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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
9 *> Download DPOEQU + dependencies
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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 *> \date November 2011
109 *
110 *> \ingroup doublePOcomputational
111 *
112 * =====================================================================
113  SUBROUTINE dpoequ( N, A, LDA, S, SCOND, AMAX, INFO )
114 *
115 * -- LAPACK computational routine (version 3.4.0) --
116 * -- LAPACK is a software package provided by Univ. of Tennessee, --
117 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
118 * November 2011
119 *
120 * .. Scalar Arguments ..
121  INTEGER info, lda, n
122  DOUBLE PRECISION amax, scond
123 * ..
124 * .. Array Arguments ..
125  DOUBLE PRECISION a( lda, * ), s( * )
126 * ..
127 *
128 * =====================================================================
129 *
130 * .. Parameters ..
131  DOUBLE PRECISION zero, one
132  parameter( zero = 0.0d+0, one = 1.0d+0 )
133 * ..
134 * .. Local Scalars ..
135  INTEGER i
136  DOUBLE PRECISION smin
137 * ..
138 * .. External Subroutines ..
139  EXTERNAL xerbla
140 * ..
141 * .. Intrinsic Functions ..
142  INTRINSIC max, min, sqrt
143 * ..
144 * .. Executable Statements ..
145 *
146 * Test the input parameters.
147 *
148  info = 0
149  IF( n.LT.0 ) THEN
150  info = -1
151  ELSE IF( lda.LT.max( 1, n ) ) THEN
152  info = -3
153  END IF
154  IF( info.NE.0 ) THEN
155  CALL xerbla( 'DPOEQU', -info )
156  RETURN
157  END IF
158 *
159 * Quick return if possible
160 *
161  IF( n.EQ.0 ) THEN
162  scond = one
163  amax = zero
164  RETURN
165  END IF
166 *
167 * Find the minimum and maximum diagonal elements.
168 *
169  s( 1 ) = a( 1, 1 )
170  smin = s( 1 )
171  amax = s( 1 )
172  DO 10 i = 2, n
173  s( i ) = a( i, i )
174  smin = min( smin, s( i ) )
175  amax = max( amax, s( i ) )
176  10 CONTINUE
177 *
178  IF( smin.LE.zero ) THEN
179 *
180 * Find the first non-positive diagonal element and return.
181 *
182  DO 20 i = 1, n
183  IF( s( i ).LE.zero ) THEN
184  info = i
185  RETURN
186  END IF
187  20 CONTINUE
188  ELSE
189 *
190 * Set the scale factors to the reciprocals
191 * of the diagonal elements.
192 *
193  DO 30 i = 1, n
194  s( i ) = one / sqrt( s( i ) )
195  30 CONTINUE
196 *
197 * Compute SCOND = min(S(I)) / max(S(I))
198 *
199  scond = sqrt( smin ) / sqrt( amax )
200  END IF
201  RETURN
202 *
203 * End of DPOEQU
204 *
205  END