Annotation of rpl/lapack/lapack/dlals0.f, revision 1.14
1.11 bertrand 1: *> \brief \b DLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.
1.8 bertrand 2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLALS0 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlals0.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlals0.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlals0.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX,
22: * PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM,
23: * POLES, DIFL, DIFR, Z, K, C, S, WORK, INFO )
24: *
25: * .. Scalar Arguments ..
26: * INTEGER GIVPTR, ICOMPQ, INFO, K, LDB, LDBX, LDGCOL,
27: * $ LDGNUM, NL, NR, NRHS, SQRE
28: * DOUBLE PRECISION C, S
29: * ..
30: * .. Array Arguments ..
31: * INTEGER GIVCOL( LDGCOL, * ), PERM( * )
32: * DOUBLE PRECISION B( LDB, * ), BX( LDBX, * ), DIFL( * ),
33: * $ DIFR( LDGNUM, * ), GIVNUM( LDGNUM, * ),
34: * $ POLES( LDGNUM, * ), WORK( * ), Z( * )
35: * ..
36: *
37: *
38: *> \par Purpose:
39: * =============
40: *>
41: *> \verbatim
42: *>
43: *> DLALS0 applies back the multiplying factors of either the left or the
44: *> right singular vector matrix of a diagonal matrix appended by a row
45: *> to the right hand side matrix B in solving the least squares problem
46: *> using the divide-and-conquer SVD approach.
47: *>
48: *> For the left singular vector matrix, three types of orthogonal
49: *> matrices are involved:
50: *>
51: *> (1L) Givens rotations: the number of such rotations is GIVPTR; the
52: *> pairs of columns/rows they were applied to are stored in GIVCOL;
53: *> and the C- and S-values of these rotations are stored in GIVNUM.
54: *>
55: *> (2L) Permutation. The (NL+1)-st row of B is to be moved to the first
56: *> row, and for J=2:N, PERM(J)-th row of B is to be moved to the
57: *> J-th row.
58: *>
59: *> (3L) The left singular vector matrix of the remaining matrix.
60: *>
61: *> For the right singular vector matrix, four types of orthogonal
62: *> matrices are involved:
63: *>
64: *> (1R) The right singular vector matrix of the remaining matrix.
65: *>
66: *> (2R) If SQRE = 1, one extra Givens rotation to generate the right
67: *> null space.
68: *>
69: *> (3R) The inverse transformation of (2L).
70: *>
71: *> (4R) The inverse transformation of (1L).
72: *> \endverbatim
73: *
74: * Arguments:
75: * ==========
76: *
77: *> \param[in] ICOMPQ
78: *> \verbatim
79: *> ICOMPQ is INTEGER
80: *> Specifies whether singular vectors are to be computed in
81: *> factored form:
82: *> = 0: Left singular vector matrix.
83: *> = 1: Right singular vector matrix.
84: *> \endverbatim
85: *>
86: *> \param[in] NL
87: *> \verbatim
88: *> NL is INTEGER
89: *> The row dimension of the upper block. NL >= 1.
90: *> \endverbatim
91: *>
92: *> \param[in] NR
93: *> \verbatim
94: *> NR is INTEGER
95: *> The row dimension of the lower block. NR >= 1.
96: *> \endverbatim
97: *>
98: *> \param[in] SQRE
99: *> \verbatim
100: *> SQRE is INTEGER
101: *> = 0: the lower block is an NR-by-NR square matrix.
102: *> = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
103: *>
104: *> The bidiagonal matrix has row dimension N = NL + NR + 1,
105: *> and column dimension M = N + SQRE.
106: *> \endverbatim
107: *>
108: *> \param[in] NRHS
109: *> \verbatim
110: *> NRHS is INTEGER
111: *> The number of columns of B and BX. NRHS must be at least 1.
112: *> \endverbatim
113: *>
114: *> \param[in,out] B
115: *> \verbatim
116: *> B is DOUBLE PRECISION array, dimension ( LDB, NRHS )
117: *> On input, B contains the right hand sides of the least
118: *> squares problem in rows 1 through M. On output, B contains
119: *> the solution X in rows 1 through N.
120: *> \endverbatim
121: *>
122: *> \param[in] LDB
123: *> \verbatim
124: *> LDB is INTEGER
125: *> The leading dimension of B. LDB must be at least
126: *> max(1,MAX( M, N ) ).
127: *> \endverbatim
128: *>
129: *> \param[out] BX
130: *> \verbatim
131: *> BX is DOUBLE PRECISION array, dimension ( LDBX, NRHS )
132: *> \endverbatim
133: *>
134: *> \param[in] LDBX
135: *> \verbatim
136: *> LDBX is INTEGER
137: *> The leading dimension of BX.
138: *> \endverbatim
139: *>
140: *> \param[in] PERM
141: *> \verbatim
142: *> PERM is INTEGER array, dimension ( N )
143: *> The permutations (from deflation and sorting) applied
144: *> to the two blocks.
145: *> \endverbatim
146: *>
147: *> \param[in] GIVPTR
148: *> \verbatim
149: *> GIVPTR is INTEGER
150: *> The number of Givens rotations which took place in this
151: *> subproblem.
152: *> \endverbatim
153: *>
154: *> \param[in] GIVCOL
155: *> \verbatim
156: *> GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
157: *> Each pair of numbers indicates a pair of rows/columns
158: *> involved in a Givens rotation.
159: *> \endverbatim
160: *>
161: *> \param[in] LDGCOL
162: *> \verbatim
163: *> LDGCOL is INTEGER
164: *> The leading dimension of GIVCOL, must be at least N.
165: *> \endverbatim
166: *>
167: *> \param[in] GIVNUM
168: *> \verbatim
169: *> GIVNUM is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
170: *> Each number indicates the C or S value used in the
171: *> corresponding Givens rotation.
172: *> \endverbatim
173: *>
174: *> \param[in] LDGNUM
175: *> \verbatim
176: *> LDGNUM is INTEGER
177: *> The leading dimension of arrays DIFR, POLES and
178: *> GIVNUM, must be at least K.
179: *> \endverbatim
180: *>
181: *> \param[in] POLES
182: *> \verbatim
183: *> POLES is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
184: *> On entry, POLES(1:K, 1) contains the new singular
185: *> values obtained from solving the secular equation, and
186: *> POLES(1:K, 2) is an array containing the poles in the secular
187: *> equation.
188: *> \endverbatim
189: *>
190: *> \param[in] DIFL
191: *> \verbatim
192: *> DIFL is DOUBLE PRECISION array, dimension ( K ).
193: *> On entry, DIFL(I) is the distance between I-th updated
194: *> (undeflated) singular value and the I-th (undeflated) old
195: *> singular value.
196: *> \endverbatim
197: *>
198: *> \param[in] DIFR
199: *> \verbatim
200: *> DIFR is DOUBLE PRECISION array, dimension ( LDGNUM, 2 ).
201: *> On entry, DIFR(I, 1) contains the distances between I-th
202: *> updated (undeflated) singular value and the I+1-th
203: *> (undeflated) old singular value. And DIFR(I, 2) is the
204: *> normalizing factor for the I-th right singular vector.
205: *> \endverbatim
206: *>
207: *> \param[in] Z
208: *> \verbatim
209: *> Z is DOUBLE PRECISION array, dimension ( K )
210: *> Contain the components of the deflation-adjusted updating row
211: *> vector.
212: *> \endverbatim
213: *>
214: *> \param[in] K
215: *> \verbatim
216: *> K is INTEGER
217: *> Contains the dimension of the non-deflated matrix,
218: *> This is the order of the related secular equation. 1 <= K <=N.
219: *> \endverbatim
220: *>
221: *> \param[in] C
222: *> \verbatim
223: *> C is DOUBLE PRECISION
224: *> C contains garbage if SQRE =0 and the C-value of a Givens
225: *> rotation related to the right null space if SQRE = 1.
226: *> \endverbatim
227: *>
228: *> \param[in] S
229: *> \verbatim
230: *> S is DOUBLE PRECISION
231: *> S contains garbage if SQRE =0 and the S-value of a Givens
232: *> rotation related to the right null space if SQRE = 1.
233: *> \endverbatim
234: *>
235: *> \param[out] WORK
236: *> \verbatim
237: *> WORK is DOUBLE PRECISION array, dimension ( K )
238: *> \endverbatim
239: *>
240: *> \param[out] INFO
241: *> \verbatim
242: *> INFO is INTEGER
243: *> = 0: successful exit.
244: *> < 0: if INFO = -i, the i-th argument had an illegal value.
245: *> \endverbatim
246: *
247: * Authors:
248: * ========
249: *
250: *> \author Univ. of Tennessee
251: *> \author Univ. of California Berkeley
252: *> \author Univ. of Colorado Denver
253: *> \author NAG Ltd.
254: *
1.14 ! bertrand 255: *> \date November 2015
1.8 bertrand 256: *
257: *> \ingroup doubleOTHERcomputational
258: *
259: *> \par Contributors:
260: * ==================
261: *>
262: *> Ming Gu and Ren-Cang Li, Computer Science Division, University of
263: *> California at Berkeley, USA \n
264: *> Osni Marques, LBNL/NERSC, USA \n
265: *
266: * =====================================================================
1.1 bertrand 267: SUBROUTINE DLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX,
268: $ PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM,
269: $ POLES, DIFL, DIFR, Z, K, C, S, WORK, INFO )
270: *
1.14 ! bertrand 271: * -- LAPACK computational routine (version 3.6.0) --
1.1 bertrand 272: * -- LAPACK is a software package provided by Univ. of Tennessee, --
273: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.14 ! bertrand 274: * November 2015
1.1 bertrand 275: *
276: * .. Scalar Arguments ..
277: INTEGER GIVPTR, ICOMPQ, INFO, K, LDB, LDBX, LDGCOL,
278: $ LDGNUM, NL, NR, NRHS, SQRE
279: DOUBLE PRECISION C, S
280: * ..
281: * .. Array Arguments ..
282: INTEGER GIVCOL( LDGCOL, * ), PERM( * )
283: DOUBLE PRECISION B( LDB, * ), BX( LDBX, * ), DIFL( * ),
284: $ DIFR( LDGNUM, * ), GIVNUM( LDGNUM, * ),
285: $ POLES( LDGNUM, * ), WORK( * ), Z( * )
286: * ..
287: *
288: * =====================================================================
289: *
290: * .. Parameters ..
291: DOUBLE PRECISION ONE, ZERO, NEGONE
292: PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0, NEGONE = -1.0D0 )
293: * ..
294: * .. Local Scalars ..
295: INTEGER I, J, M, N, NLP1
296: DOUBLE PRECISION DIFLJ, DIFRJ, DJ, DSIGJ, DSIGJP, TEMP
297: * ..
298: * .. External Subroutines ..
299: EXTERNAL DCOPY, DGEMV, DLACPY, DLASCL, DROT, DSCAL,
300: $ XERBLA
301: * ..
302: * .. External Functions ..
303: DOUBLE PRECISION DLAMC3, DNRM2
304: EXTERNAL DLAMC3, DNRM2
305: * ..
306: * .. Intrinsic Functions ..
307: INTRINSIC MAX
308: * ..
309: * .. Executable Statements ..
310: *
311: * Test the input parameters.
312: *
313: INFO = 0
1.14 ! bertrand 314: N = NL + NR + 1
1.1 bertrand 315: *
316: IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
317: INFO = -1
318: ELSE IF( NL.LT.1 ) THEN
319: INFO = -2
320: ELSE IF( NR.LT.1 ) THEN
321: INFO = -3
322: ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
323: INFO = -4
1.14 ! bertrand 324: ELSE IF( NRHS.LT.1 ) THEN
1.1 bertrand 325: INFO = -5
326: ELSE IF( LDB.LT.N ) THEN
327: INFO = -7
328: ELSE IF( LDBX.LT.N ) THEN
329: INFO = -9
330: ELSE IF( GIVPTR.LT.0 ) THEN
331: INFO = -11
332: ELSE IF( LDGCOL.LT.N ) THEN
333: INFO = -13
334: ELSE IF( LDGNUM.LT.N ) THEN
335: INFO = -15
336: ELSE IF( K.LT.1 ) THEN
337: INFO = -20
338: END IF
339: IF( INFO.NE.0 ) THEN
340: CALL XERBLA( 'DLALS0', -INFO )
341: RETURN
342: END IF
343: *
344: M = N + SQRE
345: NLP1 = NL + 1
346: *
347: IF( ICOMPQ.EQ.0 ) THEN
348: *
349: * Apply back orthogonal transformations from the left.
350: *
351: * Step (1L): apply back the Givens rotations performed.
352: *
353: DO 10 I = 1, GIVPTR
354: CALL DROT( NRHS, B( GIVCOL( I, 2 ), 1 ), LDB,
355: $ B( GIVCOL( I, 1 ), 1 ), LDB, GIVNUM( I, 2 ),
356: $ GIVNUM( I, 1 ) )
357: 10 CONTINUE
358: *
359: * Step (2L): permute rows of B.
360: *
361: CALL DCOPY( NRHS, B( NLP1, 1 ), LDB, BX( 1, 1 ), LDBX )
362: DO 20 I = 2, N
363: CALL DCOPY( NRHS, B( PERM( I ), 1 ), LDB, BX( I, 1 ), LDBX )
364: 20 CONTINUE
365: *
366: * Step (3L): apply the inverse of the left singular vector
367: * matrix to BX.
368: *
369: IF( K.EQ.1 ) THEN
370: CALL DCOPY( NRHS, BX, LDBX, B, LDB )
371: IF( Z( 1 ).LT.ZERO ) THEN
372: CALL DSCAL( NRHS, NEGONE, B, LDB )
373: END IF
374: ELSE
375: DO 50 J = 1, K
376: DIFLJ = DIFL( J )
377: DJ = POLES( J, 1 )
378: DSIGJ = -POLES( J, 2 )
379: IF( J.LT.K ) THEN
380: DIFRJ = -DIFR( J, 1 )
381: DSIGJP = -POLES( J+1, 2 )
382: END IF
383: IF( ( Z( J ).EQ.ZERO ) .OR. ( POLES( J, 2 ).EQ.ZERO ) )
384: $ THEN
385: WORK( J ) = ZERO
386: ELSE
387: WORK( J ) = -POLES( J, 2 )*Z( J ) / DIFLJ /
388: $ ( POLES( J, 2 )+DJ )
389: END IF
390: DO 30 I = 1, J - 1
391: IF( ( Z( I ).EQ.ZERO ) .OR.
392: $ ( POLES( I, 2 ).EQ.ZERO ) ) THEN
393: WORK( I ) = ZERO
394: ELSE
395: WORK( I ) = POLES( I, 2 )*Z( I ) /
396: $ ( DLAMC3( POLES( I, 2 ), DSIGJ )-
397: $ DIFLJ ) / ( POLES( I, 2 )+DJ )
398: END IF
399: 30 CONTINUE
400: DO 40 I = J + 1, K
401: IF( ( Z( I ).EQ.ZERO ) .OR.
402: $ ( POLES( I, 2 ).EQ.ZERO ) ) THEN
403: WORK( I ) = ZERO
404: ELSE
405: WORK( I ) = POLES( I, 2 )*Z( I ) /
406: $ ( DLAMC3( POLES( I, 2 ), DSIGJP )+
407: $ DIFRJ ) / ( POLES( I, 2 )+DJ )
408: END IF
409: 40 CONTINUE
410: WORK( 1 ) = NEGONE
411: TEMP = DNRM2( K, WORK, 1 )
412: CALL DGEMV( 'T', K, NRHS, ONE, BX, LDBX, WORK, 1, ZERO,
413: $ B( J, 1 ), LDB )
414: CALL DLASCL( 'G', 0, 0, TEMP, ONE, 1, NRHS, B( J, 1 ),
415: $ LDB, INFO )
416: 50 CONTINUE
417: END IF
418: *
419: * Move the deflated rows of BX to B also.
420: *
421: IF( K.LT.MAX( M, N ) )
422: $ CALL DLACPY( 'A', N-K, NRHS, BX( K+1, 1 ), LDBX,
423: $ B( K+1, 1 ), LDB )
424: ELSE
425: *
426: * Apply back the right orthogonal transformations.
427: *
428: * Step (1R): apply back the new right singular vector matrix
429: * to B.
430: *
431: IF( K.EQ.1 ) THEN
432: CALL DCOPY( NRHS, B, LDB, BX, LDBX )
433: ELSE
434: DO 80 J = 1, K
435: DSIGJ = POLES( J, 2 )
436: IF( Z( J ).EQ.ZERO ) THEN
437: WORK( J ) = ZERO
438: ELSE
439: WORK( J ) = -Z( J ) / DIFL( J ) /
440: $ ( DSIGJ+POLES( J, 1 ) ) / DIFR( J, 2 )
441: END IF
442: DO 60 I = 1, J - 1
443: IF( Z( J ).EQ.ZERO ) THEN
444: WORK( I ) = ZERO
445: ELSE
446: WORK( I ) = Z( J ) / ( DLAMC3( DSIGJ, -POLES( I+1,
447: $ 2 ) )-DIFR( I, 1 ) ) /
448: $ ( DSIGJ+POLES( I, 1 ) ) / DIFR( I, 2 )
449: END IF
450: 60 CONTINUE
451: DO 70 I = J + 1, K
452: IF( Z( J ).EQ.ZERO ) THEN
453: WORK( I ) = ZERO
454: ELSE
455: WORK( I ) = Z( J ) / ( DLAMC3( DSIGJ, -POLES( I,
456: $ 2 ) )-DIFL( I ) ) /
457: $ ( DSIGJ+POLES( I, 1 ) ) / DIFR( I, 2 )
458: END IF
459: 70 CONTINUE
460: CALL DGEMV( 'T', K, NRHS, ONE, B, LDB, WORK, 1, ZERO,
461: $ BX( J, 1 ), LDBX )
462: 80 CONTINUE
463: END IF
464: *
465: * Step (2R): if SQRE = 1, apply back the rotation that is
466: * related to the right null space of the subproblem.
467: *
468: IF( SQRE.EQ.1 ) THEN
469: CALL DCOPY( NRHS, B( M, 1 ), LDB, BX( M, 1 ), LDBX )
470: CALL DROT( NRHS, BX( 1, 1 ), LDBX, BX( M, 1 ), LDBX, C, S )
471: END IF
472: IF( K.LT.MAX( M, N ) )
473: $ CALL DLACPY( 'A', N-K, NRHS, B( K+1, 1 ), LDB, BX( K+1, 1 ),
474: $ LDBX )
475: *
476: * Step (3R): permute rows of B.
477: *
478: CALL DCOPY( NRHS, BX( 1, 1 ), LDBX, B( NLP1, 1 ), LDB )
479: IF( SQRE.EQ.1 ) THEN
480: CALL DCOPY( NRHS, BX( M, 1 ), LDBX, B( M, 1 ), LDB )
481: END IF
482: DO 90 I = 2, N
483: CALL DCOPY( NRHS, BX( I, 1 ), LDBX, B( PERM( I ), 1 ), LDB )
484: 90 CONTINUE
485: *
486: * Step (4R): apply back the Givens rotations performed.
487: *
488: DO 100 I = GIVPTR, 1, -1
489: CALL DROT( NRHS, B( GIVCOL( I, 2 ), 1 ), LDB,
490: $ B( GIVCOL( I, 1 ), 1 ), LDB, GIVNUM( I, 2 ),
491: $ -GIVNUM( I, 1 ) )
492: 100 CONTINUE
493: END IF
494: *
495: RETURN
496: *
497: * End of DLALS0
498: *
499: END
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