1: *> \brief \b DGSVJ0 pre-processor for the routine sgesvj.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DGSVJ0 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgsvj0.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgsvj0.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgsvj0.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DGSVJ0( JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS,
22: * SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * INTEGER INFO, LDA, LDV, LWORK, M, MV, N, NSWEEP
26: * DOUBLE PRECISION EPS, SFMIN, TOL
27: * CHARACTER*1 JOBV
28: * ..
29: * .. Array Arguments ..
30: * DOUBLE PRECISION A( LDA, * ), SVA( N ), D( N ), V( LDV, * ),
31: * $ WORK( LWORK )
32: * ..
33: *
34: *
35: *> \par Purpose:
36: * =============
37: *>
38: *> \verbatim
39: *>
40: *> DGSVJ0 is called from DGESVJ as a pre-processor and that is its main
41: *> purpose. It applies Jacobi rotations in the same way as DGESVJ does, but
42: *> it does not check convergence (stopping criterion). Few tuning
43: *> parameters (marked by [TP]) are available for the implementer.
44: *> \endverbatim
45: *
46: * Arguments:
47: * ==========
48: *
49: *> \param[in] JOBV
50: *> \verbatim
51: *> JOBV is CHARACTER*1
52: *> Specifies whether the output from this procedure is used
53: *> to compute the matrix V:
54: *> = 'V': the product of the Jacobi rotations is accumulated
55: *> by postmulyiplying the N-by-N array V.
56: *> (See the description of V.)
57: *> = 'A': the product of the Jacobi rotations is accumulated
58: *> by postmulyiplying the MV-by-N array V.
59: *> (See the descriptions of MV and V.)
60: *> = 'N': the Jacobi rotations are not accumulated.
61: *> \endverbatim
62: *>
63: *> \param[in] M
64: *> \verbatim
65: *> M is INTEGER
66: *> The number of rows of the input matrix A. M >= 0.
67: *> \endverbatim
68: *>
69: *> \param[in] N
70: *> \verbatim
71: *> N is INTEGER
72: *> The number of columns of the input matrix A.
73: *> M >= N >= 0.
74: *> \endverbatim
75: *>
76: *> \param[in,out] A
77: *> \verbatim
78: *> A is DOUBLE PRECISION array, dimension (LDA,N)
79: *> On entry, M-by-N matrix A, such that A*diag(D) represents
80: *> the input matrix.
81: *> On exit,
82: *> A_onexit * D_onexit represents the input matrix A*diag(D)
83: *> post-multiplied by a sequence of Jacobi rotations, where the
84: *> rotation threshold and the total number of sweeps are given in
85: *> TOL and NSWEEP, respectively.
86: *> (See the descriptions of D, TOL and NSWEEP.)
87: *> \endverbatim
88: *>
89: *> \param[in] LDA
90: *> \verbatim
91: *> LDA is INTEGER
92: *> The leading dimension of the array A. LDA >= max(1,M).
93: *> \endverbatim
94: *>
95: *> \param[in,out] D
96: *> \verbatim
97: *> D is DOUBLE PRECISION array, dimension (N)
98: *> The array D accumulates the scaling factors from the fast scaled
99: *> Jacobi rotations.
100: *> On entry, A*diag(D) represents the input matrix.
101: *> On exit, A_onexit*diag(D_onexit) represents the input matrix
102: *> post-multiplied by a sequence of Jacobi rotations, where the
103: *> rotation threshold and the total number of sweeps are given in
104: *> TOL and NSWEEP, respectively.
105: *> (See the descriptions of A, TOL and NSWEEP.)
106: *> \endverbatim
107: *>
108: *> \param[in,out] SVA
109: *> \verbatim
110: *> SVA is DOUBLE PRECISION array, dimension (N)
111: *> On entry, SVA contains the Euclidean norms of the columns of
112: *> the matrix A*diag(D).
113: *> On exit, SVA contains the Euclidean norms of the columns of
114: *> the matrix onexit*diag(D_onexit).
115: *> \endverbatim
116: *>
117: *> \param[in] MV
118: *> \verbatim
119: *> MV is INTEGER
120: *> If JOBV .EQ. 'A', then MV rows of V are post-multipled by a
121: *> sequence of Jacobi rotations.
122: *> If JOBV = 'N', then MV is not referenced.
123: *> \endverbatim
124: *>
125: *> \param[in,out] V
126: *> \verbatim
127: *> V is DOUBLE PRECISION array, dimension (LDV,N)
128: *> If JOBV .EQ. 'V' then N rows of V are post-multipled by a
129: *> sequence of Jacobi rotations.
130: *> If JOBV .EQ. 'A' then MV rows of V are post-multipled by a
131: *> sequence of Jacobi rotations.
132: *> If JOBV = 'N', then V is not referenced.
133: *> \endverbatim
134: *>
135: *> \param[in] LDV
136: *> \verbatim
137: *> LDV is INTEGER
138: *> The leading dimension of the array V, LDV >= 1.
139: *> If JOBV = 'V', LDV .GE. N.
140: *> If JOBV = 'A', LDV .GE. MV.
141: *> \endverbatim
142: *>
143: *> \param[in] EPS
144: *> \verbatim
145: *> EPS is DOUBLE PRECISION
146: *> EPS = DLAMCH('Epsilon')
147: *> \endverbatim
148: *>
149: *> \param[in] SFMIN
150: *> \verbatim
151: *> SFMIN is DOUBLE PRECISION
152: *> SFMIN = DLAMCH('Safe Minimum')
153: *> \endverbatim
154: *>
155: *> \param[in] TOL
156: *> \verbatim
157: *> TOL is DOUBLE PRECISION
158: *> TOL is the threshold for Jacobi rotations. For a pair
159: *> A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
160: *> applied only if DABS(COS(angle(A(:,p),A(:,q)))) .GT. TOL.
161: *> \endverbatim
162: *>
163: *> \param[in] NSWEEP
164: *> \verbatim
165: *> NSWEEP is INTEGER
166: *> NSWEEP is the number of sweeps of Jacobi rotations to be
167: *> performed.
168: *> \endverbatim
169: *>
170: *> \param[out] WORK
171: *> \verbatim
172: *> WORK is DOUBLE PRECISION array, dimension (LWORK)
173: *> \endverbatim
174: *>
175: *> \param[in] LWORK
176: *> \verbatim
177: *> LWORK is INTEGER
178: *> LWORK is the dimension of WORK. LWORK .GE. M.
179: *> \endverbatim
180: *>
181: *> \param[out] INFO
182: *> \verbatim
183: *> INFO is INTEGER
184: *> = 0 : successful exit.
185: *> < 0 : if INFO = -i, then the i-th argument had an illegal value
186: *> \endverbatim
187: *
188: * Authors:
189: * ========
190: *
191: *> \author Univ. of Tennessee
192: *> \author Univ. of California Berkeley
193: *> \author Univ. of Colorado Denver
194: *> \author NAG Ltd.
195: *
196: *> \date September 2012
197: *
198: *> \ingroup doubleOTHERcomputational
199: *
200: *> \par Further Details:
201: * =====================
202: *>
203: *> DGSVJ0 is used just to enable DGESVJ to call a simplified version of
204: *> itself to work on a submatrix of the original matrix.
205: *>
206: *> \par Contributors:
207: * ==================
208: *>
209: *> Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
210: *>
211: *> \par Bugs, Examples and Comments:
212: * =================================
213: *>
214: *> Please report all bugs and send interesting test examples and comments to
215: *> drmac@math.hr. Thank you.
216: *
217: * =====================================================================
218: SUBROUTINE DGSVJ0( JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS,
219: $ SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
220: *
221: * -- LAPACK computational routine (version 3.4.2) --
222: * -- LAPACK is a software package provided by Univ. of Tennessee, --
223: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
224: * September 2012
225: *
226: * .. Scalar Arguments ..
227: INTEGER INFO, LDA, LDV, LWORK, M, MV, N, NSWEEP
228: DOUBLE PRECISION EPS, SFMIN, TOL
229: CHARACTER*1 JOBV
230: * ..
231: * .. Array Arguments ..
232: DOUBLE PRECISION A( LDA, * ), SVA( N ), D( N ), V( LDV, * ),
233: $ WORK( LWORK )
234: * ..
235: *
236: * =====================================================================
237: *
238: * .. Local Parameters ..
239: DOUBLE PRECISION ZERO, HALF, ONE
240: PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0)
241: * ..
242: * .. Local Scalars ..
243: DOUBLE PRECISION AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG,
244: $ BIGTHETA, CS, MXAAPQ, MXSINJ, ROOTBIG, ROOTEPS,
245: $ ROOTSFMIN, ROOTTOL, SMALL, SN, T, TEMP1, THETA,
246: $ THSIGN
247: INTEGER BLSKIP, EMPTSW, i, ibr, IERR, igl, IJBLSK, ir1,
248: $ ISWROT, jbc, jgl, KBL, LKAHEAD, MVL, NBL,
249: $ NOTROT, p, PSKIPPED, q, ROWSKIP, SWBAND
250: LOGICAL APPLV, ROTOK, RSVEC
251: * ..
252: * .. Local Arrays ..
253: DOUBLE PRECISION FASTR( 5 )
254: * ..
255: * .. Intrinsic Functions ..
256: INTRINSIC DABS, DMAX1, DBLE, MIN0, DSIGN, DSQRT
257: * ..
258: * .. External Functions ..
259: DOUBLE PRECISION DDOT, DNRM2
260: INTEGER IDAMAX
261: LOGICAL LSAME
262: EXTERNAL IDAMAX, LSAME, DDOT, DNRM2
263: * ..
264: * .. External Subroutines ..
265: EXTERNAL DAXPY, DCOPY, DLASCL, DLASSQ, DROTM, DSWAP
266: * ..
267: * .. Executable Statements ..
268: *
269: * Test the input parameters.
270: *
271: APPLV = LSAME( JOBV, 'A' )
272: RSVEC = LSAME( JOBV, 'V' )
273: IF( .NOT.( RSVEC .OR. APPLV .OR. LSAME( JOBV, 'N' ) ) ) THEN
274: INFO = -1
275: ELSE IF( M.LT.0 ) THEN
276: INFO = -2
277: ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN
278: INFO = -3
279: ELSE IF( LDA.LT.M ) THEN
280: INFO = -5
281: ELSE IF( ( RSVEC.OR.APPLV ) .AND. ( MV.LT.0 ) ) THEN
282: INFO = -8
283: ELSE IF( ( RSVEC.AND.( LDV.LT.N ) ).OR.
284: $ ( APPLV.AND.( LDV.LT.MV ) ) ) THEN
285: INFO = -10
286: ELSE IF( TOL.LE.EPS ) THEN
287: INFO = -13
288: ELSE IF( NSWEEP.LT.0 ) THEN
289: INFO = -14
290: ELSE IF( LWORK.LT.M ) THEN
291: INFO = -16
292: ELSE
293: INFO = 0
294: END IF
295: *
296: * #:(
297: IF( INFO.NE.0 ) THEN
298: CALL XERBLA( 'DGSVJ0', -INFO )
299: RETURN
300: END IF
301: *
302: IF( RSVEC ) THEN
303: MVL = N
304: ELSE IF( APPLV ) THEN
305: MVL = MV
306: END IF
307: RSVEC = RSVEC .OR. APPLV
308:
309: ROOTEPS = DSQRT( EPS )
310: ROOTSFMIN = DSQRT( SFMIN )
311: SMALL = SFMIN / EPS
312: BIG = ONE / SFMIN
313: ROOTBIG = ONE / ROOTSFMIN
314: BIGTHETA = ONE / ROOTEPS
315: ROOTTOL = DSQRT( TOL )
316: *
317: * -#- Row-cyclic Jacobi SVD algorithm with column pivoting -#-
318: *
319: EMPTSW = ( N*( N-1 ) ) / 2
320: NOTROT = 0
321: FASTR( 1 ) = ZERO
322: *
323: * -#- Row-cyclic pivot strategy with de Rijk's pivoting -#-
324: *
325:
326: SWBAND = 0
327: *[TP] SWBAND is a tuning parameter. It is meaningful and effective
328: * if SGESVJ is used as a computational routine in the preconditioned
329: * Jacobi SVD algorithm SGESVJ. For sweeps i=1:SWBAND the procedure
330: * ......
331:
332: KBL = MIN0( 8, N )
333: *[TP] KBL is a tuning parameter that defines the tile size in the
334: * tiling of the p-q loops of pivot pairs. In general, an optimal
335: * value of KBL depends on the matrix dimensions and on the
336: * parameters of the computer's memory.
337: *
338: NBL = N / KBL
339: IF( ( NBL*KBL ).NE.N )NBL = NBL + 1
340:
341: BLSKIP = ( KBL**2 ) + 1
342: *[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.
343:
344: ROWSKIP = MIN0( 5, KBL )
345: *[TP] ROWSKIP is a tuning parameter.
346:
347: LKAHEAD = 1
348: *[TP] LKAHEAD is a tuning parameter.
349: SWBAND = 0
350: PSKIPPED = 0
351: *
352: DO 1993 i = 1, NSWEEP
353: * .. go go go ...
354: *
355: MXAAPQ = ZERO
356: MXSINJ = ZERO
357: ISWROT = 0
358: *
359: NOTROT = 0
360: PSKIPPED = 0
361: *
362: DO 2000 ibr = 1, NBL
363:
364: igl = ( ibr-1 )*KBL + 1
365: *
366: DO 1002 ir1 = 0, MIN0( LKAHEAD, NBL-ibr )
367: *
368: igl = igl + ir1*KBL
369: *
370: DO 2001 p = igl, MIN0( igl+KBL-1, N-1 )
371:
372: * .. de Rijk's pivoting
373: q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
374: IF( p.NE.q ) THEN
375: CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
376: IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1,
377: $ V( 1, q ), 1 )
378: TEMP1 = SVA( p )
379: SVA( p ) = SVA( q )
380: SVA( q ) = TEMP1
381: TEMP1 = D( p )
382: D( p ) = D( q )
383: D( q ) = TEMP1
384: END IF
385: *
386: IF( ir1.EQ.0 ) THEN
387: *
388: * Column norms are periodically updated by explicit
389: * norm computation.
390: * Caveat:
391: * Some BLAS implementations compute DNRM2(M,A(1,p),1)
392: * as DSQRT(DDOT(M,A(1,p),1,A(1,p),1)), which may result in
393: * overflow for ||A(:,p)||_2 > DSQRT(overflow_threshold), and
394: * undeflow for ||A(:,p)||_2 < DSQRT(underflow_threshold).
395: * Hence, DNRM2 cannot be trusted, not even in the case when
396: * the true norm is far from the under(over)flow boundaries.
397: * If properly implemented DNRM2 is available, the IF-THEN-ELSE
398: * below should read "AAPP = DNRM2( M, A(1,p), 1 ) * D(p)".
399: *
400: IF( ( SVA( p ).LT.ROOTBIG ) .AND.
401: $ ( SVA( p ).GT.ROOTSFMIN ) ) THEN
402: SVA( p ) = DNRM2( M, A( 1, p ), 1 )*D( p )
403: ELSE
404: TEMP1 = ZERO
405: AAPP = ONE
406: CALL DLASSQ( M, A( 1, p ), 1, TEMP1, AAPP )
407: SVA( p ) = TEMP1*DSQRT( AAPP )*D( p )
408: END IF
409: AAPP = SVA( p )
410: ELSE
411: AAPP = SVA( p )
412: END IF
413:
414: *
415: IF( AAPP.GT.ZERO ) THEN
416: *
417: PSKIPPED = 0
418: *
419: DO 2002 q = p + 1, MIN0( igl+KBL-1, N )
420: *
421: AAQQ = SVA( q )
422:
423: IF( AAQQ.GT.ZERO ) THEN
424: *
425: AAPP0 = AAPP
426: IF( AAQQ.GE.ONE ) THEN
427: ROTOK = ( SMALL*AAPP ).LE.AAQQ
428: IF( AAPP.LT.( BIG / AAQQ ) ) THEN
429: AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
430: $ q ), 1 )*D( p )*D( q ) / AAQQ )
431: $ / AAPP
432: ELSE
433: CALL DCOPY( M, A( 1, p ), 1, WORK, 1 )
434: CALL DLASCL( 'G', 0, 0, AAPP, D( p ),
435: $ M, 1, WORK, LDA, IERR )
436: AAPQ = DDOT( M, WORK, 1, A( 1, q ),
437: $ 1 )*D( q ) / AAQQ
438: END IF
439: ELSE
440: ROTOK = AAPP.LE.( AAQQ / SMALL )
441: IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
442: AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
443: $ q ), 1 )*D( p )*D( q ) / AAQQ )
444: $ / AAPP
445: ELSE
446: CALL DCOPY( M, A( 1, q ), 1, WORK, 1 )
447: CALL DLASCL( 'G', 0, 0, AAQQ, D( q ),
448: $ M, 1, WORK, LDA, IERR )
449: AAPQ = DDOT( M, WORK, 1, A( 1, p ),
450: $ 1 )*D( p ) / AAPP
451: END IF
452: END IF
453: *
454: MXAAPQ = DMAX1( MXAAPQ, DABS( AAPQ ) )
455: *
456: * TO rotate or NOT to rotate, THAT is the question ...
457: *
458: IF( DABS( AAPQ ).GT.TOL ) THEN
459: *
460: * .. rotate
461: * ROTATED = ROTATED + ONE
462: *
463: IF( ir1.EQ.0 ) THEN
464: NOTROT = 0
465: PSKIPPED = 0
466: ISWROT = ISWROT + 1
467: END IF
468: *
469: IF( ROTOK ) THEN
470: *
471: AQOAP = AAQQ / AAPP
472: APOAQ = AAPP / AAQQ
473: THETA = -HALF*DABS( AQOAP-APOAQ )/AAPQ
474: *
475: IF( DABS( THETA ).GT.BIGTHETA ) THEN
476: *
477: T = HALF / THETA
478: FASTR( 3 ) = T*D( p ) / D( q )
479: FASTR( 4 ) = -T*D( q ) / D( p )
480: CALL DROTM( M, A( 1, p ), 1,
481: $ A( 1, q ), 1, FASTR )
482: IF( RSVEC )CALL DROTM( MVL,
483: $ V( 1, p ), 1,
484: $ V( 1, q ), 1,
485: $ FASTR )
486: SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
487: $ ONE+T*APOAQ*AAPQ ) )
488: AAPP = AAPP*DSQRT( DMAX1( ZERO,
489: $ ONE-T*AQOAP*AAPQ ) )
490: MXSINJ = DMAX1( MXSINJ, DABS( T ) )
491: *
492: ELSE
493: *
494: * .. choose correct signum for THETA and rotate
495: *
496: THSIGN = -DSIGN( ONE, AAPQ )
497: T = ONE / ( THETA+THSIGN*
498: $ DSQRT( ONE+THETA*THETA ) )
499: CS = DSQRT( ONE / ( ONE+T*T ) )
500: SN = T*CS
501: *
502: MXSINJ = DMAX1( MXSINJ, DABS( SN ) )
503: SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
504: $ ONE+T*APOAQ*AAPQ ) )
505: AAPP = AAPP*DSQRT( DMAX1( ZERO,
506: $ ONE-T*AQOAP*AAPQ ) )
507: *
508: APOAQ = D( p ) / D( q )
509: AQOAP = D( q ) / D( p )
510: IF( D( p ).GE.ONE ) THEN
511: IF( D( q ).GE.ONE ) THEN
512: FASTR( 3 ) = T*APOAQ
513: FASTR( 4 ) = -T*AQOAP
514: D( p ) = D( p )*CS
515: D( q ) = D( q )*CS
516: CALL DROTM( M, A( 1, p ), 1,
517: $ A( 1, q ), 1,
518: $ FASTR )
519: IF( RSVEC )CALL DROTM( MVL,
520: $ V( 1, p ), 1, V( 1, q ),
521: $ 1, FASTR )
522: ELSE
523: CALL DAXPY( M, -T*AQOAP,
524: $ A( 1, q ), 1,
525: $ A( 1, p ), 1 )
526: CALL DAXPY( M, CS*SN*APOAQ,
527: $ A( 1, p ), 1,
528: $ A( 1, q ), 1 )
529: D( p ) = D( p )*CS
530: D( q ) = D( q ) / CS
531: IF( RSVEC ) THEN
532: CALL DAXPY( MVL, -T*AQOAP,
533: $ V( 1, q ), 1,
534: $ V( 1, p ), 1 )
535: CALL DAXPY( MVL,
536: $ CS*SN*APOAQ,
537: $ V( 1, p ), 1,
538: $ V( 1, q ), 1 )
539: END IF
540: END IF
541: ELSE
542: IF( D( q ).GE.ONE ) THEN
543: CALL DAXPY( M, T*APOAQ,
544: $ A( 1, p ), 1,
545: $ A( 1, q ), 1 )
546: CALL DAXPY( M, -CS*SN*AQOAP,
547: $ A( 1, q ), 1,
548: $ A( 1, p ), 1 )
549: D( p ) = D( p ) / CS
550: D( q ) = D( q )*CS
551: IF( RSVEC ) THEN
552: CALL DAXPY( MVL, T*APOAQ,
553: $ V( 1, p ), 1,
554: $ V( 1, q ), 1 )
555: CALL DAXPY( MVL,
556: $ -CS*SN*AQOAP,
557: $ V( 1, q ), 1,
558: $ V( 1, p ), 1 )
559: END IF
560: ELSE
561: IF( D( p ).GE.D( q ) ) THEN
562: CALL DAXPY( M, -T*AQOAP,
563: $ A( 1, q ), 1,
564: $ A( 1, p ), 1 )
565: CALL DAXPY( M, CS*SN*APOAQ,
566: $ A( 1, p ), 1,
567: $ A( 1, q ), 1 )
568: D( p ) = D( p )*CS
569: D( q ) = D( q ) / CS
570: IF( RSVEC ) THEN
571: CALL DAXPY( MVL,
572: $ -T*AQOAP,
573: $ V( 1, q ), 1,
574: $ V( 1, p ), 1 )
575: CALL DAXPY( MVL,
576: $ CS*SN*APOAQ,
577: $ V( 1, p ), 1,
578: $ V( 1, q ), 1 )
579: END IF
580: ELSE
581: CALL DAXPY( M, T*APOAQ,
582: $ A( 1, p ), 1,
583: $ A( 1, q ), 1 )
584: CALL DAXPY( M,
585: $ -CS*SN*AQOAP,
586: $ A( 1, q ), 1,
587: $ A( 1, p ), 1 )
588: D( p ) = D( p ) / CS
589: D( q ) = D( q )*CS
590: IF( RSVEC ) THEN
591: CALL DAXPY( MVL,
592: $ T*APOAQ, V( 1, p ),
593: $ 1, V( 1, q ), 1 )
594: CALL DAXPY( MVL,
595: $ -CS*SN*AQOAP,
596: $ V( 1, q ), 1,
597: $ V( 1, p ), 1 )
598: END IF
599: END IF
600: END IF
601: END IF
602: END IF
603: *
604: ELSE
605: * .. have to use modified Gram-Schmidt like transformation
606: CALL DCOPY( M, A( 1, p ), 1, WORK, 1 )
607: CALL DLASCL( 'G', 0, 0, AAPP, ONE, M,
608: $ 1, WORK, LDA, IERR )
609: CALL DLASCL( 'G', 0, 0, AAQQ, ONE, M,
610: $ 1, A( 1, q ), LDA, IERR )
611: TEMP1 = -AAPQ*D( p ) / D( q )
612: CALL DAXPY( M, TEMP1, WORK, 1,
613: $ A( 1, q ), 1 )
614: CALL DLASCL( 'G', 0, 0, ONE, AAQQ, M,
615: $ 1, A( 1, q ), LDA, IERR )
616: SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
617: $ ONE-AAPQ*AAPQ ) )
618: MXSINJ = DMAX1( MXSINJ, SFMIN )
619: END IF
620: * END IF ROTOK THEN ... ELSE
621: *
622: * In the case of cancellation in updating SVA(q), SVA(p)
623: * recompute SVA(q), SVA(p).
624: IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
625: $ THEN
626: IF( ( AAQQ.LT.ROOTBIG ) .AND.
627: $ ( AAQQ.GT.ROOTSFMIN ) ) THEN
628: SVA( q ) = DNRM2( M, A( 1, q ), 1 )*
629: $ D( q )
630: ELSE
631: T = ZERO
632: AAQQ = ONE
633: CALL DLASSQ( M, A( 1, q ), 1, T,
634: $ AAQQ )
635: SVA( q ) = T*DSQRT( AAQQ )*D( q )
636: END IF
637: END IF
638: IF( ( AAPP / AAPP0 ).LE.ROOTEPS ) THEN
639: IF( ( AAPP.LT.ROOTBIG ) .AND.
640: $ ( AAPP.GT.ROOTSFMIN ) ) THEN
641: AAPP = DNRM2( M, A( 1, p ), 1 )*
642: $ D( p )
643: ELSE
644: T = ZERO
645: AAPP = ONE
646: CALL DLASSQ( M, A( 1, p ), 1, T,
647: $ AAPP )
648: AAPP = T*DSQRT( AAPP )*D( p )
649: END IF
650: SVA( p ) = AAPP
651: END IF
652: *
653: ELSE
654: * A(:,p) and A(:,q) already numerically orthogonal
655: IF( ir1.EQ.0 )NOTROT = NOTROT + 1
656: PSKIPPED = PSKIPPED + 1
657: END IF
658: ELSE
659: * A(:,q) is zero column
660: IF( ir1.EQ.0 )NOTROT = NOTROT + 1
661: PSKIPPED = PSKIPPED + 1
662: END IF
663: *
664: IF( ( i.LE.SWBAND ) .AND.
665: $ ( PSKIPPED.GT.ROWSKIP ) ) THEN
666: IF( ir1.EQ.0 )AAPP = -AAPP
667: NOTROT = 0
668: GO TO 2103
669: END IF
670: *
671: 2002 CONTINUE
672: * END q-LOOP
673: *
674: 2103 CONTINUE
675: * bailed out of q-loop
676:
677: SVA( p ) = AAPP
678:
679: ELSE
680: SVA( p ) = AAPP
681: IF( ( ir1.EQ.0 ) .AND. ( AAPP.EQ.ZERO ) )
682: $ NOTROT = NOTROT + MIN0( igl+KBL-1, N ) - p
683: END IF
684: *
685: 2001 CONTINUE
686: * end of the p-loop
687: * end of doing the block ( ibr, ibr )
688: 1002 CONTINUE
689: * end of ir1-loop
690: *
691: *........................................................
692: * ... go to the off diagonal blocks
693: *
694: igl = ( ibr-1 )*KBL + 1
695: *
696: DO 2010 jbc = ibr + 1, NBL
697: *
698: jgl = ( jbc-1 )*KBL + 1
699: *
700: * doing the block at ( ibr, jbc )
701: *
702: IJBLSK = 0
703: DO 2100 p = igl, MIN0( igl+KBL-1, N )
704: *
705: AAPP = SVA( p )
706: *
707: IF( AAPP.GT.ZERO ) THEN
708: *
709: PSKIPPED = 0
710: *
711: DO 2200 q = jgl, MIN0( jgl+KBL-1, N )
712: *
713: AAQQ = SVA( q )
714: *
715: IF( AAQQ.GT.ZERO ) THEN
716: AAPP0 = AAPP
717: *
718: * -#- M x 2 Jacobi SVD -#-
719: *
720: * -#- Safe Gram matrix computation -#-
721: *
722: IF( AAQQ.GE.ONE ) THEN
723: IF( AAPP.GE.AAQQ ) THEN
724: ROTOK = ( SMALL*AAPP ).LE.AAQQ
725: ELSE
726: ROTOK = ( SMALL*AAQQ ).LE.AAPP
727: END IF
728: IF( AAPP.LT.( BIG / AAQQ ) ) THEN
729: AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
730: $ q ), 1 )*D( p )*D( q ) / AAQQ )
731: $ / AAPP
732: ELSE
733: CALL DCOPY( M, A( 1, p ), 1, WORK, 1 )
734: CALL DLASCL( 'G', 0, 0, AAPP, D( p ),
735: $ M, 1, WORK, LDA, IERR )
736: AAPQ = DDOT( M, WORK, 1, A( 1, q ),
737: $ 1 )*D( q ) / AAQQ
738: END IF
739: ELSE
740: IF( AAPP.GE.AAQQ ) THEN
741: ROTOK = AAPP.LE.( AAQQ / SMALL )
742: ELSE
743: ROTOK = AAQQ.LE.( AAPP / SMALL )
744: END IF
745: IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
746: AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
747: $ q ), 1 )*D( p )*D( q ) / AAQQ )
748: $ / AAPP
749: ELSE
750: CALL DCOPY( M, A( 1, q ), 1, WORK, 1 )
751: CALL DLASCL( 'G', 0, 0, AAQQ, D( q ),
752: $ M, 1, WORK, LDA, IERR )
753: AAPQ = DDOT( M, WORK, 1, A( 1, p ),
754: $ 1 )*D( p ) / AAPP
755: END IF
756: END IF
757: *
758: MXAAPQ = DMAX1( MXAAPQ, DABS( AAPQ ) )
759: *
760: * TO rotate or NOT to rotate, THAT is the question ...
761: *
762: IF( DABS( AAPQ ).GT.TOL ) THEN
763: NOTROT = 0
764: * ROTATED = ROTATED + 1
765: PSKIPPED = 0
766: ISWROT = ISWROT + 1
767: *
768: IF( ROTOK ) THEN
769: *
770: AQOAP = AAQQ / AAPP
771: APOAQ = AAPP / AAQQ
772: THETA = -HALF*DABS( AQOAP-APOAQ )/AAPQ
773: IF( AAQQ.GT.AAPP0 )THETA = -THETA
774: *
775: IF( DABS( THETA ).GT.BIGTHETA ) THEN
776: T = HALF / THETA
777: FASTR( 3 ) = T*D( p ) / D( q )
778: FASTR( 4 ) = -T*D( q ) / D( p )
779: CALL DROTM( M, A( 1, p ), 1,
780: $ A( 1, q ), 1, FASTR )
781: IF( RSVEC )CALL DROTM( MVL,
782: $ V( 1, p ), 1,
783: $ V( 1, q ), 1,
784: $ FASTR )
785: SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
786: $ ONE+T*APOAQ*AAPQ ) )
787: AAPP = AAPP*DSQRT( DMAX1( ZERO,
788: $ ONE-T*AQOAP*AAPQ ) )
789: MXSINJ = DMAX1( MXSINJ, DABS( T ) )
790: ELSE
791: *
792: * .. choose correct signum for THETA and rotate
793: *
794: THSIGN = -DSIGN( ONE, AAPQ )
795: IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN
796: T = ONE / ( THETA+THSIGN*
797: $ DSQRT( ONE+THETA*THETA ) )
798: CS = DSQRT( ONE / ( ONE+T*T ) )
799: SN = T*CS
800: MXSINJ = DMAX1( MXSINJ, DABS( SN ) )
801: SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
802: $ ONE+T*APOAQ*AAPQ ) )
803: AAPP = AAPP*DSQRT( DMAX1( ZERO,
804: $ ONE-T*AQOAP*AAPQ ) )
805: *
806: APOAQ = D( p ) / D( q )
807: AQOAP = D( q ) / D( p )
808: IF( D( p ).GE.ONE ) THEN
809: *
810: IF( D( q ).GE.ONE ) THEN
811: FASTR( 3 ) = T*APOAQ
812: FASTR( 4 ) = -T*AQOAP
813: D( p ) = D( p )*CS
814: D( q ) = D( q )*CS
815: CALL DROTM( M, A( 1, p ), 1,
816: $ A( 1, q ), 1,
817: $ FASTR )
818: IF( RSVEC )CALL DROTM( MVL,
819: $ V( 1, p ), 1, V( 1, q ),
820: $ 1, FASTR )
821: ELSE
822: CALL DAXPY( M, -T*AQOAP,
823: $ A( 1, q ), 1,
824: $ A( 1, p ), 1 )
825: CALL DAXPY( M, CS*SN*APOAQ,
826: $ A( 1, p ), 1,
827: $ A( 1, q ), 1 )
828: IF( RSVEC ) THEN
829: CALL DAXPY( MVL, -T*AQOAP,
830: $ V( 1, q ), 1,
831: $ V( 1, p ), 1 )
832: CALL DAXPY( MVL,
833: $ CS*SN*APOAQ,
834: $ V( 1, p ), 1,
835: $ V( 1, q ), 1 )
836: END IF
837: D( p ) = D( p )*CS
838: D( q ) = D( q ) / CS
839: END IF
840: ELSE
841: IF( D( q ).GE.ONE ) THEN
842: CALL DAXPY( M, T*APOAQ,
843: $ A( 1, p ), 1,
844: $ A( 1, q ), 1 )
845: CALL DAXPY( M, -CS*SN*AQOAP,
846: $ A( 1, q ), 1,
847: $ A( 1, p ), 1 )
848: IF( RSVEC ) THEN
849: CALL DAXPY( MVL, T*APOAQ,
850: $ V( 1, p ), 1,
851: $ V( 1, q ), 1 )
852: CALL DAXPY( MVL,
853: $ -CS*SN*AQOAP,
854: $ V( 1, q ), 1,
855: $ V( 1, p ), 1 )
856: END IF
857: D( p ) = D( p ) / CS
858: D( q ) = D( q )*CS
859: ELSE
860: IF( D( p ).GE.D( q ) ) THEN
861: CALL DAXPY( M, -T*AQOAP,
862: $ A( 1, q ), 1,
863: $ A( 1, p ), 1 )
864: CALL DAXPY( M, CS*SN*APOAQ,
865: $ A( 1, p ), 1,
866: $ A( 1, q ), 1 )
867: D( p ) = D( p )*CS
868: D( q ) = D( q ) / CS
869: IF( RSVEC ) THEN
870: CALL DAXPY( MVL,
871: $ -T*AQOAP,
872: $ V( 1, q ), 1,
873: $ V( 1, p ), 1 )
874: CALL DAXPY( MVL,
875: $ CS*SN*APOAQ,
876: $ V( 1, p ), 1,
877: $ V( 1, q ), 1 )
878: END IF
879: ELSE
880: CALL DAXPY( M, T*APOAQ,
881: $ A( 1, p ), 1,
882: $ A( 1, q ), 1 )
883: CALL DAXPY( M,
884: $ -CS*SN*AQOAP,
885: $ A( 1, q ), 1,
886: $ A( 1, p ), 1 )
887: D( p ) = D( p ) / CS
888: D( q ) = D( q )*CS
889: IF( RSVEC ) THEN
890: CALL DAXPY( MVL,
891: $ T*APOAQ, V( 1, p ),
892: $ 1, V( 1, q ), 1 )
893: CALL DAXPY( MVL,
894: $ -CS*SN*AQOAP,
895: $ V( 1, q ), 1,
896: $ V( 1, p ), 1 )
897: END IF
898: END IF
899: END IF
900: END IF
901: END IF
902: *
903: ELSE
904: IF( AAPP.GT.AAQQ ) THEN
905: CALL DCOPY( M, A( 1, p ), 1, WORK,
906: $ 1 )
907: CALL DLASCL( 'G', 0, 0, AAPP, ONE,
908: $ M, 1, WORK, LDA, IERR )
909: CALL DLASCL( 'G', 0, 0, AAQQ, ONE,
910: $ M, 1, A( 1, q ), LDA,
911: $ IERR )
912: TEMP1 = -AAPQ*D( p ) / D( q )
913: CALL DAXPY( M, TEMP1, WORK, 1,
914: $ A( 1, q ), 1 )
915: CALL DLASCL( 'G', 0, 0, ONE, AAQQ,
916: $ M, 1, A( 1, q ), LDA,
917: $ IERR )
918: SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
919: $ ONE-AAPQ*AAPQ ) )
920: MXSINJ = DMAX1( MXSINJ, SFMIN )
921: ELSE
922: CALL DCOPY( M, A( 1, q ), 1, WORK,
923: $ 1 )
924: CALL DLASCL( 'G', 0, 0, AAQQ, ONE,
925: $ M, 1, WORK, LDA, IERR )
926: CALL DLASCL( 'G', 0, 0, AAPP, ONE,
927: $ M, 1, A( 1, p ), LDA,
928: $ IERR )
929: TEMP1 = -AAPQ*D( q ) / D( p )
930: CALL DAXPY( M, TEMP1, WORK, 1,
931: $ A( 1, p ), 1 )
932: CALL DLASCL( 'G', 0, 0, ONE, AAPP,
933: $ M, 1, A( 1, p ), LDA,
934: $ IERR )
935: SVA( p ) = AAPP*DSQRT( DMAX1( ZERO,
936: $ ONE-AAPQ*AAPQ ) )
937: MXSINJ = DMAX1( MXSINJ, SFMIN )
938: END IF
939: END IF
940: * END IF ROTOK THEN ... ELSE
941: *
942: * In the case of cancellation in updating SVA(q)
943: * .. recompute SVA(q)
944: IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
945: $ THEN
946: IF( ( AAQQ.LT.ROOTBIG ) .AND.
947: $ ( AAQQ.GT.ROOTSFMIN ) ) THEN
948: SVA( q ) = DNRM2( M, A( 1, q ), 1 )*
949: $ D( q )
950: ELSE
951: T = ZERO
952: AAQQ = ONE
953: CALL DLASSQ( M, A( 1, q ), 1, T,
954: $ AAQQ )
955: SVA( q ) = T*DSQRT( AAQQ )*D( q )
956: END IF
957: END IF
958: IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN
959: IF( ( AAPP.LT.ROOTBIG ) .AND.
960: $ ( AAPP.GT.ROOTSFMIN ) ) THEN
961: AAPP = DNRM2( M, A( 1, p ), 1 )*
962: $ D( p )
963: ELSE
964: T = ZERO
965: AAPP = ONE
966: CALL DLASSQ( M, A( 1, p ), 1, T,
967: $ AAPP )
968: AAPP = T*DSQRT( AAPP )*D( p )
969: END IF
970: SVA( p ) = AAPP
971: END IF
972: * end of OK rotation
973: ELSE
974: NOTROT = NOTROT + 1
975: PSKIPPED = PSKIPPED + 1
976: IJBLSK = IJBLSK + 1
977: END IF
978: ELSE
979: NOTROT = NOTROT + 1
980: PSKIPPED = PSKIPPED + 1
981: IJBLSK = IJBLSK + 1
982: END IF
983: *
984: IF( ( i.LE.SWBAND ) .AND. ( IJBLSK.GE.BLSKIP ) )
985: $ THEN
986: SVA( p ) = AAPP
987: NOTROT = 0
988: GO TO 2011
989: END IF
990: IF( ( i.LE.SWBAND ) .AND.
991: $ ( PSKIPPED.GT.ROWSKIP ) ) THEN
992: AAPP = -AAPP
993: NOTROT = 0
994: GO TO 2203
995: END IF
996: *
997: 2200 CONTINUE
998: * end of the q-loop
999: 2203 CONTINUE
1000: *
1001: SVA( p ) = AAPP
1002: *
1003: ELSE
1004: IF( AAPP.EQ.ZERO )NOTROT = NOTROT +
1005: $ MIN0( jgl+KBL-1, N ) - jgl + 1
1006: IF( AAPP.LT.ZERO )NOTROT = 0
1007: END IF
1008:
1009: 2100 CONTINUE
1010: * end of the p-loop
1011: 2010 CONTINUE
1012: * end of the jbc-loop
1013: 2011 CONTINUE
1014: *2011 bailed out of the jbc-loop
1015: DO 2012 p = igl, MIN0( igl+KBL-1, N )
1016: SVA( p ) = DABS( SVA( p ) )
1017: 2012 CONTINUE
1018: *
1019: 2000 CONTINUE
1020: *2000 :: end of the ibr-loop
1021: *
1022: * .. update SVA(N)
1023: IF( ( SVA( N ).LT.ROOTBIG ) .AND. ( SVA( N ).GT.ROOTSFMIN ) )
1024: $ THEN
1025: SVA( N ) = DNRM2( M, A( 1, N ), 1 )*D( N )
1026: ELSE
1027: T = ZERO
1028: AAPP = ONE
1029: CALL DLASSQ( M, A( 1, N ), 1, T, AAPP )
1030: SVA( N ) = T*DSQRT( AAPP )*D( N )
1031: END IF
1032: *
1033: * Additional steering devices
1034: *
1035: IF( ( i.LT.SWBAND ) .AND. ( ( MXAAPQ.LE.ROOTTOL ) .OR.
1036: $ ( ISWROT.LE.N ) ) )SWBAND = i
1037: *
1038: IF( ( i.GT.SWBAND+1 ) .AND. ( MXAAPQ.LT.DBLE( N )*TOL ) .AND.
1039: $ ( DBLE( N )*MXAAPQ*MXSINJ.LT.TOL ) ) THEN
1040: GO TO 1994
1041: END IF
1042: *
1043: IF( NOTROT.GE.EMPTSW )GO TO 1994
1044:
1045: 1993 CONTINUE
1046: * end i=1:NSWEEP loop
1047: * #:) Reaching this point means that the procedure has comleted the given
1048: * number of iterations.
1049: INFO = NSWEEP - 1
1050: GO TO 1995
1051: 1994 CONTINUE
1052: * #:) Reaching this point means that during the i-th sweep all pivots were
1053: * below the given tolerance, causing early exit.
1054: *
1055: INFO = 0
1056: * #:) INFO = 0 confirms successful iterations.
1057: 1995 CONTINUE
1058: *
1059: * Sort the vector D.
1060: DO 5991 p = 1, N - 1
1061: q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
1062: IF( p.NE.q ) THEN
1063: TEMP1 = SVA( p )
1064: SVA( p ) = SVA( q )
1065: SVA( q ) = TEMP1
1066: TEMP1 = D( p )
1067: D( p ) = D( q )
1068: D( q ) = TEMP1
1069: CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
1070: IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 )
1071: END IF
1072: 5991 CONTINUE
1073: *
1074: RETURN
1075: * ..
1076: * .. END OF DGSVJ0
1077: * ..
1078: END
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