LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
dpoequ.f
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1 *> \brief \b DPOEQU
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
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14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpoequ.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DPOEQU( N, A, LDA, S, SCOND, AMAX, INFO )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER INFO, LDA, N
25 * DOUBLE PRECISION AMAX, SCOND
26 * ..
27 * .. Array Arguments ..
28 * DOUBLE PRECISION A( LDA, * ), S( * )
29 * ..
30 *
31 *
32 *> \par Purpose:
33 * =============
34 *>
35 *> \verbatim
36 *>
37 *> DPOEQU computes row and column scalings intended to equilibrate a
38 *> symmetric positive definite matrix A and reduce its condition number
39 *> (with respect to the two-norm). S contains the scale factors,
40 *> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
41 *> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
42 *> choice of S puts the condition number of B within a factor N of the
43 *> smallest possible condition number over all possible diagonal
44 *> scalings.
45 *> \endverbatim
46 *
47 * Arguments:
48 * ==========
49 *
50 *> \param[in] N
51 *> \verbatim
52 *> N is INTEGER
53 *> The order of the matrix A. N >= 0.
54 *> \endverbatim
55 *>
56 *> \param[in] A
57 *> \verbatim
58 *> A is DOUBLE PRECISION array, dimension (LDA,N)
59 *> The N-by-N symmetric positive definite matrix whose scaling
60 *> factors are to be computed. Only the diagonal elements of A
61 *> are referenced.
62 *> \endverbatim
63 *>
64 *> \param[in] LDA
65 *> \verbatim
66 *> LDA is INTEGER
67 *> The leading dimension of the array A. LDA >= max(1,N).
68 *> \endverbatim
69 *>
70 *> \param[out] S
71 *> \verbatim
72 *> S is DOUBLE PRECISION array, dimension (N)
73 *> If INFO = 0, S contains the scale factors for A.
74 *> \endverbatim
75 *>
76 *> \param[out] SCOND
77 *> \verbatim
78 *> SCOND is DOUBLE PRECISION
79 *> If INFO = 0, S contains the ratio of the smallest S(i) to
80 *> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
81 *> large nor too small, it is not worth scaling by S.
82 *> \endverbatim
83 *>
84 *> \param[out] AMAX
85 *> \verbatim
86 *> AMAX is DOUBLE PRECISION
87 *> Absolute value of largest matrix element. If AMAX is very
88 *> close to overflow or very close to underflow, the matrix
89 *> should be scaled.
90 *> \endverbatim
91 *>
92 *> \param[out] INFO
93 *> \verbatim
94 *> INFO is INTEGER
95 *> = 0: successful exit
96 *> < 0: if INFO = -i, the i-th argument had an illegal value
97 *> > 0: if INFO = i, the i-th diagonal element is nonpositive.
98 *> \endverbatim
99 *
100 * Authors:
101 * ========
102 *
103 *> \author Univ. of Tennessee
104 *> \author Univ. of California Berkeley
105 *> \author Univ. of Colorado Denver
106 *> \author NAG Ltd.
107 *
108 *> \ingroup doublePOcomputational
109 *
110 * =====================================================================
111  SUBROUTINE dpoequ( N, A, LDA, S, SCOND, AMAX, INFO )
112 *
113 * -- LAPACK computational routine --
114 * -- LAPACK is a software package provided by Univ. of Tennessee, --
115 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
116 *
117 * .. Scalar Arguments ..
118  INTEGER INFO, LDA, N
119  DOUBLE PRECISION AMAX, SCOND
120 * ..
121 * .. Array Arguments ..
122  DOUBLE PRECISION A( LDA, * ), S( * )
123 * ..
124 *
125 * =====================================================================
126 *
127 * .. Parameters ..
128  DOUBLE PRECISION ZERO, ONE
129  parameter( zero = 0.0d+0, one = 1.0d+0 )
130 * ..
131 * .. Local Scalars ..
132  INTEGER I
133  DOUBLE PRECISION SMIN
134 * ..
135 * .. External Subroutines ..
136  EXTERNAL xerbla
137 * ..
138 * .. Intrinsic Functions ..
139  INTRINSIC max, min, sqrt
140 * ..
141 * .. Executable Statements ..
142 *
143 * Test the input parameters.
144 *
145  info = 0
146  IF( n.LT.0 ) THEN
147  info = -1
148  ELSE IF( lda.LT.max( 1, n ) ) THEN
149  info = -3
150  END IF
151  IF( info.NE.0 ) THEN
152  CALL xerbla( 'DPOEQU', -info )
153  RETURN
154  END IF
155 *
156 * Quick return if possible
157 *
158  IF( n.EQ.0 ) THEN
159  scond = one
160  amax = zero
161  RETURN
162  END IF
163 *
164 * Find the minimum and maximum diagonal elements.
165 *
166  s( 1 ) = a( 1, 1 )
167  smin = s( 1 )
168  amax = s( 1 )
169  DO 10 i = 2, n
170  s( i ) = a( i, i )
171  smin = min( smin, s( i ) )
172  amax = max( amax, s( i ) )
173  10 CONTINUE
174 *
175  IF( smin.LE.zero ) THEN
176 *
177 * Find the first non-positive diagonal element and return.
178 *
179  DO 20 i = 1, n
180  IF( s( i ).LE.zero ) THEN
181  info = i
182  RETURN
183  END IF
184  20 CONTINUE
185  ELSE
186 *
187 * Set the scale factors to the reciprocals
188 * of the diagonal elements.
189 *
190  DO 30 i = 1, n
191  s( i ) = one / sqrt( s( i ) )
192  30 CONTINUE
193 *
194 * Compute SCOND = min(S(I)) / max(S(I))
195 *
196  scond = sqrt( smin ) / sqrt( amax )
197  END IF
198  RETURN
199 *
200 * End of DPOEQU
201 *
202  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dpoequ(N, A, LDA, S, SCOND, AMAX, INFO)
DPOEQU
Definition: dpoequ.f:112