Annotation of rpl/lapack/lapack/dlasd3.f, revision 1.15
1.13 bertrand 1: *> \brief \b DLASD3 finds all square roots of the roots of the secular equation, as defined by the values in D and Z, and then updates the singular vectors by matrix multiplication. Used by sbdsdc.
1.10 bertrand 2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLASD3 + dependencies
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11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd3.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd3.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLASD3( NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2,
22: * LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z,
23: * INFO )
24: *
25: * .. Scalar Arguments ..
26: * INTEGER INFO, K, LDQ, LDU, LDU2, LDVT, LDVT2, NL, NR,
27: * $ SQRE
28: * ..
29: * .. Array Arguments ..
30: * INTEGER CTOT( * ), IDXC( * )
31: * DOUBLE PRECISION D( * ), DSIGMA( * ), Q( LDQ, * ), U( LDU, * ),
32: * $ U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
33: * $ Z( * )
34: * ..
35: *
36: *
37: *> \par Purpose:
38: * =============
39: *>
40: *> \verbatim
41: *>
42: *> DLASD3 finds all the square roots of the roots of the secular
43: *> equation, as defined by the values in D and Z. It makes the
44: *> appropriate calls to DLASD4 and then updates the singular
45: *> vectors by matrix multiplication.
46: *>
47: *> This code makes very mild assumptions about floating point
48: *> arithmetic. It will work on machines with a guard digit in
49: *> add/subtract, or on those binary machines without guard digits
50: *> which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.
51: *> It could conceivably fail on hexadecimal or decimal machines
52: *> without guard digits, but we know of none.
53: *>
54: *> DLASD3 is called from DLASD1.
55: *> \endverbatim
56: *
57: * Arguments:
58: * ==========
59: *
60: *> \param[in] NL
61: *> \verbatim
62: *> NL is INTEGER
63: *> The row dimension of the upper block. NL >= 1.
64: *> \endverbatim
65: *>
66: *> \param[in] NR
67: *> \verbatim
68: *> NR is INTEGER
69: *> The row dimension of the lower block. NR >= 1.
70: *> \endverbatim
71: *>
72: *> \param[in] SQRE
73: *> \verbatim
74: *> SQRE is INTEGER
75: *> = 0: the lower block is an NR-by-NR square matrix.
76: *> = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
77: *>
78: *> The bidiagonal matrix has N = NL + NR + 1 rows and
79: *> M = N + SQRE >= N columns.
80: *> \endverbatim
81: *>
82: *> \param[in] K
83: *> \verbatim
84: *> K is INTEGER
85: *> The size of the secular equation, 1 =< K = < N.
86: *> \endverbatim
87: *>
88: *> \param[out] D
89: *> \verbatim
90: *> D is DOUBLE PRECISION array, dimension(K)
91: *> On exit the square roots of the roots of the secular equation,
92: *> in ascending order.
93: *> \endverbatim
94: *>
95: *> \param[out] Q
96: *> \verbatim
97: *> Q is DOUBLE PRECISION array,
98: *> dimension at least (LDQ,K).
99: *> \endverbatim
100: *>
101: *> \param[in] LDQ
102: *> \verbatim
103: *> LDQ is INTEGER
104: *> The leading dimension of the array Q. LDQ >= K.
105: *> \endverbatim
106: *>
107: *> \param[in] DSIGMA
108: *> \verbatim
109: *> DSIGMA is DOUBLE PRECISION array, dimension(K)
110: *> The first K elements of this array contain the old roots
111: *> of the deflated updating problem. These are the poles
112: *> of the secular equation.
113: *> \endverbatim
114: *>
115: *> \param[out] U
116: *> \verbatim
117: *> U is DOUBLE PRECISION array, dimension (LDU, N)
118: *> The last N - K columns of this matrix contain the deflated
119: *> left singular vectors.
120: *> \endverbatim
121: *>
122: *> \param[in] LDU
123: *> \verbatim
124: *> LDU is INTEGER
125: *> The leading dimension of the array U. LDU >= N.
126: *> \endverbatim
127: *>
128: *> \param[in,out] U2
129: *> \verbatim
130: *> U2 is DOUBLE PRECISION array, dimension (LDU2, N)
131: *> The first K columns of this matrix contain the non-deflated
132: *> left singular vectors for the split problem.
133: *> \endverbatim
134: *>
135: *> \param[in] LDU2
136: *> \verbatim
137: *> LDU2 is INTEGER
138: *> The leading dimension of the array U2. LDU2 >= N.
139: *> \endverbatim
140: *>
141: *> \param[out] VT
142: *> \verbatim
143: *> VT is DOUBLE PRECISION array, dimension (LDVT, M)
144: *> The last M - K columns of VT**T contain the deflated
145: *> right singular vectors.
146: *> \endverbatim
147: *>
148: *> \param[in] LDVT
149: *> \verbatim
150: *> LDVT is INTEGER
151: *> The leading dimension of the array VT. LDVT >= N.
152: *> \endverbatim
153: *>
154: *> \param[in,out] VT2
155: *> \verbatim
156: *> VT2 is DOUBLE PRECISION array, dimension (LDVT2, N)
157: *> The first K columns of VT2**T contain the non-deflated
158: *> right singular vectors for the split problem.
159: *> \endverbatim
160: *>
161: *> \param[in] LDVT2
162: *> \verbatim
163: *> LDVT2 is INTEGER
164: *> The leading dimension of the array VT2. LDVT2 >= N.
165: *> \endverbatim
166: *>
167: *> \param[in] IDXC
168: *> \verbatim
169: *> IDXC is INTEGER array, dimension ( N )
170: *> The permutation used to arrange the columns of U (and rows of
171: *> VT) into three groups: the first group contains non-zero
172: *> entries only at and above (or before) NL +1; the second
173: *> contains non-zero entries only at and below (or after) NL+2;
174: *> and the third is dense. The first column of U and the row of
175: *> VT are treated separately, however.
176: *>
177: *> The rows of the singular vectors found by DLASD4
178: *> must be likewise permuted before the matrix multiplies can
179: *> take place.
180: *> \endverbatim
181: *>
182: *> \param[in] CTOT
183: *> \verbatim
184: *> CTOT is INTEGER array, dimension ( 4 )
185: *> A count of the total number of the various types of columns
186: *> in U (or rows in VT), as described in IDXC. The fourth column
187: *> type is any column which has been deflated.
188: *> \endverbatim
189: *>
190: *> \param[in] Z
191: *> \verbatim
192: *> Z is DOUBLE PRECISION array, dimension (K)
193: *> The first K elements of this array contain the components
194: *> of the deflation-adjusted updating row vector.
195: *> \endverbatim
196: *>
197: *> \param[out] INFO
198: *> \verbatim
199: *> INFO is INTEGER
200: *> = 0: successful exit.
201: *> < 0: if INFO = -i, the i-th argument had an illegal value.
202: *> > 0: if INFO = 1, a singular value did not converge
203: *> \endverbatim
204: *
205: * Authors:
206: * ========
207: *
208: *> \author Univ. of Tennessee
209: *> \author Univ. of California Berkeley
210: *> \author Univ. of Colorado Denver
211: *> \author NAG Ltd.
212: *
1.13 bertrand 213: *> \date September 2012
1.10 bertrand 214: *
215: *> \ingroup auxOTHERauxiliary
216: *
217: *> \par Contributors:
218: * ==================
219: *>
220: *> Ming Gu and Huan Ren, Computer Science Division, University of
221: *> California at Berkeley, USA
222: *>
223: * =====================================================================
1.1 bertrand 224: SUBROUTINE DLASD3( NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2,
225: $ LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z,
226: $ INFO )
227: *
1.13 bertrand 228: * -- LAPACK auxiliary routine (version 3.4.2) --
1.1 bertrand 229: * -- LAPACK is a software package provided by Univ. of Tennessee, --
230: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.13 bertrand 231: * September 2012
1.1 bertrand 232: *
233: * .. Scalar Arguments ..
234: INTEGER INFO, K, LDQ, LDU, LDU2, LDVT, LDVT2, NL, NR,
235: $ SQRE
236: * ..
237: * .. Array Arguments ..
238: INTEGER CTOT( * ), IDXC( * )
239: DOUBLE PRECISION D( * ), DSIGMA( * ), Q( LDQ, * ), U( LDU, * ),
240: $ U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
241: $ Z( * )
242: * ..
243: *
244: * =====================================================================
245: *
246: * .. Parameters ..
247: DOUBLE PRECISION ONE, ZERO, NEGONE
248: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0,
249: $ NEGONE = -1.0D+0 )
250: * ..
251: * .. Local Scalars ..
252: INTEGER CTEMP, I, J, JC, KTEMP, M, N, NLP1, NLP2, NRP1
253: DOUBLE PRECISION RHO, TEMP
254: * ..
255: * .. External Functions ..
256: DOUBLE PRECISION DLAMC3, DNRM2
257: EXTERNAL DLAMC3, DNRM2
258: * ..
259: * .. External Subroutines ..
260: EXTERNAL DCOPY, DGEMM, DLACPY, DLASCL, DLASD4, XERBLA
261: * ..
262: * .. Intrinsic Functions ..
263: INTRINSIC ABS, SIGN, SQRT
264: * ..
265: * .. Executable Statements ..
266: *
267: * Test the input parameters.
268: *
269: INFO = 0
270: *
271: IF( NL.LT.1 ) THEN
272: INFO = -1
273: ELSE IF( NR.LT.1 ) THEN
274: INFO = -2
275: ELSE IF( ( SQRE.NE.1 ) .AND. ( SQRE.NE.0 ) ) THEN
276: INFO = -3
277: END IF
278: *
279: N = NL + NR + 1
280: M = N + SQRE
281: NLP1 = NL + 1
282: NLP2 = NL + 2
283: *
284: IF( ( K.LT.1 ) .OR. ( K.GT.N ) ) THEN
285: INFO = -4
286: ELSE IF( LDQ.LT.K ) THEN
287: INFO = -7
288: ELSE IF( LDU.LT.N ) THEN
289: INFO = -10
290: ELSE IF( LDU2.LT.N ) THEN
291: INFO = -12
292: ELSE IF( LDVT.LT.M ) THEN
293: INFO = -14
294: ELSE IF( LDVT2.LT.M ) THEN
295: INFO = -16
296: END IF
297: IF( INFO.NE.0 ) THEN
298: CALL XERBLA( 'DLASD3', -INFO )
299: RETURN
300: END IF
301: *
302: * Quick return if possible
303: *
304: IF( K.EQ.1 ) THEN
305: D( 1 ) = ABS( Z( 1 ) )
306: CALL DCOPY( M, VT2( 1, 1 ), LDVT2, VT( 1, 1 ), LDVT )
307: IF( Z( 1 ).GT.ZERO ) THEN
308: CALL DCOPY( N, U2( 1, 1 ), 1, U( 1, 1 ), 1 )
309: ELSE
310: DO 10 I = 1, N
311: U( I, 1 ) = -U2( I, 1 )
312: 10 CONTINUE
313: END IF
314: RETURN
315: END IF
316: *
317: * Modify values DSIGMA(i) to make sure all DSIGMA(i)-DSIGMA(j) can
318: * be computed with high relative accuracy (barring over/underflow).
319: * This is a problem on machines without a guard digit in
320: * add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
321: * The following code replaces DSIGMA(I) by 2*DSIGMA(I)-DSIGMA(I),
322: * which on any of these machines zeros out the bottommost
323: * bit of DSIGMA(I) if it is 1; this makes the subsequent
324: * subtractions DSIGMA(I)-DSIGMA(J) unproblematic when cancellation
325: * occurs. On binary machines with a guard digit (almost all
326: * machines) it does not change DSIGMA(I) at all. On hexadecimal
327: * and decimal machines with a guard digit, it slightly
328: * changes the bottommost bits of DSIGMA(I). It does not account
329: * for hexadecimal or decimal machines without guard digits
330: * (we know of none). We use a subroutine call to compute
331: * 2*DSIGMA(I) to prevent optimizing compilers from eliminating
332: * this code.
333: *
334: DO 20 I = 1, K
335: DSIGMA( I ) = DLAMC3( DSIGMA( I ), DSIGMA( I ) ) - DSIGMA( I )
336: 20 CONTINUE
337: *
338: * Keep a copy of Z.
339: *
340: CALL DCOPY( K, Z, 1, Q, 1 )
341: *
342: * Normalize Z.
343: *
344: RHO = DNRM2( K, Z, 1 )
345: CALL DLASCL( 'G', 0, 0, RHO, ONE, K, 1, Z, K, INFO )
346: RHO = RHO*RHO
347: *
348: * Find the new singular values.
349: *
350: DO 30 J = 1, K
351: CALL DLASD4( K, J, DSIGMA, Z, U( 1, J ), RHO, D( J ),
352: $ VT( 1, J ), INFO )
353: *
354: * If the zero finder fails, the computation is terminated.
355: *
356: IF( INFO.NE.0 ) THEN
357: RETURN
358: END IF
359: 30 CONTINUE
360: *
361: * Compute updated Z.
362: *
363: DO 60 I = 1, K
364: Z( I ) = U( I, K )*VT( I, K )
365: DO 40 J = 1, I - 1
366: Z( I ) = Z( I )*( U( I, J )*VT( I, J ) /
367: $ ( DSIGMA( I )-DSIGMA( J ) ) /
368: $ ( DSIGMA( I )+DSIGMA( J ) ) )
369: 40 CONTINUE
370: DO 50 J = I, K - 1
371: Z( I ) = Z( I )*( U( I, J )*VT( I, J ) /
372: $ ( DSIGMA( I )-DSIGMA( J+1 ) ) /
373: $ ( DSIGMA( I )+DSIGMA( J+1 ) ) )
374: 50 CONTINUE
375: Z( I ) = SIGN( SQRT( ABS( Z( I ) ) ), Q( I, 1 ) )
376: 60 CONTINUE
377: *
378: * Compute left singular vectors of the modified diagonal matrix,
379: * and store related information for the right singular vectors.
380: *
381: DO 90 I = 1, K
382: VT( 1, I ) = Z( 1 ) / U( 1, I ) / VT( 1, I )
383: U( 1, I ) = NEGONE
384: DO 70 J = 2, K
385: VT( J, I ) = Z( J ) / U( J, I ) / VT( J, I )
386: U( J, I ) = DSIGMA( J )*VT( J, I )
387: 70 CONTINUE
388: TEMP = DNRM2( K, U( 1, I ), 1 )
389: Q( 1, I ) = U( 1, I ) / TEMP
390: DO 80 J = 2, K
391: JC = IDXC( J )
392: Q( J, I ) = U( JC, I ) / TEMP
393: 80 CONTINUE
394: 90 CONTINUE
395: *
396: * Update the left singular vector matrix.
397: *
398: IF( K.EQ.2 ) THEN
399: CALL DGEMM( 'N', 'N', N, K, K, ONE, U2, LDU2, Q, LDQ, ZERO, U,
400: $ LDU )
401: GO TO 100
402: END IF
403: IF( CTOT( 1 ).GT.0 ) THEN
404: CALL DGEMM( 'N', 'N', NL, K, CTOT( 1 ), ONE, U2( 1, 2 ), LDU2,
405: $ Q( 2, 1 ), LDQ, ZERO, U( 1, 1 ), LDU )
406: IF( CTOT( 3 ).GT.0 ) THEN
407: KTEMP = 2 + CTOT( 1 ) + CTOT( 2 )
408: CALL DGEMM( 'N', 'N', NL, K, CTOT( 3 ), ONE, U2( 1, KTEMP ),
409: $ LDU2, Q( KTEMP, 1 ), LDQ, ONE, U( 1, 1 ), LDU )
410: END IF
411: ELSE IF( CTOT( 3 ).GT.0 ) THEN
412: KTEMP = 2 + CTOT( 1 ) + CTOT( 2 )
413: CALL DGEMM( 'N', 'N', NL, K, CTOT( 3 ), ONE, U2( 1, KTEMP ),
414: $ LDU2, Q( KTEMP, 1 ), LDQ, ZERO, U( 1, 1 ), LDU )
415: ELSE
416: CALL DLACPY( 'F', NL, K, U2, LDU2, U, LDU )
417: END IF
418: CALL DCOPY( K, Q( 1, 1 ), LDQ, U( NLP1, 1 ), LDU )
419: KTEMP = 2 + CTOT( 1 )
420: CTEMP = CTOT( 2 ) + CTOT( 3 )
421: CALL DGEMM( 'N', 'N', NR, K, CTEMP, ONE, U2( NLP2, KTEMP ), LDU2,
422: $ Q( KTEMP, 1 ), LDQ, ZERO, U( NLP2, 1 ), LDU )
423: *
424: * Generate the right singular vectors.
425: *
426: 100 CONTINUE
427: DO 120 I = 1, K
428: TEMP = DNRM2( K, VT( 1, I ), 1 )
429: Q( I, 1 ) = VT( 1, I ) / TEMP
430: DO 110 J = 2, K
431: JC = IDXC( J )
432: Q( I, J ) = VT( JC, I ) / TEMP
433: 110 CONTINUE
434: 120 CONTINUE
435: *
436: * Update the right singular vector matrix.
437: *
438: IF( K.EQ.2 ) THEN
439: CALL DGEMM( 'N', 'N', K, M, K, ONE, Q, LDQ, VT2, LDVT2, ZERO,
440: $ VT, LDVT )
441: RETURN
442: END IF
443: KTEMP = 1 + CTOT( 1 )
444: CALL DGEMM( 'N', 'N', K, NLP1, KTEMP, ONE, Q( 1, 1 ), LDQ,
445: $ VT2( 1, 1 ), LDVT2, ZERO, VT( 1, 1 ), LDVT )
446: KTEMP = 2 + CTOT( 1 ) + CTOT( 2 )
447: IF( KTEMP.LE.LDVT2 )
448: $ CALL DGEMM( 'N', 'N', K, NLP1, CTOT( 3 ), ONE, Q( 1, KTEMP ),
449: $ LDQ, VT2( KTEMP, 1 ), LDVT2, ONE, VT( 1, 1 ),
450: $ LDVT )
451: *
452: KTEMP = CTOT( 1 ) + 1
453: NRP1 = NR + SQRE
454: IF( KTEMP.GT.1 ) THEN
455: DO 130 I = 1, K
456: Q( I, KTEMP ) = Q( I, 1 )
457: 130 CONTINUE
458: DO 140 I = NLP2, M
459: VT2( KTEMP, I ) = VT2( 1, I )
460: 140 CONTINUE
461: END IF
462: CTEMP = 1 + CTOT( 2 ) + CTOT( 3 )
463: CALL DGEMM( 'N', 'N', K, NRP1, CTEMP, ONE, Q( 1, KTEMP ), LDQ,
464: $ VT2( KTEMP, NLP2 ), LDVT2, ZERO, VT( 1, NLP2 ), LDVT )
465: *
466: RETURN
467: *
468: * End of DLASD3
469: *
470: END
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