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
|
SUBROUTINE DLASD0( N, SQRE, D, E, U, LDU, VT, LDVT, SMLSIZ, IWORK,
$ WORK, INFO )
*
* -- LAPACK auxiliary routine (instrumented to count ops, version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* June 30, 1999
*
* .. Scalar Arguments ..
INTEGER INFO, LDU, LDVT, N, SMLSIZ, SQRE
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION D( * ), E( * ), U( LDU, * ), VT( LDVT, * ),
$ WORK( * )
* ..
* .. Common block to return operation count ..
COMMON / LATIME / OPS, ITCNT
* ..
* .. Scalars in Common ..
DOUBLE PRECISION ITCNT, OPS
* ..
*
* Purpose
* =======
*
* Using a divide and conquer approach, DLASD0 computes the singular
* value decomposition (SVD) of a real upper bidiagonal N-by-M
* matrix B with diagonal D and offdiagonal E, where M = N + SQRE.
* The algorithm computes orthogonal matrices U and VT such that
* B = U * S * VT. The singular values S are overwritten on D.
*
* A related subroutine, DLASDA, computes only the singular values,
* and optionally, the singular vectors in compact form.
*
* Arguments
* =========
*
* N (input) INTEGER
* On entry, the row dimension of the upper bidiagonal matrix.
* This is also the dimension of the main diagonal array D.
*
* SQRE (input) INTEGER
* Specifies the column dimension of the bidiagonal matrix.
* = 0: The bidiagonal matrix has column dimension M = N;
* = 1: The bidiagonal matrix has column dimension M = N+1;
*
* D (input/output) DOUBLE PRECISION array, dimension (N)
* On entry D contains the main diagonal of the bidiagonal
* matrix.
* On exit D, if INFO = 0, contains its singular values.
*
* E (input) DOUBLE PRECISION array, dimension (M-1)
* Contains the subdiagonal entries of the bidiagonal matrix.
* On exit, E has been destroyed.
*
* U (output) DOUBLE PRECISION array, dimension at least (LDQ, N)
* On exit, U contains the left singular vectors.
*
* LDU (input) INTEGER
* On entry, leading dimension of U.
*
* VT (output) DOUBLE PRECISION array, dimension at least (LDVT, M)
* On exit, VT' contains the right singular vectors.
*
* LDVT (input) INTEGER
* On entry, leading dimension of VT.
*
* SMLSIZ (input) INTEGER
* On entry, maximum size of the subproblems at the
* bottom of the computation tree.
*
* IWORK INTEGER work array.
* Dimension must be at least (8 * N)
*
* WORK DOUBLE PRECISION work array.
* Dimension must be at least (3 * M**2 + 2 * M)
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: if INFO = 1, an singular value did not converge
*
* Further Details
* ===============
*
* Based on contributions by
* Ming Gu and Huan Ren, Computer Science Division, University of
* California at Berkeley, USA
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, I1, IC, IDXQ, IDXQC, IM1, INODE, ITEMP, IWK,
$ J, LF, LL, LVL, M, NCC, ND, NDB1, NDIML, NDIMR,
$ NL, NLF, NLP1, NLVL, NR, NRF, NRP1, SQREI
DOUBLE PRECISION ALPHA, BETA
* ..
* .. External Subroutines ..
EXTERNAL DLASD1, DLASDQ, DLASDT, XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
INFO = -2
END IF
*
M = N + SQRE
*
IF( LDU.LT.N ) THEN
INFO = -6
ELSE IF( LDVT.LT.M ) THEN
INFO = -8
ELSE IF( SMLSIZ.LT.3 ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASD0', -INFO )
RETURN
END IF
*
* If the input matrix is too small, call DLASDQ to find the SVD.
*
IF( N.LE.SMLSIZ ) THEN
CALL DLASDQ( 'U', SQRE, N, M, N, 0, D, E, VT, LDVT, U, LDU, U,
$ LDU, WORK, INFO )
RETURN
END IF
*
* Set up the computation tree.
*
INODE = 1
NDIML = INODE + N
NDIMR = NDIML + N
IDXQ = NDIMR + N
IWK = IDXQ + N
CALL DLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ),
$ IWORK( NDIMR ), SMLSIZ )
*
* For the nodes on bottom level of the tree, solve
* their subproblems by DLASDQ.
*
NDB1 = ( ND+1 ) / 2
NCC = 0
DO 30 I = NDB1, ND
*
* IC : center row of each node
* NL : number of rows of left subproblem
* NR : number of rows of right subproblem
* NLF: starting row of the left subproblem
* NRF: starting row of the right subproblem
*
I1 = I - 1
IC = IWORK( INODE+I1 )
NL = IWORK( NDIML+I1 )
NLP1 = NL + 1
NR = IWORK( NDIMR+I1 )
NRP1 = NR + 1
NLF = IC - NL
NRF = IC + 1
SQREI = 1
CALL DLASDQ( 'U', SQREI, NL, NLP1, NL, NCC, D( NLF ), E( NLF ),
$ VT( NLF, NLF ), LDVT, U( NLF, NLF ), LDU,
$ U( NLF, NLF ), LDU, WORK, INFO )
IF( INFO.NE.0 ) THEN
RETURN
END IF
ITEMP = IDXQ + NLF - 2
DO 10 J = 1, NL
IWORK( ITEMP+J ) = J
10 CONTINUE
IF( I.EQ.ND ) THEN
SQREI = SQRE
ELSE
SQREI = 1
END IF
NRP1 = NR + SQREI
CALL DLASDQ( 'U', SQREI, NR, NRP1, NR, NCC, D( NRF ), E( NRF ),
$ VT( NRF, NRF ), LDVT, U( NRF, NRF ), LDU,
$ U( NRF, NRF ), LDU, WORK, INFO )
IF( INFO.NE.0 ) THEN
RETURN
END IF
ITEMP = IDXQ + IC
DO 20 J = 1, NR
IWORK( ITEMP+J-1 ) = J
20 CONTINUE
30 CONTINUE
*
* Now conquer each subproblem bottom-up.
*
DO 50 LVL = NLVL, 1, -1
*
* Find the first node LF and last node LL on the
* current level LVL.
*
IF( LVL.EQ.1 ) THEN
LF = 1
LL = 1
ELSE
LF = 2**( LVL-1 )
LL = 2*LF - 1
END IF
DO 40 I = LF, LL
IM1 = I - 1
IC = IWORK( INODE+IM1 )
NL = IWORK( NDIML+IM1 )
NR = IWORK( NDIMR+IM1 )
NLF = IC - NL
IF( ( SQRE.EQ.0 ) .AND. ( I.EQ.LL ) ) THEN
SQREI = SQRE
ELSE
SQREI = 1
END IF
IDXQC = IDXQ + NLF - 1
ALPHA = D( IC )
BETA = E( IC )
CALL DLASD1( NL, NR, SQREI, D( NLF ), ALPHA, BETA,
$ U( NLF, NLF ), LDU, VT( NLF, NLF ), LDVT,
$ IWORK( IDXQC ), IWORK( IWK ), WORK, INFO )
IF( INFO.NE.0 ) THEN
RETURN
END IF
40 CONTINUE
50 CONTINUE
*
RETURN
*
* End of DLASD0
*
END
|
