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
|
REAL FUNCTION SQRT12( M, N, A, LDA, S, WORK, LWORK )
*
* -- LAPACK test routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
INTEGER LDA, LWORK, M, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), S( * ), WORK( LWORK )
* ..
*
* Purpose
* =======
*
* SQRT12 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))
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A.
*
* N (input) INTEGER
* The number of columns of the matrix A.
*
* A (input) REAL array, dimension (LDA,N)
* The M-by-N matrix A. Only the upper trapezoid is referenced.
*
* LDA (input) INTEGER
* The leading dimension of the array A.
*
* S (input) REAL array, dimension (min(M,N))
* The singular values of the matrix A.
*
* WORK (workspace) REAL array, dimension (LWORK)
*
* LWORK (input) INTEGER
* The length of the array WORK. LWORK >= M*N + 4*min(M,N) +
* max(M,N).
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
* ..
* .. Local Scalars ..
INTEGER I, INFO, ISCL, J, MN
REAL ANRM, BIGNUM, NRMSVL, SMLNUM
* ..
* .. External Functions ..
REAL SASUM, SLAMCH, SLANGE, SNRM2
EXTERNAL SASUM, SLAMCH, SLANGE, SNRM2
* ..
* .. External Subroutines ..
EXTERNAL SAXPY, SBDSQR, SGEBD2, SLABAD, SLASCL, SLASET,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, REAL
* ..
* .. Local Arrays ..
REAL DUMMY( 1 )
* ..
* .. Executable Statements ..
*
SQRT12 = ZERO
*
* Test that enough workspace is supplied
*
IF( LWORK.LT.M*N+4*MIN( M, N )+MAX( M, N ) ) THEN
CALL XERBLA( 'SQRT12', 7 )
RETURN
END IF
*
* Quick return if possible
*
MN = MIN( M, N )
IF( MN.LE.ZERO )
$ RETURN
*
NRMSVL = SNRM2( MN, S, 1 )
*
* Copy upper triangle of A into work
*
CALL SLASET( 'Full', M, N, ZERO, 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 = SLAMCH( 'S' ) / SLAMCH( 'P' )
BIGNUM = ONE / SMLNUM
CALL SLABAD( SMLNUM, BIGNUM )
*
* Scale work if max entry outside range [SMLNUM,BIGNUM]
*
ANRM = SLANGE( 'M', M, N, WORK, M, DUMMY )
ISCL = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
*
* Scale matrix norm up to SMLNUM
*
CALL SLASCL( '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 SLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, WORK, M, INFO )
ISCL = 1
END IF
*
IF( ANRM.NE.ZERO ) THEN
*
* Compute SVD of work
*
CALL SGEBD2( M, N, WORK, M, WORK( M*N+1 ), WORK( M*N+MN+1 ),
$ WORK( M*N+2*MN+1 ), WORK( M*N+3*MN+1 ),
$ WORK( M*N+4*MN+1 ), INFO )
CALL SBDSQR( 'Upper', MN, 0, 0, 0, WORK( M*N+1 ),
$ WORK( M*N+MN+1 ), DUMMY, MN, DUMMY, 1, DUMMY, MN,
$ WORK( M*N+2*MN+1 ), INFO )
*
IF( ISCL.EQ.1 ) THEN
IF( ANRM.GT.BIGNUM ) THEN
CALL SLASCL( 'G', 0, 0, BIGNUM, ANRM, MN, 1,
$ WORK( M*N+1 ), MN, INFO )
END IF
IF( ANRM.LT.SMLNUM ) THEN
CALL SLASCL( 'G', 0, 0, SMLNUM, ANRM, MN, 1,
$ WORK( M*N+1 ), MN, INFO )
END IF
END IF
*
ELSE
*
DO 30 I = 1, MN
WORK( M*N+I ) = ZERO
30 CONTINUE
END IF
*
* Compare s and singular values of work
*
CALL SAXPY( MN, -ONE, S, 1, WORK( M*N+1 ), 1 )
SQRT12 = SASUM( MN, WORK( M*N+1 ), 1 ) /
$ ( SLAMCH( 'Epsilon' )*REAL( MAX( M, N ) ) )
IF( NRMSVL.NE.ZERO )
$ SQRT12 = SQRT12 / NRMSVL
*
RETURN
*
* End of SQRT12
*
END
|
