1: *> \brief \b DLASD8
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
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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 November 2011
156: *
157: *> \ingroup auxOTHERauxiliary
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 3.4.0) --
170: * -- LAPACK is a software package provided by Univ. of Tennessee, --
171: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
172: * November 2011
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, the computation is terminated.
280: *
281: IF( INFO.NE.0 ) THEN
282: CALL XERBLA( 'DLASD4', -INFO )
283: RETURN
284: END IF
285: WORK( IWK3I+J ) = WORK( IWK3I+J )*WORK( J )*WORK( IWK2I+J )
286: DIFL( J ) = -WORK( J )
287: DIFR( J, 1 ) = -WORK( J+1 )
288: DO 20 I = 1, J - 1
289: WORK( IWK3I+I ) = WORK( IWK3I+I )*WORK( I )*
290: $ WORK( IWK2I+I ) / ( DSIGMA( I )-
291: $ DSIGMA( J ) ) / ( DSIGMA( I )+
292: $ DSIGMA( J ) )
293: 20 CONTINUE
294: DO 30 I = J + 1, K
295: WORK( IWK3I+I ) = WORK( IWK3I+I )*WORK( I )*
296: $ WORK( IWK2I+I ) / ( DSIGMA( I )-
297: $ DSIGMA( J ) ) / ( DSIGMA( I )+
298: $ DSIGMA( J ) )
299: 30 CONTINUE
300: 40 CONTINUE
301: *
302: * Compute updated Z.
303: *
304: DO 50 I = 1, K
305: Z( I ) = SIGN( SQRT( ABS( WORK( IWK3I+I ) ) ), Z( I ) )
306: 50 CONTINUE
307: *
308: * Update VF and VL.
309: *
310: DO 80 J = 1, K
311: DIFLJ = DIFL( J )
312: DJ = D( J )
313: DSIGJ = -DSIGMA( J )
314: IF( J.LT.K ) THEN
315: DIFRJ = -DIFR( J, 1 )
316: DSIGJP = -DSIGMA( J+1 )
317: END IF
318: WORK( J ) = -Z( J ) / DIFLJ / ( DSIGMA( J )+DJ )
319: DO 60 I = 1, J - 1
320: WORK( I ) = Z( I ) / ( DLAMC3( DSIGMA( I ), DSIGJ )-DIFLJ )
321: $ / ( DSIGMA( I )+DJ )
322: 60 CONTINUE
323: DO 70 I = J + 1, K
324: WORK( I ) = Z( I ) / ( DLAMC3( DSIGMA( I ), DSIGJP )+DIFRJ )
325: $ / ( DSIGMA( I )+DJ )
326: 70 CONTINUE
327: TEMP = DNRM2( K, WORK, 1 )
328: WORK( IWK2I+J ) = DDOT( K, WORK, 1, VF, 1 ) / TEMP
329: WORK( IWK3I+J ) = DDOT( K, WORK, 1, VL, 1 ) / TEMP
330: IF( ICOMPQ.EQ.1 ) THEN
331: DIFR( J, 2 ) = TEMP
332: END IF
333: 80 CONTINUE
334: *
335: CALL DCOPY( K, WORK( IWK2 ), 1, VF, 1 )
336: CALL DCOPY( K, WORK( IWK3 ), 1, VL, 1 )
337: *
338: RETURN
339: *
340: * End of DLASD8
341: *
342: END
343:
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