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.
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
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd3.f">
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, dimension (LDQ,K)
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: *>
106: *> \param[in,out] DSIGMA
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: *>
127: *> \param[in] U2
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: *>
189: *> \param[in,out] Z
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: *
207: *> \author Univ. of Tennessee
208: *> \author Univ. of California Berkeley
209: *> \author Univ. of Colorado Denver
210: *> \author NAG Ltd.
211: *
212: *> \date June 2017
213: *
214: *> \ingroup OTHERauxiliary
215: *
216: *> \par Contributors:
217: * ==================
218: *>
219: *> Ming Gu and Huan Ren, Computer Science Division, University of
220: *> California at Berkeley, USA
221: *>
222: * =====================================================================
223: SUBROUTINE DLASD3( NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2,
224: $ LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z,
225: $ INFO )
226: *
227: * -- LAPACK auxiliary routine (version 3.7.1) --
228: * -- LAPACK is a software package provided by Univ. of Tennessee, --
229: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
230: * June 2017
231: *
232: * .. Scalar Arguments ..
233: INTEGER INFO, K, LDQ, LDU, LDU2, LDVT, LDVT2, NL, NR,
234: $ SQRE
235: * ..
236: * .. Array Arguments ..
237: INTEGER CTOT( * ), IDXC( * )
238: DOUBLE PRECISION D( * ), DSIGMA( * ), Q( LDQ, * ), U( LDU, * ),
239: $ U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
240: $ Z( * )
241: * ..
242: *
243: * =====================================================================
244: *
245: * .. Parameters ..
246: DOUBLE PRECISION ONE, ZERO, NEGONE
247: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0,
248: $ NEGONE = -1.0D+0 )
249: * ..
250: * .. Local Scalars ..
251: INTEGER CTEMP, I, J, JC, KTEMP, M, N, NLP1, NLP2, NRP1
252: DOUBLE PRECISION RHO, TEMP
253: * ..
254: * .. External Functions ..
255: DOUBLE PRECISION DLAMC3, DNRM2
256: EXTERNAL DLAMC3, DNRM2
257: * ..
258: * .. External Subroutines ..
259: EXTERNAL DCOPY, DGEMM, DLACPY, DLASCL, DLASD4, XERBLA
260: * ..
261: * .. Intrinsic Functions ..
262: INTRINSIC ABS, SIGN, SQRT
263: * ..
264: * .. Executable Statements ..
265: *
266: * Test the input parameters.
267: *
268: INFO = 0
269: *
270: IF( NL.LT.1 ) THEN
271: INFO = -1
272: ELSE IF( NR.LT.1 ) THEN
273: INFO = -2
274: ELSE IF( ( SQRE.NE.1 ) .AND. ( SQRE.NE.0 ) ) THEN
275: INFO = -3
276: END IF
277: *
278: N = NL + NR + 1
279: M = N + SQRE
280: NLP1 = NL + 1
281: NLP2 = NL + 2
282: *
283: IF( ( K.LT.1 ) .OR. ( K.GT.N ) ) THEN
284: INFO = -4
285: ELSE IF( LDQ.LT.K ) THEN
286: INFO = -7
287: ELSE IF( LDU.LT.N ) THEN
288: INFO = -10
289: ELSE IF( LDU2.LT.N ) THEN
290: INFO = -12
291: ELSE IF( LDVT.LT.M ) THEN
292: INFO = -14
293: ELSE IF( LDVT2.LT.M ) THEN
294: INFO = -16
295: END IF
296: IF( INFO.NE.0 ) THEN
297: CALL XERBLA( 'DLASD3', -INFO )
298: RETURN
299: END IF
300: *
301: * Quick return if possible
302: *
303: IF( K.EQ.1 ) THEN
304: D( 1 ) = ABS( Z( 1 ) )
305: CALL DCOPY( M, VT2( 1, 1 ), LDVT2, VT( 1, 1 ), LDVT )
306: IF( Z( 1 ).GT.ZERO ) THEN
307: CALL DCOPY( N, U2( 1, 1 ), 1, U( 1, 1 ), 1 )
308: ELSE
309: DO 10 I = 1, N
310: U( I, 1 ) = -U2( I, 1 )
311: 10 CONTINUE
312: END IF
313: RETURN
314: END IF
315: *
316: * Modify values DSIGMA(i) to make sure all DSIGMA(i)-DSIGMA(j) can
317: * be computed with high relative accuracy (barring over/underflow).
318: * This is a problem on machines without a guard digit in
319: * add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
320: * The following code replaces DSIGMA(I) by 2*DSIGMA(I)-DSIGMA(I),
321: * which on any of these machines zeros out the bottommost
322: * bit of DSIGMA(I) if it is 1; this makes the subsequent
323: * subtractions DSIGMA(I)-DSIGMA(J) unproblematic when cancellation
324: * occurs. On binary machines with a guard digit (almost all
325: * machines) it does not change DSIGMA(I) at all. On hexadecimal
326: * and decimal machines with a guard digit, it slightly
327: * changes the bottommost bits of DSIGMA(I). It does not account
328: * for hexadecimal or decimal machines without guard digits
329: * (we know of none). We use a subroutine call to compute
330: * 2*DSIGMA(I) to prevent optimizing compilers from eliminating
331: * this code.
332: *
333: DO 20 I = 1, K
334: DSIGMA( I ) = DLAMC3( DSIGMA( I ), DSIGMA( I ) ) - DSIGMA( I )
335: 20 CONTINUE
336: *
337: * Keep a copy of Z.
338: *
339: CALL DCOPY( K, Z, 1, Q, 1 )
340: *
341: * Normalize Z.
342: *
343: RHO = DNRM2( K, Z, 1 )
344: CALL DLASCL( 'G', 0, 0, RHO, ONE, K, 1, Z, K, INFO )
345: RHO = RHO*RHO
346: *
347: * Find the new singular values.
348: *
349: DO 30 J = 1, K
350: CALL DLASD4( K, J, DSIGMA, Z, U( 1, J ), RHO, D( J ),
351: $ VT( 1, J ), INFO )
352: *
353: * If the zero finder fails, report the convergence failure.
354: *
355: IF( INFO.NE.0 ) THEN
356: RETURN
357: END IF
358: 30 CONTINUE
359: *
360: * Compute updated Z.
361: *
362: DO 60 I = 1, K
363: Z( I ) = U( I, K )*VT( I, K )
364: DO 40 J = 1, I - 1
365: Z( I ) = Z( I )*( U( I, J )*VT( I, J ) /
366: $ ( DSIGMA( I )-DSIGMA( J ) ) /
367: $ ( DSIGMA( I )+DSIGMA( J ) ) )
368: 40 CONTINUE
369: DO 50 J = I, K - 1
370: Z( I ) = Z( I )*( U( I, J )*VT( I, J ) /
371: $ ( DSIGMA( I )-DSIGMA( J+1 ) ) /
372: $ ( DSIGMA( I )+DSIGMA( J+1 ) ) )
373: 50 CONTINUE
374: Z( I ) = SIGN( SQRT( ABS( Z( I ) ) ), Q( I, 1 ) )
375: 60 CONTINUE
376: *
377: * Compute left singular vectors of the modified diagonal matrix,
378: * and store related information for the right singular vectors.
379: *
380: DO 90 I = 1, K
381: VT( 1, I ) = Z( 1 ) / U( 1, I ) / VT( 1, I )
382: U( 1, I ) = NEGONE
383: DO 70 J = 2, K
384: VT( J, I ) = Z( J ) / U( J, I ) / VT( J, I )
385: U( J, I ) = DSIGMA( J )*VT( J, I )
386: 70 CONTINUE
387: TEMP = DNRM2( K, U( 1, I ), 1 )
388: Q( 1, I ) = U( 1, I ) / TEMP
389: DO 80 J = 2, K
390: JC = IDXC( J )
391: Q( J, I ) = U( JC, I ) / TEMP
392: 80 CONTINUE
393: 90 CONTINUE
394: *
395: * Update the left singular vector matrix.
396: *
397: IF( K.EQ.2 ) THEN
398: CALL DGEMM( 'N', 'N', N, K, K, ONE, U2, LDU2, Q, LDQ, ZERO, U,
399: $ LDU )
400: GO TO 100
401: END IF
402: IF( CTOT( 1 ).GT.0 ) THEN
403: CALL DGEMM( 'N', 'N', NL, K, CTOT( 1 ), ONE, U2( 1, 2 ), LDU2,
404: $ Q( 2, 1 ), LDQ, ZERO, U( 1, 1 ), LDU )
405: IF( CTOT( 3 ).GT.0 ) THEN
406: KTEMP = 2 + CTOT( 1 ) + CTOT( 2 )
407: CALL DGEMM( 'N', 'N', NL, K, CTOT( 3 ), ONE, U2( 1, KTEMP ),
408: $ LDU2, Q( KTEMP, 1 ), LDQ, ONE, U( 1, 1 ), LDU )
409: END IF
410: ELSE IF( CTOT( 3 ).GT.0 ) THEN
411: KTEMP = 2 + CTOT( 1 ) + CTOT( 2 )
412: CALL DGEMM( 'N', 'N', NL, K, CTOT( 3 ), ONE, U2( 1, KTEMP ),
413: $ LDU2, Q( KTEMP, 1 ), LDQ, ZERO, U( 1, 1 ), LDU )
414: ELSE
415: CALL DLACPY( 'F', NL, K, U2, LDU2, U, LDU )
416: END IF
417: CALL DCOPY( K, Q( 1, 1 ), LDQ, U( NLP1, 1 ), LDU )
418: KTEMP = 2 + CTOT( 1 )
419: CTEMP = CTOT( 2 ) + CTOT( 3 )
420: CALL DGEMM( 'N', 'N', NR, K, CTEMP, ONE, U2( NLP2, KTEMP ), LDU2,
421: $ Q( KTEMP, 1 ), LDQ, ZERO, U( NLP2, 1 ), LDU )
422: *
423: * Generate the right singular vectors.
424: *
425: 100 CONTINUE
426: DO 120 I = 1, K
427: TEMP = DNRM2( K, VT( 1, I ), 1 )
428: Q( I, 1 ) = VT( 1, I ) / TEMP
429: DO 110 J = 2, K
430: JC = IDXC( J )
431: Q( I, J ) = VT( JC, I ) / TEMP
432: 110 CONTINUE
433: 120 CONTINUE
434: *
435: * Update the right singular vector matrix.
436: *
437: IF( K.EQ.2 ) THEN
438: CALL DGEMM( 'N', 'N', K, M, K, ONE, Q, LDQ, VT2, LDVT2, ZERO,
439: $ VT, LDVT )
440: RETURN
441: END IF
442: KTEMP = 1 + CTOT( 1 )
443: CALL DGEMM( 'N', 'N', K, NLP1, KTEMP, ONE, Q( 1, 1 ), LDQ,
444: $ VT2( 1, 1 ), LDVT2, ZERO, VT( 1, 1 ), LDVT )
445: KTEMP = 2 + CTOT( 1 ) + CTOT( 2 )
446: IF( KTEMP.LE.LDVT2 )
447: $ CALL DGEMM( 'N', 'N', K, NLP1, CTOT( 3 ), ONE, Q( 1, KTEMP ),
448: $ LDQ, VT2( KTEMP, 1 ), LDVT2, ONE, VT( 1, 1 ),
449: $ LDVT )
450: *
451: KTEMP = CTOT( 1 ) + 1
452: NRP1 = NR + SQRE
453: IF( KTEMP.GT.1 ) THEN
454: DO 130 I = 1, K
455: Q( I, KTEMP ) = Q( I, 1 )
456: 130 CONTINUE
457: DO 140 I = NLP2, M
458: VT2( KTEMP, I ) = VT2( 1, I )
459: 140 CONTINUE
460: END IF
461: CTEMP = 1 + CTOT( 2 ) + CTOT( 3 )
462: CALL DGEMM( 'N', 'N', K, NRP1, CTEMP, ONE, Q( 1, KTEMP ), LDQ,
463: $ VT2( KTEMP, NLP2 ), LDVT2, ZERO, VT( 1, NLP2 ), LDVT )
464: *
465: RETURN
466: *
467: * End of DLASD3
468: *
469: END
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