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
|
SUBROUTINE DPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO )
*
* -- LAPACK routine (version 3.2) --
* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
* -- Jason Riedy of Univ. of California Berkeley. --
* -- November 2008 --
*
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley and NAG Ltd. --
*
IMPLICIT NONE
* ..
* .. Scalar Arguments ..
INTEGER INFO, LDA, N
DOUBLE PRECISION AMAX, SCOND
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), S( * )
* ..
*
* Purpose
* =======
*
* DPOEQU computes row and column scalings intended to equilibrate a
* symmetric positive definite matrix A and reduce its condition number
* (with respect to the two-norm). S contains the scale factors,
* S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
* elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
* choice of S puts the condition number of B within a factor N of the
* smallest possible condition number over all possible diagonal
* scalings.
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input) DOUBLE PRECISION array, dimension (LDA,N)
* The N-by-N symmetric positive definite matrix whose scaling
* factors are to be computed. Only the diagonal elements of A
* are referenced.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* S (output) DOUBLE PRECISION array, dimension (N)
* If INFO = 0, S contains the scale factors for A.
*
* SCOND (output) DOUBLE PRECISION
* If INFO = 0, S contains the ratio of the smallest S(i) to
* the largest S(i). If SCOND >= 0.1 and AMAX is neither too
* large nor too small, it is not worth scaling by S.
*
* AMAX (output) DOUBLE PRECISION
* Absolute value of largest matrix element. If AMAX is very
* close to overflow or very close to underflow, the matrix
* should be scaled.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, the i-th diagonal element is nonpositive.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
DOUBLE PRECISION SMIN, BASE, TMP
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT, LOG, INT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
* Positive definite only performs 1 pass of equilibration.
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -3
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPOEQUB', -INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( N.EQ.0 ) THEN
SCOND = ONE
AMAX = ZERO
RETURN
END IF
BASE = DLAMCH( 'B' )
TMP = -0.5D+0 / LOG ( BASE )
*
* Find the minimum and maximum diagonal elements.
*
S( 1 ) = A( 1, 1 )
SMIN = S( 1 )
AMAX = S( 1 )
DO 10 I = 2, N
S( I ) = A( I, I )
SMIN = MIN( SMIN, S( I ) )
AMAX = MAX( AMAX, S( I ) )
10 CONTINUE
*
IF( SMIN.LE.ZERO ) THEN
*
* Find the first non-positive diagonal element and return.
*
DO 20 I = 1, N
IF( S( I ).LE.ZERO ) THEN
INFO = I
RETURN
END IF
20 CONTINUE
ELSE
*
* Set the scale factors to the reciprocals
* of the diagonal elements.
*
DO 30 I = 1, N
S( I ) = BASE ** INT( TMP * LOG( S( I ) ) )
30 CONTINUE
*
* Compute SCOND = min(S(I)) / max(S(I)).
*
SCOND = SQRT( SMIN ) / SQRT( AMAX )
END IF
*
RETURN
*
* End of DPOEQUB
*
END
|