Annotation of rpl/lapack/lapack/dlasd3.f, revision 1.22
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: *
1.18 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.10 bertrand 7: *
8: *> \htmlonly
1.18 bertrand 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">
1.10 bertrand 15: *> [TXT]</a>
1.18 bertrand 16: *> \endhtmlonly
1.10 bertrand 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 )
1.18 bertrand 24: *
1.10 bertrand 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: * ..
1.18 bertrand 35: *
1.10 bertrand 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
1.20 bertrand 97: *> Q is DOUBLE PRECISION array, dimension (LDQ,K)
1.10 bertrand 98: *> \endverbatim
99: *>
100: *> \param[in] LDQ
101: *> \verbatim
102: *> LDQ is INTEGER
103: *> The leading dimension of the array Q. LDQ >= K.
104: *> \endverbatim
105: *>
1.20 bertrand 106: *> \param[in,out] DSIGMA
1.10 bertrand 107: *> \verbatim
108: *> DSIGMA is DOUBLE PRECISION array, dimension(K)
109: *> The first K elements of this array contain the old roots
110: *> of the deflated updating problem. These are the poles
111: *> of the secular equation.
112: *> \endverbatim
113: *>
114: *> \param[out] U
115: *> \verbatim
116: *> U is DOUBLE PRECISION array, dimension (LDU, N)
117: *> The last N - K columns of this matrix contain the deflated
118: *> left singular vectors.
119: *> \endverbatim
120: *>
121: *> \param[in] LDU
122: *> \verbatim
123: *> LDU is INTEGER
124: *> The leading dimension of the array U. LDU >= N.
125: *> \endverbatim
126: *>
1.20 bertrand 127: *> \param[in] U2
1.10 bertrand 128: *> \verbatim
129: *> U2 is DOUBLE PRECISION array, dimension (LDU2, N)
130: *> The first K columns of this matrix contain the non-deflated
131: *> left singular vectors for the split problem.
132: *> \endverbatim
133: *>
134: *> \param[in] LDU2
135: *> \verbatim
136: *> LDU2 is INTEGER
137: *> The leading dimension of the array U2. LDU2 >= N.
138: *> \endverbatim
139: *>
140: *> \param[out] VT
141: *> \verbatim
142: *> VT is DOUBLE PRECISION array, dimension (LDVT, M)
143: *> The last M - K columns of VT**T contain the deflated
144: *> right singular vectors.
145: *> \endverbatim
146: *>
147: *> \param[in] LDVT
148: *> \verbatim
149: *> LDVT is INTEGER
150: *> The leading dimension of the array VT. LDVT >= N.
151: *> \endverbatim
152: *>
153: *> \param[in,out] VT2
154: *> \verbatim
155: *> VT2 is DOUBLE PRECISION array, dimension (LDVT2, N)
156: *> The first K columns of VT2**T contain the non-deflated
157: *> right singular vectors for the split problem.
158: *> \endverbatim
159: *>
160: *> \param[in] LDVT2
161: *> \verbatim
162: *> LDVT2 is INTEGER
163: *> The leading dimension of the array VT2. LDVT2 >= N.
164: *> \endverbatim
165: *>
166: *> \param[in] IDXC
167: *> \verbatim
168: *> IDXC is INTEGER array, dimension ( N )
169: *> The permutation used to arrange the columns of U (and rows of
170: *> VT) into three groups: the first group contains non-zero
171: *> entries only at and above (or before) NL +1; the second
172: *> contains non-zero entries only at and below (or after) NL+2;
173: *> and the third is dense. The first column of U and the row of
174: *> VT are treated separately, however.
175: *>
176: *> The rows of the singular vectors found by DLASD4
177: *> must be likewise permuted before the matrix multiplies can
178: *> take place.
179: *> \endverbatim
180: *>
181: *> \param[in] CTOT
182: *> \verbatim
183: *> CTOT is INTEGER array, dimension ( 4 )
184: *> A count of the total number of the various types of columns
185: *> in U (or rows in VT), as described in IDXC. The fourth column
186: *> type is any column which has been deflated.
187: *> \endverbatim
188: *>
1.20 bertrand 189: *> \param[in,out] Z
1.10 bertrand 190: *> \verbatim
191: *> Z is DOUBLE PRECISION array, dimension (K)
192: *> The first K elements of this array contain the components
193: *> of the deflation-adjusted updating row vector.
194: *> \endverbatim
195: *>
196: *> \param[out] INFO
197: *> \verbatim
198: *> INFO is INTEGER
199: *> = 0: successful exit.
200: *> < 0: if INFO = -i, the i-th argument had an illegal value.
201: *> > 0: if INFO = 1, a singular value did not converge
202: *> \endverbatim
203: *
204: * Authors:
205: * ========
206: *
1.18 bertrand 207: *> \author Univ. of Tennessee
208: *> \author Univ. of California Berkeley
209: *> \author Univ. of Colorado Denver
210: *> \author NAG Ltd.
1.10 bertrand 211: *
1.18 bertrand 212: *> \ingroup OTHERauxiliary
1.10 bertrand 213: *
214: *> \par Contributors:
215: * ==================
216: *>
217: *> Ming Gu and Huan Ren, Computer Science Division, University of
218: *> California at Berkeley, USA
219: *>
220: * =====================================================================
1.1 bertrand 221: SUBROUTINE DLASD3( NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2,
222: $ LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z,
223: $ INFO )
224: *
1.22 ! bertrand 225: * -- LAPACK auxiliary routine --
1.1 bertrand 226: * -- LAPACK is a software package provided by Univ. of Tennessee, --
227: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
228: *
229: * .. Scalar Arguments ..
230: INTEGER INFO, K, LDQ, LDU, LDU2, LDVT, LDVT2, NL, NR,
231: $ SQRE
232: * ..
233: * .. Array Arguments ..
234: INTEGER CTOT( * ), IDXC( * )
235: DOUBLE PRECISION D( * ), DSIGMA( * ), Q( LDQ, * ), U( LDU, * ),
236: $ U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
237: $ Z( * )
238: * ..
239: *
240: * =====================================================================
241: *
242: * .. Parameters ..
243: DOUBLE PRECISION ONE, ZERO, NEGONE
244: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0,
245: $ NEGONE = -1.0D+0 )
246: * ..
247: * .. Local Scalars ..
248: INTEGER CTEMP, I, J, JC, KTEMP, M, N, NLP1, NLP2, NRP1
249: DOUBLE PRECISION RHO, TEMP
250: * ..
251: * .. External Functions ..
252: DOUBLE PRECISION DLAMC3, DNRM2
253: EXTERNAL DLAMC3, DNRM2
254: * ..
255: * .. External Subroutines ..
256: EXTERNAL DCOPY, DGEMM, DLACPY, DLASCL, DLASD4, XERBLA
257: * ..
258: * .. Intrinsic Functions ..
259: INTRINSIC ABS, SIGN, SQRT
260: * ..
261: * .. Executable Statements ..
262: *
263: * Test the input parameters.
264: *
265: INFO = 0
266: *
267: IF( NL.LT.1 ) THEN
268: INFO = -1
269: ELSE IF( NR.LT.1 ) THEN
270: INFO = -2
271: ELSE IF( ( SQRE.NE.1 ) .AND. ( SQRE.NE.0 ) ) THEN
272: INFO = -3
273: END IF
274: *
275: N = NL + NR + 1
276: M = N + SQRE
277: NLP1 = NL + 1
278: NLP2 = NL + 2
279: *
280: IF( ( K.LT.1 ) .OR. ( K.GT.N ) ) THEN
281: INFO = -4
282: ELSE IF( LDQ.LT.K ) THEN
283: INFO = -7
284: ELSE IF( LDU.LT.N ) THEN
285: INFO = -10
286: ELSE IF( LDU2.LT.N ) THEN
287: INFO = -12
288: ELSE IF( LDVT.LT.M ) THEN
289: INFO = -14
290: ELSE IF( LDVT2.LT.M ) THEN
291: INFO = -16
292: END IF
293: IF( INFO.NE.0 ) THEN
294: CALL XERBLA( 'DLASD3', -INFO )
295: RETURN
296: END IF
297: *
298: * Quick return if possible
299: *
300: IF( K.EQ.1 ) THEN
301: D( 1 ) = ABS( Z( 1 ) )
302: CALL DCOPY( M, VT2( 1, 1 ), LDVT2, VT( 1, 1 ), LDVT )
303: IF( Z( 1 ).GT.ZERO ) THEN
304: CALL DCOPY( N, U2( 1, 1 ), 1, U( 1, 1 ), 1 )
305: ELSE
306: DO 10 I = 1, N
307: U( I, 1 ) = -U2( I, 1 )
308: 10 CONTINUE
309: END IF
310: RETURN
311: END IF
312: *
313: * Modify values DSIGMA(i) to make sure all DSIGMA(i)-DSIGMA(j) can
314: * be computed with high relative accuracy (barring over/underflow).
315: * This is a problem on machines without a guard digit in
316: * add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
317: * The following code replaces DSIGMA(I) by 2*DSIGMA(I)-DSIGMA(I),
318: * which on any of these machines zeros out the bottommost
319: * bit of DSIGMA(I) if it is 1; this makes the subsequent
320: * subtractions DSIGMA(I)-DSIGMA(J) unproblematic when cancellation
321: * occurs. On binary machines with a guard digit (almost all
322: * machines) it does not change DSIGMA(I) at all. On hexadecimal
323: * and decimal machines with a guard digit, it slightly
324: * changes the bottommost bits of DSIGMA(I). It does not account
325: * for hexadecimal or decimal machines without guard digits
326: * (we know of none). We use a subroutine call to compute
327: * 2*DSIGMA(I) to prevent optimizing compilers from eliminating
328: * this code.
329: *
330: DO 20 I = 1, K
331: DSIGMA( I ) = DLAMC3( DSIGMA( I ), DSIGMA( I ) ) - DSIGMA( I )
332: 20 CONTINUE
333: *
334: * Keep a copy of Z.
335: *
336: CALL DCOPY( K, Z, 1, Q, 1 )
337: *
338: * Normalize Z.
339: *
340: RHO = DNRM2( K, Z, 1 )
341: CALL DLASCL( 'G', 0, 0, RHO, ONE, K, 1, Z, K, INFO )
342: RHO = RHO*RHO
343: *
344: * Find the new singular values.
345: *
346: DO 30 J = 1, K
347: CALL DLASD4( K, J, DSIGMA, Z, U( 1, J ), RHO, D( J ),
348: $ VT( 1, J ), INFO )
349: *
1.16 bertrand 350: * If the zero finder fails, report the convergence failure.
1.1 bertrand 351: *
352: IF( INFO.NE.0 ) THEN
353: RETURN
354: END IF
355: 30 CONTINUE
356: *
357: * Compute updated Z.
358: *
359: DO 60 I = 1, K
360: Z( I ) = U( I, K )*VT( I, K )
361: DO 40 J = 1, I - 1
362: Z( I ) = Z( I )*( U( I, J )*VT( I, J ) /
363: $ ( DSIGMA( I )-DSIGMA( J ) ) /
364: $ ( DSIGMA( I )+DSIGMA( J ) ) )
365: 40 CONTINUE
366: DO 50 J = I, K - 1
367: Z( I ) = Z( I )*( U( I, J )*VT( I, J ) /
368: $ ( DSIGMA( I )-DSIGMA( J+1 ) ) /
369: $ ( DSIGMA( I )+DSIGMA( J+1 ) ) )
370: 50 CONTINUE
371: Z( I ) = SIGN( SQRT( ABS( Z( I ) ) ), Q( I, 1 ) )
372: 60 CONTINUE
373: *
374: * Compute left singular vectors of the modified diagonal matrix,
375: * and store related information for the right singular vectors.
376: *
377: DO 90 I = 1, K
378: VT( 1, I ) = Z( 1 ) / U( 1, I ) / VT( 1, I )
379: U( 1, I ) = NEGONE
380: DO 70 J = 2, K
381: VT( J, I ) = Z( J ) / U( J, I ) / VT( J, I )
382: U( J, I ) = DSIGMA( J )*VT( J, I )
383: 70 CONTINUE
384: TEMP = DNRM2( K, U( 1, I ), 1 )
385: Q( 1, I ) = U( 1, I ) / TEMP
386: DO 80 J = 2, K
387: JC = IDXC( J )
388: Q( J, I ) = U( JC, I ) / TEMP
389: 80 CONTINUE
390: 90 CONTINUE
391: *
392: * Update the left singular vector matrix.
393: *
394: IF( K.EQ.2 ) THEN
395: CALL DGEMM( 'N', 'N', N, K, K, ONE, U2, LDU2, Q, LDQ, ZERO, U,
396: $ LDU )
397: GO TO 100
398: END IF
399: IF( CTOT( 1 ).GT.0 ) THEN
400: CALL DGEMM( 'N', 'N', NL, K, CTOT( 1 ), ONE, U2( 1, 2 ), LDU2,
401: $ Q( 2, 1 ), LDQ, ZERO, U( 1, 1 ), LDU )
402: IF( CTOT( 3 ).GT.0 ) THEN
403: KTEMP = 2 + CTOT( 1 ) + CTOT( 2 )
404: CALL DGEMM( 'N', 'N', NL, K, CTOT( 3 ), ONE, U2( 1, KTEMP ),
405: $ LDU2, Q( KTEMP, 1 ), LDQ, ONE, U( 1, 1 ), LDU )
406: END IF
407: ELSE IF( CTOT( 3 ).GT.0 ) THEN
408: KTEMP = 2 + CTOT( 1 ) + CTOT( 2 )
409: CALL DGEMM( 'N', 'N', NL, K, CTOT( 3 ), ONE, U2( 1, KTEMP ),
410: $ LDU2, Q( KTEMP, 1 ), LDQ, ZERO, U( 1, 1 ), LDU )
411: ELSE
412: CALL DLACPY( 'F', NL, K, U2, LDU2, U, LDU )
413: END IF
414: CALL DCOPY( K, Q( 1, 1 ), LDQ, U( NLP1, 1 ), LDU )
415: KTEMP = 2 + CTOT( 1 )
416: CTEMP = CTOT( 2 ) + CTOT( 3 )
417: CALL DGEMM( 'N', 'N', NR, K, CTEMP, ONE, U2( NLP2, KTEMP ), LDU2,
418: $ Q( KTEMP, 1 ), LDQ, ZERO, U( NLP2, 1 ), LDU )
419: *
420: * Generate the right singular vectors.
421: *
422: 100 CONTINUE
423: DO 120 I = 1, K
424: TEMP = DNRM2( K, VT( 1, I ), 1 )
425: Q( I, 1 ) = VT( 1, I ) / TEMP
426: DO 110 J = 2, K
427: JC = IDXC( J )
428: Q( I, J ) = VT( JC, I ) / TEMP
429: 110 CONTINUE
430: 120 CONTINUE
431: *
432: * Update the right singular vector matrix.
433: *
434: IF( K.EQ.2 ) THEN
435: CALL DGEMM( 'N', 'N', K, M, K, ONE, Q, LDQ, VT2, LDVT2, ZERO,
436: $ VT, LDVT )
437: RETURN
438: END IF
439: KTEMP = 1 + CTOT( 1 )
440: CALL DGEMM( 'N', 'N', K, NLP1, KTEMP, ONE, Q( 1, 1 ), LDQ,
441: $ VT2( 1, 1 ), LDVT2, ZERO, VT( 1, 1 ), LDVT )
442: KTEMP = 2 + CTOT( 1 ) + CTOT( 2 )
443: IF( KTEMP.LE.LDVT2 )
444: $ CALL DGEMM( 'N', 'N', K, NLP1, CTOT( 3 ), ONE, Q( 1, KTEMP ),
445: $ LDQ, VT2( KTEMP, 1 ), LDVT2, ONE, VT( 1, 1 ),
446: $ LDVT )
447: *
448: KTEMP = CTOT( 1 ) + 1
449: NRP1 = NR + SQRE
450: IF( KTEMP.GT.1 ) THEN
451: DO 130 I = 1, K
452: Q( I, KTEMP ) = Q( I, 1 )
453: 130 CONTINUE
454: DO 140 I = NLP2, M
455: VT2( KTEMP, I ) = VT2( 1, I )
456: 140 CONTINUE
457: END IF
458: CTEMP = 1 + CTOT( 2 ) + CTOT( 3 )
459: CALL DGEMM( 'N', 'N', K, NRP1, CTEMP, ONE, Q( 1, KTEMP ), LDQ,
460: $ VT2( KTEMP, NLP2 ), LDVT2, ZERO, VT( 1, NLP2 ), LDVT )
461: *
462: RETURN
463: *
464: * End of DLASD3
465: *
466: END
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