LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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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*
8*> \htmlonly
9*> Download DPOEQUB + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpoequb.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpoequb.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpoequb.f">
15*> [TXT]</a>
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