LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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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
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpoequ.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpoequ.f">
13*> [ZIP]</a>
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