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1.1 bertrand 1: SUBROUTINE DGSVJ0( JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS,
2: + SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
3: *
1.4 ! bertrand 4: * -- LAPACK routine (version 3.3.0) --
1.1 bertrand 5: *
6: * -- Contributed by Zlatko Drmac of the University of Zagreb and --
7: * -- Kresimir Veselic of the Fernuniversitaet Hagen --
1.4 ! bertrand 8: * November 2010
1.1 bertrand 9: *
10: * -- LAPACK is a software package provided by Univ. of Tennessee, --
11: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
12: *
13: * This routine is also part of SIGMA (version 1.23, October 23. 2008.)
14: * SIGMA is a library of algorithms for highly accurate algorithms for
15: * computation of SVD, PSVD, QSVD, (H,K)-SVD, and for solution of the
16: * eigenvalue problems Hx = lambda M x, H M x = lambda x with H, M > 0.
17: *
18: IMPLICIT NONE
19: * .. Scalar Arguments ..
20: INTEGER INFO, LDA, LDV, LWORK, M, MV, N, NSWEEP
21: DOUBLE PRECISION EPS, SFMIN, TOL
22: CHARACTER*1 JOBV
23: * ..
24: * .. Array Arguments ..
25: DOUBLE PRECISION A( LDA, * ), SVA( N ), D( N ), V( LDV, * ),
26: + WORK( LWORK )
27: * ..
28: *
29: * Purpose
30: * =======
31: *
32: * DGSVJ0 is called from DGESVJ as a pre-processor and that is its main
33: * purpose. It applies Jacobi rotations in the same way as DGESVJ does, but
34: * it does not check convergence (stopping criterion). Few tuning
35: * parameters (marked by [TP]) are available for the implementer.
36: *
37: * Further Details
38: * ~~~~~~~~~~~~~~~
39: * DGSVJ0 is used just to enable SGESVJ to call a simplified version of
40: * itself to work on a submatrix of the original matrix.
41: *
42: * Contributors
43: * ~~~~~~~~~~~~
44: * Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
45: *
46: * Bugs, Examples and Comments
47: * ~~~~~~~~~~~~~~~~~~~~~~~~~~~
48: * Please report all bugs and send interesting test examples and comments to
49: * drmac@math.hr. Thank you.
50: *
51: * Arguments
52: * =========
53: *
54: * JOBV (input) CHARACTER*1
55: * Specifies whether the output from this procedure is used
56: * to compute the matrix V:
57: * = 'V': the product of the Jacobi rotations is accumulated
58: * by postmulyiplying the N-by-N array V.
59: * (See the description of V.)
60: * = 'A': the product of the Jacobi rotations is accumulated
61: * by postmulyiplying the MV-by-N array V.
62: * (See the descriptions of MV and V.)
63: * = 'N': the Jacobi rotations are not accumulated.
64: *
65: * M (input) INTEGER
66: * The number of rows of the input matrix A. M >= 0.
67: *
68: * N (input) INTEGER
69: * The number of columns of the input matrix A.
70: * M >= N >= 0.
71: *
72: * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
73: * On entry, M-by-N matrix A, such that A*diag(D) represents
74: * the input matrix.
75: * On exit,
76: * A_onexit * D_onexit represents the input matrix A*diag(D)
77: * post-multiplied by a sequence of Jacobi rotations, where the
78: * rotation threshold and the total number of sweeps are given in
79: * TOL and NSWEEP, respectively.
80: * (See the descriptions of D, TOL and NSWEEP.)
81: *
82: * LDA (input) INTEGER
83: * The leading dimension of the array A. LDA >= max(1,M).
84: *
85: * D (input/workspace/output) DOUBLE PRECISION array, dimension (N)
86: * The array D accumulates the scaling factors from the fast scaled
87: * Jacobi rotations.
88: * On entry, A*diag(D) represents the input matrix.
89: * On exit, A_onexit*diag(D_onexit) represents the input matrix
90: * post-multiplied by a sequence of Jacobi rotations, where the
91: * rotation threshold and the total number of sweeps are given in
92: * TOL and NSWEEP, respectively.
93: * (See the descriptions of A, TOL and NSWEEP.)
94: *
95: * SVA (input/workspace/output) DOUBLE PRECISION array, dimension (N)
96: * On entry, SVA contains the Euclidean norms of the columns of
97: * the matrix A*diag(D).
98: * On exit, SVA contains the Euclidean norms of the columns of
99: * the matrix onexit*diag(D_onexit).
100: *
101: * MV (input) INTEGER
102: * If JOBV .EQ. 'A', then MV rows of V are post-multipled by a
103: * sequence of Jacobi rotations.
104: * If JOBV = 'N', then MV is not referenced.
105: *
106: * V (input/output) DOUBLE PRECISION array, dimension (LDV,N)
107: * If JOBV .EQ. 'V' then N rows of V are post-multipled by a
108: * sequence of Jacobi rotations.
109: * If JOBV .EQ. 'A' then MV rows of V are post-multipled by a
110: * sequence of Jacobi rotations.
111: * If JOBV = 'N', then V is not referenced.
112: *
113: * LDV (input) INTEGER
114: * The leading dimension of the array V, LDV >= 1.
115: * If JOBV = 'V', LDV .GE. N.
116: * If JOBV = 'A', LDV .GE. MV.
117: *
118: * EPS (input) DOUBLE PRECISION
119: * EPS = DLAMCH('Epsilon')
120: *
121: * SFMIN (input) DOUBLE PRECISION
122: * SFMIN = DLAMCH('Safe Minimum')
123: *
124: * TOL (input) DOUBLE PRECISION
125: * TOL is the threshold for Jacobi rotations. For a pair
126: * A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
127: * applied only if DABS(COS(angle(A(:,p),A(:,q)))) .GT. TOL.
128: *
129: * NSWEEP (input) INTEGER
130: * NSWEEP is the number of sweeps of Jacobi rotations to be
131: * performed.
132: *
133: * WORK (workspace) DOUBLE PRECISION array, dimension (LWORK)
134: *
135: * LWORK (input) INTEGER
136: * LWORK is the dimension of WORK. LWORK .GE. M.
137: *
138: * INFO (output) INTEGER
139: * = 0 : successful exit.
140: * < 0 : if INFO = -i, then the i-th argument had an illegal value
141: *
142: * =====================================================================
143: *
144: * .. Local Parameters ..
145: DOUBLE PRECISION ZERO, HALF, ONE, TWO
146: PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0,
147: + TWO = 2.0D0 )
148: * ..
149: * .. Local Scalars ..
150: DOUBLE PRECISION AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG,
151: + BIGTHETA, CS, MXAAPQ, MXSINJ, ROOTBIG, ROOTEPS,
152: + ROOTSFMIN, ROOTTOL, SMALL, SN, T, TEMP1, THETA,
153: + THSIGN
154: INTEGER BLSKIP, EMPTSW, i, ibr, IERR, igl, IJBLSK, ir1,
155: + ISWROT, jbc, jgl, KBL, LKAHEAD, MVL, NBL,
156: + NOTROT, p, PSKIPPED, q, ROWSKIP, SWBAND
157: LOGICAL APPLV, ROTOK, RSVEC
158: * ..
159: * .. Local Arrays ..
160: DOUBLE PRECISION FASTR( 5 )
161: * ..
162: * .. Intrinsic Functions ..
163: INTRINSIC DABS, DMAX1, DBLE, MIN0, DSIGN, DSQRT
164: * ..
165: * .. External Functions ..
166: DOUBLE PRECISION DDOT, DNRM2
167: INTEGER IDAMAX
168: LOGICAL LSAME
169: EXTERNAL IDAMAX, LSAME, DDOT, DNRM2
170: * ..
171: * .. External Subroutines ..
172: EXTERNAL DAXPY, DCOPY, DLASCL, DLASSQ, DROTM, DSWAP
173: * ..
174: * .. Executable Statements ..
175: *
176: APPLV = LSAME( JOBV, 'A' )
177: RSVEC = LSAME( JOBV, 'V' )
178: IF( .NOT.( RSVEC .OR. APPLV .OR. LSAME( JOBV, 'N' ) ) ) THEN
179: INFO = -1
180: ELSE IF( M.LT.0 ) THEN
181: INFO = -2
182: ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN
183: INFO = -3
184: ELSE IF( LDA.LT.M ) THEN
185: INFO = -5
1.4 ! bertrand 186: ELSE IF( ( RSVEC.OR.APPLV ) .AND. ( MV.LT.0 ) ) THEN
1.1 bertrand 187: INFO = -8
1.4 ! bertrand 188: ELSE IF( ( RSVEC.AND.( LDV.LT.N ) ).OR.
! 189: & ( APPLV.AND.( LDV.LT.MV ) ) ) THEN
1.1 bertrand 190: INFO = -10
191: ELSE IF( TOL.LE.EPS ) THEN
192: INFO = -13
193: ELSE IF( NSWEEP.LT.0 ) THEN
194: INFO = -14
195: ELSE IF( LWORK.LT.M ) THEN
196: INFO = -16
197: ELSE
198: INFO = 0
199: END IF
200: *
201: * #:(
202: IF( INFO.NE.0 ) THEN
203: CALL XERBLA( 'DGSVJ0', -INFO )
204: RETURN
205: END IF
206: *
207: IF( RSVEC ) THEN
208: MVL = N
209: ELSE IF( APPLV ) THEN
210: MVL = MV
211: END IF
212: RSVEC = RSVEC .OR. APPLV
213:
214: ROOTEPS = DSQRT( EPS )
215: ROOTSFMIN = DSQRT( SFMIN )
216: SMALL = SFMIN / EPS
217: BIG = ONE / SFMIN
218: ROOTBIG = ONE / ROOTSFMIN
219: BIGTHETA = ONE / ROOTEPS
220: ROOTTOL = DSQRT( TOL )
221: *
222: *
223: * -#- Row-cyclic Jacobi SVD algorithm with column pivoting -#-
224: *
225: EMPTSW = ( N*( N-1 ) ) / 2
226: NOTROT = 0
227: FASTR( 1 ) = ZERO
228: *
229: * -#- Row-cyclic pivot strategy with de Rijk's pivoting -#-
230: *
231:
232: SWBAND = 0
233: *[TP] SWBAND is a tuning parameter. It is meaningful and effective
234: * if SGESVJ is used as a computational routine in the preconditioned
235: * Jacobi SVD algorithm SGESVJ. For sweeps i=1:SWBAND the procedure
236: * ......
237:
238: KBL = MIN0( 8, N )
239: *[TP] KBL is a tuning parameter that defines the tile size in the
240: * tiling of the p-q loops of pivot pairs. In general, an optimal
241: * value of KBL depends on the matrix dimensions and on the
242: * parameters of the computer's memory.
243: *
244: NBL = N / KBL
245: IF( ( NBL*KBL ).NE.N )NBL = NBL + 1
246:
247: BLSKIP = ( KBL**2 ) + 1
248: *[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.
249:
250: ROWSKIP = MIN0( 5, KBL )
251: *[TP] ROWSKIP is a tuning parameter.
252:
253: LKAHEAD = 1
254: *[TP] LKAHEAD is a tuning parameter.
255: SWBAND = 0
256: PSKIPPED = 0
257: *
258: DO 1993 i = 1, NSWEEP
259: * .. go go go ...
260: *
261: MXAAPQ = ZERO
262: MXSINJ = ZERO
263: ISWROT = 0
264: *
265: NOTROT = 0
266: PSKIPPED = 0
267: *
268: DO 2000 ibr = 1, NBL
269:
270: igl = ( ibr-1 )*KBL + 1
271: *
272: DO 1002 ir1 = 0, MIN0( LKAHEAD, NBL-ibr )
273: *
274: igl = igl + ir1*KBL
275: *
276: DO 2001 p = igl, MIN0( igl+KBL-1, N-1 )
277:
278: * .. de Rijk's pivoting
279: q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
280: IF( p.NE.q ) THEN
281: CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
282: IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1,
283: + V( 1, q ), 1 )
284: TEMP1 = SVA( p )
285: SVA( p ) = SVA( q )
286: SVA( q ) = TEMP1
287: TEMP1 = D( p )
288: D( p ) = D( q )
289: D( q ) = TEMP1
290: END IF
291: *
292: IF( ir1.EQ.0 ) THEN
293: *
294: * Column norms are periodically updated by explicit
295: * norm computation.
296: * Caveat:
297: * Some BLAS implementations compute DNRM2(M,A(1,p),1)
298: * as DSQRT(DDOT(M,A(1,p),1,A(1,p),1)), which may result in
299: * overflow for ||A(:,p)||_2 > DSQRT(overflow_threshold), and
300: * undeflow for ||A(:,p)||_2 < DSQRT(underflow_threshold).
301: * Hence, DNRM2 cannot be trusted, not even in the case when
302: * the true norm is far from the under(over)flow boundaries.
303: * If properly implemented DNRM2 is available, the IF-THEN-ELSE
304: * below should read "AAPP = DNRM2( M, A(1,p), 1 ) * D(p)".
305: *
306: IF( ( SVA( p ).LT.ROOTBIG ) .AND.
307: + ( SVA( p ).GT.ROOTSFMIN ) ) THEN
308: SVA( p ) = DNRM2( M, A( 1, p ), 1 )*D( p )
309: ELSE
310: TEMP1 = ZERO
1.4 ! bertrand 311: AAPP = ONE
1.1 bertrand 312: CALL DLASSQ( M, A( 1, p ), 1, TEMP1, AAPP )
313: SVA( p ) = TEMP1*DSQRT( AAPP )*D( p )
314: END IF
315: AAPP = SVA( p )
316: ELSE
317: AAPP = SVA( p )
318: END IF
319:
320: *
321: IF( AAPP.GT.ZERO ) THEN
322: *
323: PSKIPPED = 0
324: *
325: DO 2002 q = p + 1, MIN0( igl+KBL-1, N )
326: *
327: AAQQ = SVA( q )
328:
329: IF( AAQQ.GT.ZERO ) THEN
330: *
331: AAPP0 = AAPP
332: IF( AAQQ.GE.ONE ) THEN
333: ROTOK = ( SMALL*AAPP ).LE.AAQQ
334: IF( AAPP.LT.( BIG / AAQQ ) ) THEN
335: AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
336: + q ), 1 )*D( p )*D( q ) / AAQQ )
337: + / AAPP
338: ELSE
339: CALL DCOPY( M, A( 1, p ), 1, WORK, 1 )
340: CALL DLASCL( 'G', 0, 0, AAPP, D( p ),
341: + M, 1, WORK, LDA, IERR )
342: AAPQ = DDOT( M, WORK, 1, A( 1, q ),
343: + 1 )*D( q ) / AAQQ
344: END IF
345: ELSE
346: ROTOK = AAPP.LE.( AAQQ / SMALL )
347: IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
348: AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
349: + q ), 1 )*D( p )*D( q ) / AAQQ )
350: + / AAPP
351: ELSE
352: CALL DCOPY( M, A( 1, q ), 1, WORK, 1 )
353: CALL DLASCL( 'G', 0, 0, AAQQ, D( q ),
354: + M, 1, WORK, LDA, IERR )
355: AAPQ = DDOT( M, WORK, 1, A( 1, p ),
356: + 1 )*D( p ) / AAPP
357: END IF
358: END IF
359: *
360: MXAAPQ = DMAX1( MXAAPQ, DABS( AAPQ ) )
361: *
362: * TO rotate or NOT to rotate, THAT is the question ...
363: *
364: IF( DABS( AAPQ ).GT.TOL ) THEN
365: *
366: * .. rotate
367: * ROTATED = ROTATED + ONE
368: *
369: IF( ir1.EQ.0 ) THEN
370: NOTROT = 0
371: PSKIPPED = 0
372: ISWROT = ISWROT + 1
373: END IF
374: *
375: IF( ROTOK ) THEN
376: *
377: AQOAP = AAQQ / AAPP
378: APOAQ = AAPP / AAQQ
379: THETA = -HALF*DABS( AQOAP-APOAQ ) /
380: + AAPQ
381: *
382: IF( DABS( THETA ).GT.BIGTHETA ) THEN
383: *
384: T = HALF / THETA
385: FASTR( 3 ) = T*D( p ) / D( q )
386: FASTR( 4 ) = -T*D( q ) / D( p )
387: CALL DROTM( M, A( 1, p ), 1,
388: + A( 1, q ), 1, FASTR )
389: IF( RSVEC )CALL DROTM( MVL,
390: + V( 1, p ), 1,
391: + V( 1, q ), 1,
392: + FASTR )
393: SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
394: + ONE+T*APOAQ*AAPQ ) )
1.4 ! bertrand 395: AAPP = AAPP*DSQRT( DMAX1( ZERO,
! 396: + ONE-T*AQOAP*AAPQ ) )
1.1 bertrand 397: MXSINJ = DMAX1( MXSINJ, DABS( T ) )
398: *
399: ELSE
400: *
401: * .. choose correct signum for THETA and rotate
402: *
403: THSIGN = -DSIGN( ONE, AAPQ )
404: T = ONE / ( THETA+THSIGN*
405: + DSQRT( ONE+THETA*THETA ) )
406: CS = DSQRT( ONE / ( ONE+T*T ) )
407: SN = T*CS
408: *
409: MXSINJ = DMAX1( MXSINJ, DABS( SN ) )
410: SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
411: + ONE+T*APOAQ*AAPQ ) )
412: AAPP = AAPP*DSQRT( DMAX1( ZERO,
413: + ONE-T*AQOAP*AAPQ ) )
414: *
415: APOAQ = D( p ) / D( q )
416: AQOAP = D( q ) / D( p )
417: IF( D( p ).GE.ONE ) THEN
418: IF( D( q ).GE.ONE ) THEN
419: FASTR( 3 ) = T*APOAQ
420: FASTR( 4 ) = -T*AQOAP
421: D( p ) = D( p )*CS
422: D( q ) = D( q )*CS
423: CALL DROTM( M, A( 1, p ), 1,
424: + A( 1, q ), 1,
425: + FASTR )
426: IF( RSVEC )CALL DROTM( MVL,
427: + V( 1, p ), 1, V( 1, q ),
428: + 1, FASTR )
429: ELSE
430: CALL DAXPY( M, -T*AQOAP,
431: + A( 1, q ), 1,
432: + A( 1, p ), 1 )
433: CALL DAXPY( M, CS*SN*APOAQ,
434: + A( 1, p ), 1,
435: + A( 1, q ), 1 )
436: D( p ) = D( p )*CS
437: D( q ) = D( q ) / CS
438: IF( RSVEC ) THEN
439: CALL DAXPY( MVL, -T*AQOAP,
440: + V( 1, q ), 1,
441: + V( 1, p ), 1 )
442: CALL DAXPY( MVL,
443: + CS*SN*APOAQ,
444: + V( 1, p ), 1,
445: + V( 1, q ), 1 )
446: END IF
447: END IF
448: ELSE
449: IF( D( q ).GE.ONE ) THEN
450: CALL DAXPY( M, T*APOAQ,
451: + A( 1, p ), 1,
452: + A( 1, q ), 1 )
453: CALL DAXPY( M, -CS*SN*AQOAP,
454: + A( 1, q ), 1,
455: + A( 1, p ), 1 )
456: D( p ) = D( p ) / CS
457: D( q ) = D( q )*CS
458: IF( RSVEC ) THEN
459: CALL DAXPY( MVL, T*APOAQ,
460: + V( 1, p ), 1,
461: + V( 1, q ), 1 )
462: CALL DAXPY( MVL,
463: + -CS*SN*AQOAP,
464: + V( 1, q ), 1,
465: + V( 1, p ), 1 )
466: END IF
467: ELSE
468: IF( D( p ).GE.D( q ) ) THEN
469: CALL DAXPY( M, -T*AQOAP,
470: + A( 1, q ), 1,
471: + A( 1, p ), 1 )
472: CALL DAXPY( M, CS*SN*APOAQ,
473: + A( 1, p ), 1,
474: + A( 1, q ), 1 )
475: D( p ) = D( p )*CS
476: D( q ) = D( q ) / CS
477: IF( RSVEC ) THEN
478: CALL DAXPY( MVL,
479: + -T*AQOAP,
480: + V( 1, q ), 1,
481: + V( 1, p ), 1 )
482: CALL DAXPY( MVL,
483: + CS*SN*APOAQ,
484: + V( 1, p ), 1,
485: + V( 1, q ), 1 )
486: END IF
487: ELSE
488: CALL DAXPY( M, T*APOAQ,
489: + A( 1, p ), 1,
490: + A( 1, q ), 1 )
491: CALL DAXPY( M,
492: + -CS*SN*AQOAP,
493: + A( 1, q ), 1,
494: + A( 1, p ), 1 )
495: D( p ) = D( p ) / CS
496: D( q ) = D( q )*CS
497: IF( RSVEC ) THEN
498: CALL DAXPY( MVL,
499: + T*APOAQ, V( 1, p ),
500: + 1, V( 1, q ), 1 )
501: CALL DAXPY( MVL,
502: + -CS*SN*AQOAP,
503: + V( 1, q ), 1,
504: + V( 1, p ), 1 )
505: END IF
506: END IF
507: END IF
508: END IF
509: END IF
510: *
511: ELSE
512: * .. have to use modified Gram-Schmidt like transformation
513: CALL DCOPY( M, A( 1, p ), 1, WORK, 1 )
514: CALL DLASCL( 'G', 0, 0, AAPP, ONE, M,
515: + 1, WORK, LDA, IERR )
516: CALL DLASCL( 'G', 0, 0, AAQQ, ONE, M,
517: + 1, A( 1, q ), LDA, IERR )
518: TEMP1 = -AAPQ*D( p ) / D( q )
519: CALL DAXPY( M, TEMP1, WORK, 1,
520: + A( 1, q ), 1 )
521: CALL DLASCL( 'G', 0, 0, ONE, AAQQ, M,
522: + 1, A( 1, q ), LDA, IERR )
523: SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
524: + ONE-AAPQ*AAPQ ) )
525: MXSINJ = DMAX1( MXSINJ, SFMIN )
526: END IF
527: * END IF ROTOK THEN ... ELSE
528: *
529: * In the case of cancellation in updating SVA(q), SVA(p)
530: * recompute SVA(q), SVA(p).
531: IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
532: + THEN
533: IF( ( AAQQ.LT.ROOTBIG ) .AND.
534: + ( AAQQ.GT.ROOTSFMIN ) ) THEN
535: SVA( q ) = DNRM2( M, A( 1, q ), 1 )*
536: + D( q )
537: ELSE
538: T = ZERO
1.4 ! bertrand 539: AAQQ = ONE
1.1 bertrand 540: CALL DLASSQ( M, A( 1, q ), 1, T,
541: + AAQQ )
542: SVA( q ) = T*DSQRT( AAQQ )*D( q )
543: END IF
544: END IF
545: IF( ( AAPP / AAPP0 ).LE.ROOTEPS ) THEN
546: IF( ( AAPP.LT.ROOTBIG ) .AND.
547: + ( AAPP.GT.ROOTSFMIN ) ) THEN
548: AAPP = DNRM2( M, A( 1, p ), 1 )*
549: + D( p )
550: ELSE
551: T = ZERO
1.4 ! bertrand 552: AAPP = ONE
1.1 bertrand 553: CALL DLASSQ( M, A( 1, p ), 1, T,
554: + AAPP )
555: AAPP = T*DSQRT( AAPP )*D( p )
556: END IF
557: SVA( p ) = AAPP
558: END IF
559: *
560: ELSE
561: * A(:,p) and A(:,q) already numerically orthogonal
562: IF( ir1.EQ.0 )NOTROT = NOTROT + 1
563: PSKIPPED = PSKIPPED + 1
564: END IF
565: ELSE
566: * A(:,q) is zero column
567: IF( ir1.EQ.0 )NOTROT = NOTROT + 1
568: PSKIPPED = PSKIPPED + 1
569: END IF
570: *
571: IF( ( i.LE.SWBAND ) .AND.
572: + ( PSKIPPED.GT.ROWSKIP ) ) THEN
573: IF( ir1.EQ.0 )AAPP = -AAPP
574: NOTROT = 0
575: GO TO 2103
576: END IF
577: *
578: 2002 CONTINUE
579: * END q-LOOP
580: *
581: 2103 CONTINUE
582: * bailed out of q-loop
583:
584: SVA( p ) = AAPP
585:
586: ELSE
587: SVA( p ) = AAPP
588: IF( ( ir1.EQ.0 ) .AND. ( AAPP.EQ.ZERO ) )
589: + NOTROT = NOTROT + MIN0( igl+KBL-1, N ) - p
590: END IF
591: *
592: 2001 CONTINUE
593: * end of the p-loop
594: * end of doing the block ( ibr, ibr )
595: 1002 CONTINUE
596: * end of ir1-loop
597: *
598: *........................................................
599: * ... go to the off diagonal blocks
600: *
601: igl = ( ibr-1 )*KBL + 1
602: *
603: DO 2010 jbc = ibr + 1, NBL
604: *
605: jgl = ( jbc-1 )*KBL + 1
606: *
607: * doing the block at ( ibr, jbc )
608: *
609: IJBLSK = 0
610: DO 2100 p = igl, MIN0( igl+KBL-1, N )
611: *
612: AAPP = SVA( p )
613: *
614: IF( AAPP.GT.ZERO ) THEN
615: *
616: PSKIPPED = 0
617: *
618: DO 2200 q = jgl, MIN0( jgl+KBL-1, N )
619: *
620: AAQQ = SVA( q )
621: *
622: IF( AAQQ.GT.ZERO ) THEN
623: AAPP0 = AAPP
624: *
625: * -#- M x 2 Jacobi SVD -#-
626: *
627: * -#- Safe Gram matrix computation -#-
628: *
629: IF( AAQQ.GE.ONE ) THEN
630: IF( AAPP.GE.AAQQ ) THEN
631: ROTOK = ( SMALL*AAPP ).LE.AAQQ
632: ELSE
633: ROTOK = ( SMALL*AAQQ ).LE.AAPP
634: END IF
635: IF( AAPP.LT.( BIG / AAQQ ) ) THEN
636: AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
637: + q ), 1 )*D( p )*D( q ) / AAQQ )
638: + / AAPP
639: ELSE
640: CALL DCOPY( M, A( 1, p ), 1, WORK, 1 )
641: CALL DLASCL( 'G', 0, 0, AAPP, D( p ),
642: + M, 1, WORK, LDA, IERR )
643: AAPQ = DDOT( M, WORK, 1, A( 1, q ),
644: + 1 )*D( q ) / AAQQ
645: END IF
646: ELSE
647: IF( AAPP.GE.AAQQ ) THEN
648: ROTOK = AAPP.LE.( AAQQ / SMALL )
649: ELSE
650: ROTOK = AAQQ.LE.( AAPP / SMALL )
651: END IF
652: IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
653: AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1,
654: + q ), 1 )*D( p )*D( q ) / AAQQ )
655: + / AAPP
656: ELSE
657: CALL DCOPY( M, A( 1, q ), 1, WORK, 1 )
658: CALL DLASCL( 'G', 0, 0, AAQQ, D( q ),
659: + M, 1, WORK, LDA, IERR )
660: AAPQ = DDOT( M, WORK, 1, A( 1, p ),
661: + 1 )*D( p ) / AAPP
662: END IF
663: END IF
664: *
665: MXAAPQ = DMAX1( MXAAPQ, DABS( AAPQ ) )
666: *
667: * TO rotate or NOT to rotate, THAT is the question ...
668: *
669: IF( DABS( AAPQ ).GT.TOL ) THEN
670: NOTROT = 0
671: * ROTATED = ROTATED + 1
672: PSKIPPED = 0
673: ISWROT = ISWROT + 1
674: *
675: IF( ROTOK ) THEN
676: *
677: AQOAP = AAQQ / AAPP
678: APOAQ = AAPP / AAQQ
679: THETA = -HALF*DABS( AQOAP-APOAQ ) /
680: + AAPQ
681: IF( AAQQ.GT.AAPP0 )THETA = -THETA
682: *
683: IF( DABS( THETA ).GT.BIGTHETA ) THEN
684: T = HALF / THETA
685: FASTR( 3 ) = T*D( p ) / D( q )
686: FASTR( 4 ) = -T*D( q ) / D( p )
687: CALL DROTM( M, A( 1, p ), 1,
688: + A( 1, q ), 1, FASTR )
689: IF( RSVEC )CALL DROTM( MVL,
690: + V( 1, p ), 1,
691: + V( 1, q ), 1,
692: + FASTR )
693: SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
694: + ONE+T*APOAQ*AAPQ ) )
695: AAPP = AAPP*DSQRT( DMAX1( ZERO,
696: + ONE-T*AQOAP*AAPQ ) )
697: MXSINJ = DMAX1( MXSINJ, DABS( T ) )
698: ELSE
699: *
700: * .. choose correct signum for THETA and rotate
701: *
702: THSIGN = -DSIGN( ONE, AAPQ )
703: IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN
704: T = ONE / ( THETA+THSIGN*
705: + DSQRT( ONE+THETA*THETA ) )
706: CS = DSQRT( ONE / ( ONE+T*T ) )
707: SN = T*CS
708: MXSINJ = DMAX1( MXSINJ, DABS( SN ) )
709: SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
710: + ONE+T*APOAQ*AAPQ ) )
1.4 ! bertrand 711: AAPP = AAPP*DSQRT( DMAX1( ZERO,
! 712: + ONE-T*AQOAP*AAPQ ) )
1.1 bertrand 713: *
714: APOAQ = D( p ) / D( q )
715: AQOAP = D( q ) / D( p )
716: IF( D( p ).GE.ONE ) THEN
717: *
718: IF( D( q ).GE.ONE ) THEN
719: FASTR( 3 ) = T*APOAQ
720: FASTR( 4 ) = -T*AQOAP
721: D( p ) = D( p )*CS
722: D( q ) = D( q )*CS
723: CALL DROTM( M, A( 1, p ), 1,
724: + A( 1, q ), 1,
725: + FASTR )
726: IF( RSVEC )CALL DROTM( MVL,
727: + V( 1, p ), 1, V( 1, q ),
728: + 1, FASTR )
729: ELSE
730: CALL DAXPY( M, -T*AQOAP,
731: + A( 1, q ), 1,
732: + A( 1, p ), 1 )
733: CALL DAXPY( M, CS*SN*APOAQ,
734: + A( 1, p ), 1,
735: + A( 1, q ), 1 )
736: IF( RSVEC ) THEN
737: CALL DAXPY( MVL, -T*AQOAP,
738: + V( 1, q ), 1,
739: + V( 1, p ), 1 )
740: CALL DAXPY( MVL,
741: + CS*SN*APOAQ,
742: + V( 1, p ), 1,
743: + V( 1, q ), 1 )
744: END IF
745: D( p ) = D( p )*CS
746: D( q ) = D( q ) / CS
747: END IF
748: ELSE
749: IF( D( q ).GE.ONE ) THEN
750: CALL DAXPY( M, T*APOAQ,
751: + A( 1, p ), 1,
752: + A( 1, q ), 1 )
753: CALL DAXPY( M, -CS*SN*AQOAP,
754: + A( 1, q ), 1,
755: + A( 1, p ), 1 )
756: IF( RSVEC ) THEN
757: CALL DAXPY( MVL, T*APOAQ,
758: + V( 1, p ), 1,
759: + V( 1, q ), 1 )
760: CALL DAXPY( MVL,
761: + -CS*SN*AQOAP,
762: + V( 1, q ), 1,
763: + V( 1, p ), 1 )
764: END IF
765: D( p ) = D( p ) / CS
766: D( q ) = D( q )*CS
767: ELSE
768: IF( D( p ).GE.D( q ) ) THEN
769: CALL DAXPY( M, -T*AQOAP,
770: + A( 1, q ), 1,
771: + A( 1, p ), 1 )
772: CALL DAXPY( M, CS*SN*APOAQ,
773: + A( 1, p ), 1,
774: + A( 1, q ), 1 )
775: D( p ) = D( p )*CS
776: D( q ) = D( q ) / CS
777: IF( RSVEC ) THEN
778: CALL DAXPY( MVL,
779: + -T*AQOAP,
780: + V( 1, q ), 1,
781: + V( 1, p ), 1 )
782: CALL DAXPY( MVL,
783: + CS*SN*APOAQ,
784: + V( 1, p ), 1,
785: + V( 1, q ), 1 )
786: END IF
787: ELSE
788: CALL DAXPY( M, T*APOAQ,
789: + A( 1, p ), 1,
790: + A( 1, q ), 1 )
791: CALL DAXPY( M,
792: + -CS*SN*AQOAP,
793: + A( 1, q ), 1,
794: + A( 1, p ), 1 )
795: D( p ) = D( p ) / CS
796: D( q ) = D( q )*CS
797: IF( RSVEC ) THEN
798: CALL DAXPY( MVL,
799: + T*APOAQ, V( 1, p ),
800: + 1, V( 1, q ), 1 )
801: CALL DAXPY( MVL,
802: + -CS*SN*AQOAP,
803: + V( 1, q ), 1,
804: + V( 1, p ), 1 )
805: END IF
806: END IF
807: END IF
808: END IF
809: END IF
810: *
811: ELSE
812: IF( AAPP.GT.AAQQ ) THEN
813: CALL DCOPY( M, A( 1, p ), 1, WORK,
814: + 1 )
815: CALL DLASCL( 'G', 0, 0, AAPP, ONE,
816: + M, 1, WORK, LDA, IERR )
817: CALL DLASCL( 'G', 0, 0, AAQQ, ONE,
818: + M, 1, A( 1, q ), LDA,
819: + IERR )
820: TEMP1 = -AAPQ*D( p ) / D( q )
821: CALL DAXPY( M, TEMP1, WORK, 1,
822: + A( 1, q ), 1 )
823: CALL DLASCL( 'G', 0, 0, ONE, AAQQ,
824: + M, 1, A( 1, q ), LDA,
825: + IERR )
826: SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
827: + ONE-AAPQ*AAPQ ) )
828: MXSINJ = DMAX1( MXSINJ, SFMIN )
829: ELSE
830: CALL DCOPY( M, A( 1, q ), 1, WORK,
831: + 1 )
832: CALL DLASCL( 'G', 0, 0, AAQQ, ONE,
833: + M, 1, WORK, LDA, IERR )
834: CALL DLASCL( 'G', 0, 0, AAPP, ONE,
835: + M, 1, A( 1, p ), LDA,
836: + IERR )
837: TEMP1 = -AAPQ*D( q ) / D( p )
838: CALL DAXPY( M, TEMP1, WORK, 1,
839: + A( 1, p ), 1 )
840: CALL DLASCL( 'G', 0, 0, ONE, AAPP,
841: + M, 1, A( 1, p ), LDA,
842: + IERR )
843: SVA( p ) = AAPP*DSQRT( DMAX1( ZERO,
844: + ONE-AAPQ*AAPQ ) )
845: MXSINJ = DMAX1( MXSINJ, SFMIN )
846: END IF
847: END IF
848: * END IF ROTOK THEN ... ELSE
849: *
850: * In the case of cancellation in updating SVA(q)
851: * .. recompute SVA(q)
852: IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
853: + THEN
854: IF( ( AAQQ.LT.ROOTBIG ) .AND.
855: + ( AAQQ.GT.ROOTSFMIN ) ) THEN
856: SVA( q ) = DNRM2( M, A( 1, q ), 1 )*
857: + D( q )
858: ELSE
859: T = ZERO
1.4 ! bertrand 860: AAQQ = ONE
1.1 bertrand 861: CALL DLASSQ( M, A( 1, q ), 1, T,
862: + AAQQ )
863: SVA( q ) = T*DSQRT( AAQQ )*D( q )
864: END IF
865: END IF
866: IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN
867: IF( ( AAPP.LT.ROOTBIG ) .AND.
868: + ( AAPP.GT.ROOTSFMIN ) ) THEN
869: AAPP = DNRM2( M, A( 1, p ), 1 )*
870: + D( p )
871: ELSE
872: T = ZERO
1.4 ! bertrand 873: AAPP = ONE
1.1 bertrand 874: CALL DLASSQ( M, A( 1, p ), 1, T,
875: + AAPP )
876: AAPP = T*DSQRT( AAPP )*D( p )
877: END IF
878: SVA( p ) = AAPP
879: END IF
880: * end of OK rotation
881: ELSE
882: NOTROT = NOTROT + 1
883: PSKIPPED = PSKIPPED + 1
884: IJBLSK = IJBLSK + 1
885: END IF
886: ELSE
887: NOTROT = NOTROT + 1
888: PSKIPPED = PSKIPPED + 1
889: IJBLSK = IJBLSK + 1
890: END IF
891: *
892: IF( ( i.LE.SWBAND ) .AND. ( IJBLSK.GE.BLSKIP ) )
893: + THEN
894: SVA( p ) = AAPP
895: NOTROT = 0
896: GO TO 2011
897: END IF
898: IF( ( i.LE.SWBAND ) .AND.
899: + ( PSKIPPED.GT.ROWSKIP ) ) THEN
900: AAPP = -AAPP
901: NOTROT = 0
902: GO TO 2203
903: END IF
904: *
905: 2200 CONTINUE
906: * end of the q-loop
907: 2203 CONTINUE
908: *
909: SVA( p ) = AAPP
910: *
911: ELSE
912: IF( AAPP.EQ.ZERO )NOTROT = NOTROT +
913: + MIN0( jgl+KBL-1, N ) - jgl + 1
914: IF( AAPP.LT.ZERO )NOTROT = 0
915: END IF
916:
917: 2100 CONTINUE
918: * end of the p-loop
919: 2010 CONTINUE
920: * end of the jbc-loop
921: 2011 CONTINUE
922: *2011 bailed out of the jbc-loop
923: DO 2012 p = igl, MIN0( igl+KBL-1, N )
924: SVA( p ) = DABS( SVA( p ) )
925: 2012 CONTINUE
926: *
927: 2000 CONTINUE
928: *2000 :: end of the ibr-loop
929: *
930: * .. update SVA(N)
931: IF( ( SVA( N ).LT.ROOTBIG ) .AND. ( SVA( N ).GT.ROOTSFMIN ) )
932: + THEN
933: SVA( N ) = DNRM2( M, A( 1, N ), 1 )*D( N )
934: ELSE
935: T = ZERO
1.4 ! bertrand 936: AAPP = ONE
1.1 bertrand 937: CALL DLASSQ( M, A( 1, N ), 1, T, AAPP )
938: SVA( N ) = T*DSQRT( AAPP )*D( N )
939: END IF
940: *
941: * Additional steering devices
942: *
943: IF( ( i.LT.SWBAND ) .AND. ( ( MXAAPQ.LE.ROOTTOL ) .OR.
944: + ( ISWROT.LE.N ) ) )SWBAND = i
945: *
946: IF( ( i.GT.SWBAND+1 ) .AND. ( MXAAPQ.LT.DBLE( N )*TOL ) .AND.
947: + ( DBLE( N )*MXAAPQ*MXSINJ.LT.TOL ) ) THEN
948: GO TO 1994
949: END IF
950: *
951: IF( NOTROT.GE.EMPTSW )GO TO 1994
952:
953: 1993 CONTINUE
954: * end i=1:NSWEEP loop
955: * #:) Reaching this point means that the procedure has comleted the given
956: * number of iterations.
957: INFO = NSWEEP - 1
958: GO TO 1995
959: 1994 CONTINUE
960: * #:) Reaching this point means that during the i-th sweep all pivots were
961: * below the given tolerance, causing early exit.
962: *
963: INFO = 0
964: * #:) INFO = 0 confirms successful iterations.
965: 1995 CONTINUE
966: *
967: * Sort the vector D.
968: DO 5991 p = 1, N - 1
969: q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1
970: IF( p.NE.q ) THEN
971: TEMP1 = SVA( p )
972: SVA( p ) = SVA( q )
973: SVA( q ) = TEMP1
974: TEMP1 = D( p )
975: D( p ) = D( q )
976: D( q ) = TEMP1
977: CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
978: IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 )
979: END IF
980: 5991 CONTINUE
981: *
982: RETURN
983: * ..
984: * .. END OF DGSVJ0
985: * ..
986: END