1: *> \brief \b DLASD8 finds the square roots of the roots of the secular equation, and stores, for each element in D, the distance to its two nearest poles. 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 DLASD8 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd8.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd8.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd8.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR,
22: * DSIGMA, WORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * INTEGER ICOMPQ, INFO, K, LDDIFR
26: * ..
27: * .. Array Arguments ..
28: * DOUBLE PRECISION D( * ), DIFL( * ), DIFR( LDDIFR, * ),
29: * $ DSIGMA( * ), VF( * ), VL( * ), WORK( * ),
30: * $ Z( * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *>
39: *> DLASD8 finds the square roots of the roots of the secular equation,
40: *> as defined by the values in DSIGMA and Z. It makes the appropriate
41: *> calls to DLASD4, and stores, for each element in D, the distance
42: *> to its two nearest poles (elements in DSIGMA). It also updates
43: *> the arrays VF and VL, the first and last components of all the
44: *> right singular vectors of the original bidiagonal matrix.
45: *>
46: *> DLASD8 is called from DLASD6.
47: *> \endverbatim
48: *
49: * Arguments:
50: * ==========
51: *
52: *> \param[in] ICOMPQ
53: *> \verbatim
54: *> ICOMPQ is INTEGER
55: *> Specifies whether singular vectors are to be computed in
56: *> factored form in the calling routine:
57: *> = 0: Compute singular values only.
58: *> = 1: Compute singular vectors in factored form as well.
59: *> \endverbatim
60: *>
61: *> \param[in] K
62: *> \verbatim
63: *> K is INTEGER
64: *> The number of terms in the rational function to be solved
65: *> by DLASD4. K >= 1.
66: *> \endverbatim
67: *>
68: *> \param[out] D
69: *> \verbatim
70: *> D is DOUBLE PRECISION array, dimension ( K )
71: *> On output, D contains the updated singular values.
72: *> \endverbatim
73: *>
74: *> \param[in,out] Z
75: *> \verbatim
76: *> Z is DOUBLE PRECISION array, dimension ( K )
77: *> On entry, the first K elements of this array contain the
78: *> components of the deflation-adjusted updating row vector.
79: *> On exit, Z is updated.
80: *> \endverbatim
81: *>
82: *> \param[in,out] VF
83: *> \verbatim
84: *> VF is DOUBLE PRECISION array, dimension ( K )
85: *> On entry, VF contains information passed through DBEDE8.
86: *> On exit, VF contains the first K components of the first
87: *> components of all right singular vectors of the bidiagonal
88: *> matrix.
89: *> \endverbatim
90: *>
91: *> \param[in,out] VL
92: *> \verbatim
93: *> VL is DOUBLE PRECISION array, dimension ( K )
94: *> On entry, VL contains information passed through DBEDE8.
95: *> On exit, VL contains the first K components of the last
96: *> components of all right singular vectors of the bidiagonal
97: *> matrix.
98: *> \endverbatim
99: *>
100: *> \param[out] DIFL
101: *> \verbatim
102: *> DIFL is DOUBLE PRECISION array, dimension ( K )
103: *> On exit, DIFL(I) = D(I) - DSIGMA(I).
104: *> \endverbatim
105: *>
106: *> \param[out] DIFR
107: *> \verbatim
108: *> DIFR is DOUBLE PRECISION array,
109: *> dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and
110: *> dimension ( K ) if ICOMPQ = 0.
111: *> On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not
112: *> defined and will not be referenced.
113: *>
114: *> If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
115: *> normalizing factors for the right singular vector matrix.
116: *> \endverbatim
117: *>
118: *> \param[in] LDDIFR
119: *> \verbatim
120: *> LDDIFR is INTEGER
121: *> The leading dimension of DIFR, must be at least K.
122: *> \endverbatim
123: *>
124: *> \param[in,out] DSIGMA
125: *> \verbatim
126: *> DSIGMA is DOUBLE PRECISION array, dimension ( K )
127: *> On entry, the first K elements of this array contain the old
128: *> roots of the deflated updating problem. These are the poles
129: *> of the secular equation.
130: *> On exit, the elements of DSIGMA may be very slightly altered
131: *> in value.
132: *> \endverbatim
133: *>
134: *> \param[out] WORK
135: *> \verbatim
136: *> WORK is DOUBLE PRECISION array, dimension at least 3 * K
137: *> \endverbatim
138: *>
139: *> \param[out] INFO
140: *> \verbatim
141: *> INFO is INTEGER
142: *> = 0: successful exit.
143: *> < 0: if INFO = -i, the i-th argument had an illegal value.
144: *> > 0: if INFO = 1, a singular value did not converge
145: *> \endverbatim
146: *
147: * Authors:
148: * ========
149: *
150: *> \author Univ. of Tennessee
151: *> \author Univ. of California Berkeley
152: *> \author Univ. of Colorado Denver
153: *> \author NAG Ltd.
154: *
155: *> \date December 2016
156: *
157: *> \ingroup OTHERauxiliary
158: *
159: *> \par Contributors:
160: * ==================
161: *>
162: *> Ming Gu and Huan Ren, Computer Science Division, University of
163: *> California at Berkeley, USA
164: *>
165: * =====================================================================
166: SUBROUTINE DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR,
167: $ DSIGMA, WORK, INFO )
168: *
169: * -- LAPACK auxiliary routine (version 3.7.0) --
170: * -- LAPACK is a software package provided by Univ. of Tennessee, --
171: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
172: * December 2016
173: *
174: * .. Scalar Arguments ..
175: INTEGER ICOMPQ, INFO, K, LDDIFR
176: * ..
177: * .. Array Arguments ..
178: DOUBLE PRECISION D( * ), DIFL( * ), DIFR( LDDIFR, * ),
179: $ DSIGMA( * ), VF( * ), VL( * ), WORK( * ),
180: $ Z( * )
181: * ..
182: *
183: * =====================================================================
184: *
185: * .. Parameters ..
186: DOUBLE PRECISION ONE
187: PARAMETER ( ONE = 1.0D+0 )
188: * ..
189: * .. Local Scalars ..
190: INTEGER I, IWK1, IWK2, IWK2I, IWK3, IWK3I, J
191: DOUBLE PRECISION DIFLJ, DIFRJ, DJ, DSIGJ, DSIGJP, RHO, TEMP
192: * ..
193: * .. External Subroutines ..
194: EXTERNAL DCOPY, DLASCL, DLASD4, DLASET, XERBLA
195: * ..
196: * .. External Functions ..
197: DOUBLE PRECISION DDOT, DLAMC3, DNRM2
198: EXTERNAL DDOT, DLAMC3, DNRM2
199: * ..
200: * .. Intrinsic Functions ..
201: INTRINSIC ABS, SIGN, SQRT
202: * ..
203: * .. Executable Statements ..
204: *
205: * Test the input parameters.
206: *
207: INFO = 0
208: *
209: IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
210: INFO = -1
211: ELSE IF( K.LT.1 ) THEN
212: INFO = -2
213: ELSE IF( LDDIFR.LT.K ) THEN
214: INFO = -9
215: END IF
216: IF( INFO.NE.0 ) THEN
217: CALL XERBLA( 'DLASD8', -INFO )
218: RETURN
219: END IF
220: *
221: * Quick return if possible
222: *
223: IF( K.EQ.1 ) THEN
224: D( 1 ) = ABS( Z( 1 ) )
225: DIFL( 1 ) = D( 1 )
226: IF( ICOMPQ.EQ.1 ) THEN
227: DIFL( 2 ) = ONE
228: DIFR( 1, 2 ) = ONE
229: END IF
230: RETURN
231: END IF
232: *
233: * Modify values DSIGMA(i) to make sure all DSIGMA(i)-DSIGMA(j) can
234: * be computed with high relative accuracy (barring over/underflow).
235: * This is a problem on machines without a guard digit in
236: * add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
237: * The following code replaces DSIGMA(I) by 2*DSIGMA(I)-DSIGMA(I),
238: * which on any of these machines zeros out the bottommost
239: * bit of DSIGMA(I) if it is 1; this makes the subsequent
240: * subtractions DSIGMA(I)-DSIGMA(J) unproblematic when cancellation
241: * occurs. On binary machines with a guard digit (almost all
242: * machines) it does not change DSIGMA(I) at all. On hexadecimal
243: * and decimal machines with a guard digit, it slightly
244: * changes the bottommost bits of DSIGMA(I). It does not account
245: * for hexadecimal or decimal machines without guard digits
246: * (we know of none). We use a subroutine call to compute
247: * 2*DLAMBDA(I) to prevent optimizing compilers from eliminating
248: * this code.
249: *
250: DO 10 I = 1, K
251: DSIGMA( I ) = DLAMC3( DSIGMA( I ), DSIGMA( I ) ) - DSIGMA( I )
252: 10 CONTINUE
253: *
254: * Book keeping.
255: *
256: IWK1 = 1
257: IWK2 = IWK1 + K
258: IWK3 = IWK2 + K
259: IWK2I = IWK2 - 1
260: IWK3I = IWK3 - 1
261: *
262: * Normalize Z.
263: *
264: RHO = DNRM2( K, Z, 1 )
265: CALL DLASCL( 'G', 0, 0, RHO, ONE, K, 1, Z, K, INFO )
266: RHO = RHO*RHO
267: *
268: * Initialize WORK(IWK3).
269: *
270: CALL DLASET( 'A', K, 1, ONE, ONE, WORK( IWK3 ), K )
271: *
272: * Compute the updated singular values, the arrays DIFL, DIFR,
273: * and the updated Z.
274: *
275: DO 40 J = 1, K
276: CALL DLASD4( K, J, DSIGMA, Z, WORK( IWK1 ), RHO, D( J ),
277: $ WORK( IWK2 ), INFO )
278: *
279: * If the root finder fails, report the convergence failure.
280: *
281: IF( INFO.NE.0 ) THEN
282: RETURN
283: END IF
284: WORK( IWK3I+J ) = WORK( IWK3I+J )*WORK( J )*WORK( IWK2I+J )
285: DIFL( J ) = -WORK( J )
286: DIFR( J, 1 ) = -WORK( J+1 )
287: DO 20 I = 1, J - 1
288: WORK( IWK3I+I ) = WORK( IWK3I+I )*WORK( I )*
289: $ WORK( IWK2I+I ) / ( DSIGMA( I )-
290: $ DSIGMA( J ) ) / ( DSIGMA( I )+
291: $ DSIGMA( J ) )
292: 20 CONTINUE
293: DO 30 I = J + 1, K
294: WORK( IWK3I+I ) = WORK( IWK3I+I )*WORK( I )*
295: $ WORK( IWK2I+I ) / ( DSIGMA( I )-
296: $ DSIGMA( J ) ) / ( DSIGMA( I )+
297: $ DSIGMA( J ) )
298: 30 CONTINUE
299: 40 CONTINUE
300: *
301: * Compute updated Z.
302: *
303: DO 50 I = 1, K
304: Z( I ) = SIGN( SQRT( ABS( WORK( IWK3I+I ) ) ), Z( I ) )
305: 50 CONTINUE
306: *
307: * Update VF and VL.
308: *
309: DO 80 J = 1, K
310: DIFLJ = DIFL( J )
311: DJ = D( J )
312: DSIGJ = -DSIGMA( J )
313: IF( J.LT.K ) THEN
314: DIFRJ = -DIFR( J, 1 )
315: DSIGJP = -DSIGMA( J+1 )
316: END IF
317: WORK( J ) = -Z( J ) / DIFLJ / ( DSIGMA( J )+DJ )
318: DO 60 I = 1, J - 1
319: WORK( I ) = Z( I ) / ( DLAMC3( DSIGMA( I ), DSIGJ )-DIFLJ )
320: $ / ( DSIGMA( I )+DJ )
321: 60 CONTINUE
322: DO 70 I = J + 1, K
323: WORK( I ) = Z( I ) / ( DLAMC3( DSIGMA( I ), DSIGJP )+DIFRJ )
324: $ / ( DSIGMA( I )+DJ )
325: 70 CONTINUE
326: TEMP = DNRM2( K, WORK, 1 )
327: WORK( IWK2I+J ) = DDOT( K, WORK, 1, VF, 1 ) / TEMP
328: WORK( IWK3I+J ) = DDOT( K, WORK, 1, VL, 1 ) / TEMP
329: IF( ICOMPQ.EQ.1 ) THEN
330: DIFR( J, 2 ) = TEMP
331: END IF
332: 80 CONTINUE
333: *
334: CALL DCOPY( K, WORK( IWK2 ), 1, VF, 1 )
335: CALL DCOPY( K, WORK( IWK3 ), 1, VL, 1 )
336: *
337: RETURN
338: *
339: * End of DLASD8
340: *
341: END
342:
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