( .NOT.( RSVEC .OR. APPLV .OR. LSAME( JOBV, 'N' ) ) ) THEN
296: INFO = -1
297: ELSE IF( M.LT.0 ) THEN
298: INFO = -2
299: ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN
300: INFO = -3
301: ELSE IF( N1.LT.0 ) THEN
302: INFO = -4
303: ELSE IF( LDA.LT.M ) THEN
304: INFO = -6
305: ELSE IF( ( RSVEC.OR.APPLV ) .AND. ( MV.LT.0 ) ) THEN
306: INFO = -9
1.4 bertrand 307: ELSE IF( ( RSVEC.AND.( LDV.LT.N ) ).OR.
1.1 bertrand 308: $ ( APPLV.AND.( LDV.LT.MV ) ) ) THEN
309: INFO = -11
310: ELSE IF( TOL.LE.EPS ) THEN
311: INFO = -14
312: ELSE IF( NSWEEP.LT.0 ) THEN
313: INFO = -15
314: ELSE IF( LWORK.LT.M ) THEN
315: INFO = -17
316: ELSE
317: INFO = 0
318: END IF
319: *
320: * #:(
321: IF( INFO.NE.0 ) THEN
322: CALL XERBLA( 'ZGSVJ1', -INFO )
323: RETURN
324: END IF
325: *
326: IF( RSVEC ) THEN
327: MVL = N
328: ELSE IF( APPLV ) THEN
329: MVL = MV
330: END IF
331: RSVEC = RSVEC .OR. APPLV
332:
1.4 bertrand 333: ROOTEPS = SQRT( EPS )
334: ROOTSFMIN = SQRT( SFMIN )
1.1 bertrand 335: SMALL = SFMIN / EPS
336: BIG = ONE / SFMIN
337: ROOTBIG = ONE / ROOTSFMIN
1.4 bertrand 338: * LARGE = BIG / SQRT( DBLE( M*N ) )
1.1 bertrand 339: BIGTHETA = ONE / ROOTEPS
1.4 bertrand 340: ROOTTOL = SQRT( TOL )
1.1 bertrand 341: *
342: * .. Initialize the right singular vector matrix ..
343: *
344: * RSVEC = LSAME( JOBV, 'Y' )
345: *
346: EMPTSW = N1*( N-N1 )
347: NOTROT = 0
348: *
349: * .. Row-cyclic pivot strategy with de Rijk's pivoting ..
350: *
1.4 bertrand 351: KBL = MIN( 8, N )
1.1 bertrand 352: NBLR = N1 / KBL
353: IF( ( NBLR*KBL ).NE.N1 )NBLR = NBLR + 1
354:
355: * .. the tiling is nblr-by-nblc [tiles]
356:
357: NBLC = ( N-N1 ) / KBL
358: IF( ( NBLC*KBL ).NE.( N-N1 ) )NBLC = NBLC + 1
359: BLSKIP = ( KBL**2 ) + 1
360: *[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.
361:
1.4 bertrand 362: ROWSKIP = MIN( 5, KBL )
1.1 bertrand 363: *[TP] ROWSKIP is a tuning parameter.
364: SWBAND = 0
365: *[TP] SWBAND is a tuning parameter. It is meaningful and effective
366: * if ZGESVJ is used as a computational routine in the preconditioned
367: * Jacobi SVD algorithm ZGEJSV.
368: *
369: *
370: * | * * * [x] [x] [x]|
371: * | * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x] blocks.
372: * | * * * [x] [x] [x]| Row-cyclic pivoting inside each [x] block.
373: * |[x] [x] [x] * * * |
374: * |[x] [x] [x] * * * |
375: * |[x] [x] [x] * * * |
376: *
377: *
378: DO 1993 i = 1, NSWEEP
379: *
380: * .. go go go ...
381: *
382: MXAAPQ = ZERO
383: MXSINJ = ZERO
384: ISWROT = 0
385: *
386: NOTROT = 0
387: PSKIPPED = 0
388: *
389: * Each sweep is unrolled using KBL-by-KBL tiles over the pivot pairs
390: * 1 <= p < q <= N. This is the first step toward a blocked implementation
391: * of the rotations. New implementation, based on block transformations,
392: * is under development.
393: *
394: DO 2000 ibr = 1, NBLR
395: *
396: igl = ( ibr-1 )*KBL + 1
397: *
398:
399: *
400: * ... go to the off diagonal blocks
401: *
402: igl = ( ibr-1 )*KBL + 1
403: *
404: * DO 2010 jbc = ibr + 1, NBL
1.4 bertrand 405: DO 2010 jbc = 1, NBLC
1.1 bertrand 406: *
407: jgl = ( jbc-1 )*KBL + N1 + 1
408: *
409: * doing the block at ( ibr, jbc )
410: *
411: IJBLSK = 0
1.4 bertrand 412: DO 2100 p = igl, MIN( igl+KBL-1, N1 )
1.1 bertrand 413: *
414: AAPP = SVA( p )
415: IF( AAPP.GT.ZERO ) THEN
416: *
417: PSKIPPED = 0
418: *
1.4 bertrand 419: DO 2200 q = jgl, MIN( jgl+KBL-1, N )
1.1 bertrand 420: *
421: AAQQ = SVA( q )
422: IF( AAQQ.GT.ZERO ) THEN
423: AAPP0 = AAPP
424: *
425: * .. M x 2 Jacobi SVD ..
426: *
427: * Safe Gram matrix computation
428: *
429: IF( AAQQ.GE.ONE ) THEN
430: IF( AAPP.GE.AAQQ ) THEN
431: ROTOK = ( SMALL*AAPP ).LE.AAQQ
432: ELSE
433: ROTOK = ( SMALL*AAQQ ).LE.AAPP
434: END IF
435: IF( AAPP.LT.( BIG / AAQQ ) ) THEN
1.4 bertrand 436: AAPQ = ( ZDOTC( M, A( 1, p ), 1,
1.1 bertrand 437: $ A( 1, q ), 1 ) / AAQQ ) / AAPP
438: ELSE
439: CALL ZCOPY( M, A( 1, p ), 1,
440: $ WORK, 1 )
441: CALL ZLASCL( 'G', 0, 0, AAPP,
442: $ ONE, M, 1,
443: $ WORK, LDA, IERR )
444: AAPQ = ZDOTC( M, WORK, 1,
445: $ A( 1, q ), 1 ) / AAQQ
446: END IF
447: ELSE
448: IF( AAPP.GE.AAQQ ) THEN
449: ROTOK = AAPP.LE.( AAQQ / SMALL )
450: ELSE
451: ROTOK = AAQQ.LE.( AAPP / SMALL )
452: END IF
453: IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
1.4 bertrand 454: AAPQ = ( ZDOTC( M, A( 1, p ), 1,
455: $ A( 1, q ), 1 ) / MAX(AAQQ,AAPP) )
456: $ / MIN(AAQQ,AAPP)
1.1 bertrand 457: ELSE
458: CALL ZCOPY( M, A( 1, q ), 1,
459: $ WORK, 1 )
460: CALL ZLASCL( 'G', 0, 0, AAQQ,
461: $ ONE, M, 1,
462: $ WORK, LDA, IERR )
463: AAPQ = ZDOTC( M, A( 1, p ), 1,
464: $ WORK, 1 ) / AAPP
465: END IF
466: END IF
467: *
1.4 bertrand 468: * AAPQ = AAPQ * CONJG(CWORK(p))*CWORK(q)
1.1 bertrand 469: AAPQ1 = -ABS(AAPQ)
1.4 bertrand 470: MXAAPQ = MAX( MXAAPQ, -AAPQ1 )
1.1 bertrand 471: *
472: * TO rotate or NOT to rotate, THAT is the question ...
473: *
474: IF( ABS( AAPQ1 ).GT.TOL ) THEN
1.4 bertrand 475: OMPQ = AAPQ / ABS(AAPQ)
1.1 bertrand 476: NOTROT = 0
477: *[RTD] ROTATED = ROTATED + 1
478: PSKIPPED = 0
479: ISWROT = ISWROT + 1
480: *
481: IF( ROTOK ) THEN
482: *
483: AQOAP = AAQQ / AAPP
484: APOAQ = AAPP / AAQQ
485: THETA = -HALF*ABS( AQOAP-APOAQ )/ AAPQ1
486: IF( AAQQ.GT.AAPP0 )THETA = -THETA
487: *
488: IF( ABS( THETA ).GT.BIGTHETA ) THEN
489: T = HALF / THETA
1.4 bertrand 490: CS = ONE
1.1 bertrand 491: CALL ZROT( M, A(1,p), 1, A(1,q), 1,
1.4 bertrand 492: $ CS, CONJG(OMPQ)*T )
1.1 bertrand 493: IF( RSVEC ) THEN
1.4 bertrand 494: CALL ZROT( MVL, V(1,p), 1,
495: $ V(1,q), 1, CS, CONJG(OMPQ)*T )
1.1 bertrand 496: END IF
1.4 bertrand 497: SVA( q ) = AAQQ*SQRT( MAX( ZERO,
1.1 bertrand 498: $ ONE+T*APOAQ*AAPQ1 ) )
1.4 bertrand 499: AAPP = AAPP*SQRT( MAX( ZERO,
1.1 bertrand 500: $ ONE-T*AQOAP*AAPQ1 ) )
1.4 bertrand 501: MXSINJ = MAX( MXSINJ, ABS( T ) )
1.1 bertrand 502: ELSE
503: *
504: * .. choose correct signum for THETA and rotate
505: *
1.4 bertrand 506: THSIGN = -SIGN( ONE, AAPQ1 )
1.1 bertrand 507: IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN
508: T = ONE / ( THETA+THSIGN*
1.4 bertrand 509: $ SQRT( ONE+THETA*THETA ) )
510: CS = SQRT( ONE / ( ONE+T*T ) )
1.1 bertrand 511: SN = T*CS
1.4 bertrand 512: MXSINJ = MAX( MXSINJ, ABS( SN ) )
513: SVA( q ) = AAQQ*SQRT( MAX( ZERO,
1.1 bertrand 514: $ ONE+T*APOAQ*AAPQ1 ) )
1.4 bertrand 515: AAPP = AAPP*SQRT( MAX( ZERO,
1.1 bertrand 516: $ ONE-T*AQOAP*AAPQ1 ) )
517: *
518: CALL ZROT( M, A(1,p), 1, A(1,q), 1,
1.4 bertrand 519: $ CS, CONJG(OMPQ)*SN )
1.1 bertrand 520: IF( RSVEC ) THEN
1.4 bertrand 521: CALL ZROT( MVL, V(1,p), 1,
522: $ V(1,q), 1, CS, CONJG(OMPQ)*SN )
1.1 bertrand 523: END IF
524: END IF
525: D(p) = -D(q) * OMPQ
526: *
527: ELSE
528: * .. have to use modified Gram-Schmidt like transformation
529: IF( AAPP.GT.AAQQ ) THEN
530: CALL ZCOPY( M, A( 1, p ), 1,
531: $ WORK, 1 )
532: CALL ZLASCL( 'G', 0, 0, AAPP, ONE,
533: $ M, 1, WORK,LDA,
534: $ IERR )
535: CALL ZLASCL( 'G', 0, 0, AAQQ, ONE,
536: $ M, 1, A( 1, q ), LDA,
537: $ IERR )
538: CALL ZAXPY( M, -AAPQ, WORK,
539: $ 1, A( 1, q ), 1 )
540: CALL ZLASCL( 'G', 0, 0, ONE, AAQQ,
541: $ M, 1, A( 1, q ), LDA,
542: $ IERR )
1.4 bertrand 543: SVA( q ) = AAQQ*SQRT( MAX( ZERO,
1.1 bertrand 544: $ ONE-AAPQ1*AAPQ1 ) )
1.4 bertrand 545: MXSINJ = MAX( MXSINJ, SFMIN )
1.1 bertrand 546: ELSE
547: CALL ZCOPY( M, A( 1, q ), 1,
548: $ WORK, 1 )
549: CALL ZLASCL( 'G', 0, 0, AAQQ, ONE,
550: $ M, 1, WORK,LDA,
551: $ IERR )
552: CALL ZLASCL( 'G', 0, 0, AAPP, ONE,
553: $ M, 1, A( 1, p ), LDA,
554: $ IERR )
1.4 bertrand 555: CALL ZAXPY( M, -CONJG(AAPQ),
1.1 bertrand 556: $ WORK, 1, A( 1, p ), 1 )
557: CALL ZLASCL( 'G', 0, 0, ONE, AAPP,
558: $ M, 1, A( 1, p ), LDA,
559: $ IERR )
1.4 bertrand 560: SVA( p ) = AAPP*SQRT( MAX( ZERO,
1.1 bertrand 561: $ ONE-AAPQ1*AAPQ1 ) )
1.4 bertrand 562: MXSINJ = MAX( MXSINJ, SFMIN )
1.1 bertrand 563: END IF
564: END IF
565: * END IF ROTOK THEN ... ELSE
566: *
567: * In the case of cancellation in updating SVA(q), SVA(p)
568: * .. recompute SVA(q), SVA(p)
569: IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
570: $ THEN
571: IF( ( AAQQ.LT.ROOTBIG ) .AND.
572: $ ( AAQQ.GT.ROOTSFMIN ) ) THEN
573: SVA( q ) = DZNRM2( M, A( 1, q ), 1)
574: ELSE
575: T = ZERO
576: AAQQ = ONE
577: CALL ZLASSQ( M, A( 1, q ), 1, T,
578: $ AAQQ )
1.4 bertrand 579: SVA( q ) = T*SQRT( AAQQ )
1.1 bertrand 580: END IF
581: END IF
582: IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN
583: IF( ( AAPP.LT.ROOTBIG ) .AND.
584: $ ( AAPP.GT.ROOTSFMIN ) ) THEN
585: AAPP = DZNRM2( M, A( 1, p ), 1 )
586: ELSE
587: T = ZERO
588: AAPP = ONE
589: CALL ZLASSQ( M, A( 1, p ), 1, T,
590: $ AAPP )
1.4 bertrand 591: AAPP = T*SQRT( AAPP )
1.1 bertrand 592: END IF
593: SVA( p ) = AAPP
594: END IF
595: * end of OK rotation
596: ELSE
597: NOTROT = NOTROT + 1
598: *[RTD] SKIPPED = SKIPPED + 1
599: PSKIPPED = PSKIPPED + 1
600: IJBLSK = IJBLSK + 1
601: END IF
602: ELSE
603: NOTROT = NOTROT + 1
604: PSKIPPED = PSKIPPED + 1
605: IJBLSK = IJBLSK + 1
606: END IF
607: *
608: IF( ( i.LE.SWBAND ) .AND. ( IJBLSK.GE.BLSKIP ) )
609: $ THEN
610: SVA( p ) = AAPP
611: NOTROT = 0
612: GO TO 2011
613: END IF
614: IF( ( i.LE.SWBAND ) .AND.
615: $ ( PSKIPPED.GT.ROWSKIP ) ) THEN
616: AAPP = -AAPP
617: NOTROT = 0
618: GO TO 2203
619: END IF
620: *
621: 2200 CONTINUE
622: * end of the q-loop
623: 2203 CONTINUE
624: *
625: SVA( p ) = AAPP
626: *
627: ELSE
628: *
629: IF( AAPP.EQ.ZERO )NOTROT = NOTROT +
1.4 bertrand 630: $ MIN( jgl+KBL-1, N ) - jgl + 1
1.1 bertrand 631: IF( AAPP.LT.ZERO )NOTROT = 0
632: *
633: END IF
634: *
635: 2100 CONTINUE
636: * end of the p-loop
637: 2010 CONTINUE
638: * end of the jbc-loop
639: 2011 CONTINUE
640: *2011 bailed out of the jbc-loop
1.4 bertrand 641: DO 2012 p = igl, MIN( igl+KBL-1, N )
1.1 bertrand 642: SVA( p ) = ABS( SVA( p ) )
643: 2012 CONTINUE
644: ***
645: 2000 CONTINUE
646: *2000 :: end of the ibr-loop
647: *
648: * .. update SVA(N)
649: IF( ( SVA( N ).LT.ROOTBIG ) .AND. ( SVA( N ).GT.ROOTSFMIN ) )
650: $ THEN
651: SVA( N ) = DZNRM2( M, A( 1, N ), 1 )
652: ELSE
653: T = ZERO
654: AAPP = ONE
655: CALL ZLASSQ( M, A( 1, N ), 1, T, AAPP )
1.4 bertrand 656: SVA( N ) = T*SQRT( AAPP )
1.1 bertrand 657: END IF
658: *
659: * Additional steering devices
660: *
661: IF( ( i.LT.SWBAND ) .AND. ( ( MXAAPQ.LE.ROOTTOL ) .OR.
662: $ ( ISWROT.LE.N ) ) )SWBAND = i
663: *
1.4 bertrand 664: IF( ( i.GT.SWBAND+1 ) .AND. ( MXAAPQ.LT.SQRT( DBLE( N ) )*
1.2 bertrand 665: $ TOL ) .AND. ( DBLE( N )*MXAAPQ*MXSINJ.LT.TOL ) ) THEN
1.1 bertrand 666: GO TO 1994
667: END IF
668: *
669: IF( NOTROT.GE.EMPTSW )GO TO 1994
670: *
671: 1993 CONTINUE
672: * end i=1:NSWEEP loop
673: *
674: * #:( Reaching this point means that the procedure has not converged.
675: INFO = NSWEEP - 1
676: GO TO 1995
677: *
678: 1994 CONTINUE
679: * #:) Reaching this point means numerical convergence after the i-th
680: * sweep.
681: *
682: INFO = 0
683: * #:) INFO = 0 confirms successful iterations.
684: 1995 CONTINUE
685: *
686: * Sort the vector SVA() of column norms.
687: DO 5991 p = 1, N - 1
688: q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
689: IF( p.NE.q ) THEN
690: TEMP1 = SVA( p )
691: SVA( p ) = SVA( q )
692: SVA( q ) = TEMP1
693: AAPQ = D( p )
694: D( p ) = D( q )
695: D( q ) = AAPQ
696: CALL ZSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
697: IF( RSVEC )CALL ZSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 )
698: END IF
699: 5991 CONTINUE
700: *
701: *
702: RETURN
703: * ..
704: * .. END OF ZGSVJ1
705: * ..
706: END
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