1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
|
*> \brief \b SLAED9 used by sstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SLAED9 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slaed9.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slaed9.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slaed9.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SLAED9( K, KSTART, KSTOP, N, D, Q, LDQ, RHO, DLAMDA, W,
* S, LDS, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, KSTART, KSTOP, LDQ, LDS, N
* REAL RHO
* ..
* .. Array Arguments ..
* REAL D( * ), DLAMDA( * ), Q( LDQ, * ), S( LDS, * ),
* $ W( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SLAED9 finds the roots of the secular equation, as defined by the
*> values in D, Z, and RHO, between KSTART and KSTOP. It makes the
*> appropriate calls to SLAED4 and then stores the new matrix of
*> eigenvectors for use in calculating the next level of Z vectors.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of terms in the rational function to be solved by
*> SLAED4. K >= 0.
*> \endverbatim
*>
*> \param[in] KSTART
*> \verbatim
*> KSTART is INTEGER
*> \endverbatim
*>
*> \param[in] KSTOP
*> \verbatim
*> KSTOP is INTEGER
*> The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP
*> are to be computed. 1 <= KSTART <= KSTOP <= K.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows and columns in the Q matrix.
*> N >= K (delation may result in N > K).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is REAL array, dimension (N)
*> D(I) contains the updated eigenvalues
*> for KSTART <= I <= KSTOP.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is REAL array, dimension (LDQ,N)
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max( 1, N ).
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is REAL
*> The value of the parameter in the rank one update equation.
*> RHO >= 0 required.
*> \endverbatim
*>
*> \param[in] DLAMDA
*> \verbatim
*> DLAMDA is REAL array, dimension (K)
*> The first K elements of this array contain the old roots
*> of the deflated updating problem. These are the poles
*> of the secular equation.
*> \endverbatim
*>
*> \param[in] W
*> \verbatim
*> W is REAL array, dimension (K)
*> The first K elements of this array contain the components
*> of the deflation-adjusted updating vector.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is REAL array, dimension (LDS, K)
*> Will contain the eigenvectors of the repaired matrix which
*> will be stored for subsequent Z vector calculation and
*> multiplied by the previously accumulated eigenvectors
*> to update the system.
*> \endverbatim
*>
*> \param[in] LDS
*> \verbatim
*> LDS is INTEGER
*> The leading dimension of S. LDS >= max( 1, K ).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = 1, an eigenvalue did not converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA
*
* =====================================================================
SUBROUTINE SLAED9( K, KSTART, KSTOP, N, D, Q, LDQ, RHO, DLAMDA, W,
$ S, LDS, INFO )
*
* -- LAPACK computational routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
INTEGER INFO, K, KSTART, KSTOP, LDQ, LDS, N
REAL RHO
* ..
* .. Array Arguments ..
REAL D( * ), DLAMDA( * ), Q( LDQ, * ), S( LDS, * ),
$ W( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, J
REAL TEMP
* ..
* .. External Functions ..
REAL SLAMC3, SNRM2
EXTERNAL SLAMC3, SNRM2
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SLAED4, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( K.LT.0 ) THEN
INFO = -1
ELSE IF( KSTART.LT.1 .OR. KSTART.GT.MAX( 1, K ) ) THEN
INFO = -2
ELSE IF( MAX( 1, KSTOP ).LT.KSTART .OR. KSTOP.GT.MAX( 1, K ) )
$ THEN
INFO = -3
ELSE IF( N.LT.K ) THEN
INFO = -4
ELSE IF( LDQ.LT.MAX( 1, K ) ) THEN
INFO = -7
ELSE IF( LDS.LT.MAX( 1, K ) ) THEN
INFO = -12
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SLAED9', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( K.EQ.0 )
$ RETURN
*
* Modify values DLAMDA(i) to make sure all DLAMDA(i)-DLAMDA(j) can
* be computed with high relative accuracy (barring over/underflow).
* This is a problem on machines without a guard digit in
* add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
* The following code replaces DLAMDA(I) by 2*DLAMDA(I)-DLAMDA(I),
* which on any of these machines zeros out the bottommost
* bit of DLAMDA(I) if it is 1; this makes the subsequent
* subtractions DLAMDA(I)-DLAMDA(J) unproblematic when cancellation
* occurs. On binary machines with a guard digit (almost all
* machines) it does not change DLAMDA(I) at all. On hexadecimal
* and decimal machines with a guard digit, it slightly
* changes the bottommost bits of DLAMDA(I). It does not account
* for hexadecimal or decimal machines without guard digits
* (we know of none). We use a subroutine call to compute
* 2*DLAMBDA(I) to prevent optimizing compilers from eliminating
* this code.
*
DO 10 I = 1, N
DLAMDA( I ) = SLAMC3( DLAMDA( I ), DLAMDA( I ) ) - DLAMDA( I )
10 CONTINUE
*
DO 20 J = KSTART, KSTOP
CALL SLAED4( K, J, DLAMDA, W, Q( 1, J ), RHO, D( J ), INFO )
*
* If the zero finder fails, the computation is terminated.
*
IF( INFO.NE.0 )
$ GO TO 120
20 CONTINUE
*
IF( K.EQ.1 .OR. K.EQ.2 ) THEN
DO 40 I = 1, K
DO 30 J = 1, K
S( J, I ) = Q( J, I )
30 CONTINUE
40 CONTINUE
GO TO 120
END IF
*
* Compute updated W.
*
CALL SCOPY( K, W, 1, S, 1 )
*
* Initialize W(I) = Q(I,I)
*
CALL SCOPY( K, Q, LDQ+1, W, 1 )
DO 70 J = 1, K
DO 50 I = 1, J - 1
W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) )
50 CONTINUE
DO 60 I = J + 1, K
W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) )
60 CONTINUE
70 CONTINUE
DO 80 I = 1, K
W( I ) = SIGN( SQRT( -W( I ) ), S( I, 1 ) )
80 CONTINUE
*
* Compute eigenvectors of the modified rank-1 modification.
*
DO 110 J = 1, K
DO 90 I = 1, K
Q( I, J ) = W( I ) / Q( I, J )
90 CONTINUE
TEMP = SNRM2( K, Q( 1, J ), 1 )
DO 100 I = 1, K
S( I, J ) = Q( I, J ) / TEMP
100 CONTINUE
110 CONTINUE
*
120 CONTINUE
RETURN
*
* End of SLAED9
*
END
|
