LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
dpoequb.f
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1 *> \brief \b DPOEQUB
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
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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 *> DPOEQUB 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 *>
46 *> This routine differs from DPOEQU by restricting the scaling factors
47 *> to a power of the radix. Barring over- and underflow, scaling by
48 *> these factors introduces no additional rounding errors. However, the
49 *> scaled diagonal entries are no longer approximately 1 but lie
50 *> between sqrt(radix) and 1/sqrt(radix).
51 *> \endverbatim
52 *
53 * Arguments:
54 * ==========
55 *
56 *> \param[in] N
57 *> \verbatim
58 *> N is INTEGER
59 *> The order of the matrix A. N >= 0.
60 *> \endverbatim
61 *>
62 *> \param[in] A
63 *> \verbatim
64 *> A is DOUBLE PRECISION array, dimension (LDA,N)
65 *> The N-by-N symmetric positive definite matrix whose scaling
66 *> factors are to be computed. Only the diagonal elements of A
67 *> are referenced.
68 *> \endverbatim
69 *>
70 *> \param[in] LDA
71 *> \verbatim
72 *> LDA is INTEGER
73 *> The leading dimension of the array A. LDA >= max(1,N).
74 *> \endverbatim
75 *>
76 *> \param[out] S
77 *> \verbatim
78 *> S is DOUBLE PRECISION array, dimension (N)
79 *> If INFO = 0, S contains the scale factors for A.
80 *> \endverbatim
81 *>
82 *> \param[out] SCOND
83 *> \verbatim
84 *> SCOND is DOUBLE PRECISION
85 *> If INFO = 0, S contains the ratio of the smallest S(i) to
86 *> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
87 *> large nor too small, it is not worth scaling by S.
88 *> \endverbatim
89 *>
90 *> \param[out] AMAX
91 *> \verbatim
92 *> AMAX is DOUBLE PRECISION
93 *> Absolute value of largest matrix element. If AMAX is very
94 *> close to overflow or very close to underflow, the matrix
95 *> should be scaled.
96 *> \endverbatim
97 *>
98 *> \param[out] INFO
99 *> \verbatim
100 *> INFO is INTEGER
101 *> = 0: successful exit
102 *> < 0: if INFO = -i, the i-th argument had an illegal value
103 *> > 0: if INFO = i, the i-th diagonal element is nonpositive.
104 *> \endverbatim
105 *
106 * Authors:
107 * ========
108 *
109 *> \author Univ. of Tennessee
110 *> \author Univ. of California Berkeley
111 *> \author Univ. of Colorado Denver
112 *> \author NAG Ltd.
113 *
114 *> \ingroup doublePOcomputational
115 *
116 * =====================================================================
117  SUBROUTINE dpoequb( N, A, LDA, S, SCOND, AMAX, INFO )
118 *
119 * -- LAPACK computational routine --
120 * -- LAPACK is a software package provided by Univ. of Tennessee, --
121 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
122 *
123 * .. Scalar Arguments ..
124  INTEGER INFO, LDA, N
125  DOUBLE PRECISION AMAX, SCOND
126 * ..
127 * .. Array Arguments ..
128  DOUBLE PRECISION A( LDA, * ), S( * )
129 * ..
130 *
131 * =====================================================================
132 *
133 * .. Parameters ..
134  DOUBLE PRECISION ZERO, ONE
135  parameter( zero = 0.0d+0, one = 1.0d+0 )
136 * ..
137 * .. Local Scalars ..
138  INTEGER I
139  DOUBLE PRECISION SMIN, BASE, TMP
140 * ..
141 * .. External Functions ..
142  DOUBLE PRECISION DLAMCH
143  EXTERNAL dlamch
144 * ..
145 * .. External Subroutines ..
146  EXTERNAL xerbla
147 * ..
148 * .. Intrinsic Functions ..
149  INTRINSIC max, min, sqrt, log, int
150 * ..
151 * .. Executable Statements ..
152 *
153 * Test the input parameters.
154 *
155 * Positive definite only performs 1 pass of equilibration.
156 *
157  info = 0
158  IF( n.LT.0 ) THEN
159  info = -1
160  ELSE IF( lda.LT.max( 1, n ) ) THEN
161  info = -3
162  END IF
163  IF( info.NE.0 ) THEN
164  CALL xerbla( 'DPOEQUB', -info )
165  RETURN
166  END IF
167 *
168 * Quick return if possible.
169 *
170  IF( n.EQ.0 ) THEN
171  scond = one
172  amax = zero
173  RETURN
174  END IF
175 
176  base = dlamch( 'B' )
177  tmp = -0.5d+0 / log( base )
178 *
179 * Find the minimum and maximum diagonal elements.
180 *
181  s( 1 ) = a( 1, 1 )
182  smin = s( 1 )
183  amax = s( 1 )
184  DO 10 i = 2, n
185  s( i ) = a( i, i )
186  smin = min( smin, s( i ) )
187  amax = max( amax, s( i ) )
188  10 CONTINUE
189 *
190  IF( smin.LE.zero ) THEN
191 *
192 * Find the first non-positive diagonal element and return.
193 *
194  DO 20 i = 1, n
195  IF( s( i ).LE.zero ) THEN
196  info = i
197  RETURN
198  END IF
199  20 CONTINUE
200  ELSE
201 *
202 * Set the scale factors to the reciprocals
203 * of the diagonal elements.
204 *
205  DO 30 i = 1, n
206  s( i ) = base ** int( tmp * log( s( i ) ) )
207  30 CONTINUE
208 *
209 * Compute SCOND = min(S(I)) / max(S(I)).
210 *
211  scond = sqrt( smin ) / sqrt( amax )
212  END IF
213 *
214  RETURN
215 *
216 * End of DPOEQUB
217 *
218  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dpoequb(N, A, LDA, S, SCOND, AMAX, INFO)
DPOEQUB
Definition: dpoequb.f:118