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
|
*> \brief \b ZQRT12
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION ZQRT12( M, N, A, LDA, S, WORK, LWORK,
* RWORK )
*
* .. Scalar Arguments ..
* INTEGER LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION RWORK( * ), S( * )
* COMPLEX*16 A( LDA, * ), WORK( LWORK )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZQRT12 computes the singular values `svlues' of the upper trapezoid
*> of A(1:M,1:N) and returns the ratio
*>
*> || s - svlues||/(||svlues||*eps*max(M,N))
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is COMPLEX*16 array, dimension (LDA,N)
*> The M-by-N matrix A. Only the upper trapezoid is referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (min(M,N))
*> The singular values of the matrix A.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX*16 array, dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The length of the array WORK. LWORK >= M*N + 2*min(M,N) +
*> max(M,N).
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is DOUBLE PRECISION array, dimension (2*min(M,N))
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup complex16_lin
*
* =====================================================================
DOUBLE PRECISION FUNCTION ZQRT12( M, N, A, LDA, S, WORK, LWORK,
$ RWORK )
*
* -- LAPACK test routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION RWORK( * ), S( * )
COMPLEX*16 A( LDA, * ), WORK( LWORK )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
INTEGER I, INFO, ISCL, J, MN
DOUBLE PRECISION ANRM, BIGNUM, NRMSVL, SMLNUM
* ..
* .. Local Arrays ..
DOUBLE PRECISION DUMMY( 1 )
* ..
* .. External Functions ..
DOUBLE PRECISION DASUM, DLAMCH, DNRM2, ZLANGE
EXTERNAL DASUM, DLAMCH, DNRM2, ZLANGE
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DBDSQR, DLABAD, DLASCL, XERBLA, ZGEBD2,
$ ZLASCL, ZLASET
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, DCMPLX, MAX, MIN
* ..
* .. Executable Statements ..
*
ZQRT12 = ZERO
*
* Test that enough workspace is supplied
*
IF( LWORK.LT.M*N+2*MIN( M, N )+MAX( M, N ) ) THEN
CALL XERBLA( 'ZQRT12', 7 )
RETURN
END IF
*
* Quick return if possible
*
MN = MIN( M, N )
IF( MN.LE.ZERO )
$ RETURN
*
NRMSVL = DNRM2( MN, S, 1 )
*
* Copy upper triangle of A into work
*
CALL ZLASET( 'Full', M, N, DCMPLX( ZERO ), DCMPLX( ZERO ), WORK,
$ M )
DO 20 J = 1, N
DO 10 I = 1, MIN( J, M )
WORK( ( J-1 )*M+I ) = A( I, J )
10 CONTINUE
20 CONTINUE
*
* Get machine parameters
*
SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' )
BIGNUM = ONE / SMLNUM
CALL DLABAD( SMLNUM, BIGNUM )
*
* Scale work if max entry outside range [SMLNUM,BIGNUM]
*
ANRM = ZLANGE( 'M', M, N, WORK, M, DUMMY )
ISCL = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
*
* Scale matrix norm up to SMLNUM
*
CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, WORK, M, INFO )
ISCL = 1
ELSE IF( ANRM.GT.BIGNUM ) THEN
*
* Scale matrix norm down to BIGNUM
*
CALL ZLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, WORK, M, INFO )
ISCL = 1
END IF
*
IF( ANRM.NE.ZERO ) THEN
*
* Compute SVD of work
*
CALL ZGEBD2( M, N, WORK, M, RWORK( 1 ), RWORK( MN+1 ),
$ WORK( M*N+1 ), WORK( M*N+MN+1 ),
$ WORK( M*N+2*MN+1 ), INFO )
CALL DBDSQR( 'Upper', MN, 0, 0, 0, RWORK( 1 ), RWORK( MN+1 ),
$ DUMMY, MN, DUMMY, 1, DUMMY, MN, RWORK( 2*MN+1 ),
$ INFO )
*
IF( ISCL.EQ.1 ) THEN
IF( ANRM.GT.BIGNUM ) THEN
CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MN, 1, RWORK( 1 ),
$ MN, INFO )
END IF
IF( ANRM.LT.SMLNUM ) THEN
CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MN, 1, RWORK( 1 ),
$ MN, INFO )
END IF
END IF
*
ELSE
*
DO 30 I = 1, MN
RWORK( I ) = ZERO
30 CONTINUE
END IF
*
* Compare s and singular values of work
*
CALL DAXPY( MN, -ONE, S, 1, RWORK( 1 ), 1 )
ZQRT12 = DASUM( MN, RWORK( 1 ), 1 ) /
$ ( DLAMCH( 'Epsilon' )*DBLE( MAX( M, N ) ) )
IF( NRMSVL.NE.ZERO )
$ ZQRT12 = ZQRT12 / NRMSVL
*
RETURN
*
* End of ZQRT12
*
END
|