LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
dpoequb.f
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1 *> \brief \b DPOEQUB
2 *
3 * =========== DOCUMENTATION ===========
4 *
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16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DPOEQUB( 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 dpoequb( 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, BASE, TMP
137 * ..
138 * .. External Functions ..
139  DOUBLE PRECISION DLAMCH
140  EXTERNAL dlamch
141 * ..
142 * .. External Subroutines ..
143  EXTERNAL xerbla
144 * ..
145 * .. Intrinsic Functions ..
146  INTRINSIC max, min, sqrt, log, int
147 * ..
148 * .. Executable Statements ..
149 *
150 * Test the input parameters.
151 *
152 * Positive definite only performs 1 pass of equilibration.
153 *
154  info = 0
155  IF( n.LT.0 ) THEN
156  info = -1
157  ELSE IF( lda.LT.max( 1, n ) ) THEN
158  info = -3
159  END IF
160  IF( info.NE.0 ) THEN
161  CALL xerbla( 'DPOEQUB', -info )
162  RETURN
163  END IF
164 *
165 * Quick return if possible.
166 *
167  IF( n.EQ.0 ) THEN
168  scond = one
169  amax = zero
170  RETURN
171  END IF
172
173  base = dlamch( 'B' )
174  tmp = -0.5d+0 / log( base )
175 *
176 * Find the minimum and maximum diagonal elements.
177 *
178  s( 1 ) = a( 1, 1 )
179  smin = s( 1 )
180  amax = s( 1 )
181  DO 10 i = 2, n
182  s( i ) = a( i, i )
183  smin = min( smin, s( i ) )
184  amax = max( amax, s( i ) )
185  10 CONTINUE
186 *
187  IF( smin.LE.zero ) THEN
188 *
189 * Find the first non-positive diagonal element and return.
190 *
191  DO 20 i = 1, n
192  IF( s( i ).LE.zero ) THEN
193  info = i
194  RETURN
195  END IF
196  20 CONTINUE
197  ELSE
198 *
199 * Set the scale factors to the reciprocals
200 * of the diagonal elements.
201 *
202  DO 30 i = 1, n
203  s( i ) = base ** int( tmp * log( s( i ) ) )
204  30 CONTINUE
205 *
206 * Compute SCOND = min(S(I)) / max(S(I)).
207 *
208  scond = sqrt( smin ) / sqrt( amax )
209  END IF
210 *
211  RETURN
212 *
213 * End of DPOEQUB
214 *
215  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine dpoequb(N, A, LDA, S, SCOND, AMAX, INFO)
DPOEQUB
Definition: dpoequb.f:114