LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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cpoequ.f
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1*> \brief \b CPOEQU
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download CPOEQU + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cpoequ.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cpoequ.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cpoequ.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE CPOEQU( N, A, LDA, S, SCOND, AMAX, INFO )
22*
23* .. Scalar Arguments ..
24* INTEGER INFO, LDA, N
25* REAL AMAX, SCOND
26* ..
27* .. Array Arguments ..
28* REAL S( * )
29* COMPLEX A( LDA, * )
30* ..
31*
32*
33*> \par Purpose:
34* =============
35*>
36*> \verbatim
37*>
38*> CPOEQU computes row and column scalings intended to equilibrate a
39*> Hermitian positive definite matrix A and reduce its condition number
40*> (with respect to the two-norm). S contains the scale factors,
41*> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
42*> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
43*> choice of S puts the condition number of B within a factor N of the
44*> smallest possible condition number over all possible diagonal
45*> scalings.
46*> \endverbatim
47*
48* Arguments:
49* ==========
50*
51*> \param[in] N
52*> \verbatim
53*> N is INTEGER
54*> The order of the matrix A. N >= 0.
55*> \endverbatim
56*>
57*> \param[in] A
58*> \verbatim
59*> A is COMPLEX array, dimension (LDA,N)
60*> The N-by-N Hermitian positive definite matrix whose scaling
61*> factors are to be computed. Only the diagonal elements of A
62*> are referenced.
63*> \endverbatim
64*>
65*> \param[in] LDA
66*> \verbatim
67*> LDA is INTEGER
68*> The leading dimension of the array A. LDA >= max(1,N).
69*> \endverbatim
70*>
71*> \param[out] S
72*> \verbatim
73*> S is REAL array, dimension (N)
74*> If INFO = 0, S contains the scale factors for A.
75*> \endverbatim
76*>
77*> \param[out] SCOND
78*> \verbatim
79*> SCOND is REAL
80*> If INFO = 0, S contains the ratio of the smallest S(i) to
81*> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
82*> large nor too small, it is not worth scaling by S.
83*> \endverbatim
84*>
85*> \param[out] AMAX
86*> \verbatim
87*> AMAX is REAL
88*> Absolute value of largest matrix element. If AMAX is very
89*> close to overflow or very close to underflow, the matrix
90*> should be scaled.
91*> \endverbatim
92*>
93*> \param[out] INFO
94*> \verbatim
95*> INFO is INTEGER
96*> = 0: successful exit
97*> < 0: if INFO = -i, the i-th argument had an illegal value
98*> > 0: if INFO = i, the i-th diagonal element is nonpositive.
99*> \endverbatim
100*
101* Authors:
102* ========
103*
104*> \author Univ. of Tennessee
105*> \author Univ. of California Berkeley
106*> \author Univ. of Colorado Denver
107*> \author NAG Ltd.
108*
109*> \ingroup complexPOcomputational
110*
111* =====================================================================
112 SUBROUTINE cpoequ( N, A, LDA, S, SCOND, AMAX, INFO )
113*
114* -- LAPACK computational routine --
115* -- LAPACK is a software package provided by Univ. of Tennessee, --
116* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
117*
118* .. Scalar Arguments ..
119 INTEGER INFO, LDA, N
120 REAL AMAX, SCOND
121* ..
122* .. Array Arguments ..
123 REAL S( * )
124 COMPLEX A( LDA, * )
125* ..
126*
127* =====================================================================
128*
129* .. Parameters ..
130 REAL ZERO, ONE
131 parameter( zero = 0.0e+0, one = 1.0e+0 )
132* ..
133* .. Local Scalars ..
134 INTEGER I
135 REAL SMIN
136* ..
137* .. External Subroutines ..
138 EXTERNAL xerbla
139* ..
140* .. Intrinsic Functions ..
141 INTRINSIC max, min, real, sqrt
142* ..
143* .. Executable Statements ..
144*
145* Test the input parameters.
146*
147 info = 0
148 IF( n.LT.0 ) THEN
149 info = -1
150 ELSE IF( lda.LT.max( 1, n ) ) THEN
151 info = -3
152 END IF
153 IF( info.NE.0 ) THEN
154 CALL xerbla( 'CPOEQU', -info )
155 RETURN
156 END IF
157*
158* Quick return if possible
159*
160 IF( n.EQ.0 ) THEN
161 scond = one
162 amax = zero
163 RETURN
164 END IF
165*
166* Find the minimum and maximum diagonal elements.
167*
168 s( 1 ) = real( a( 1, 1 ) )
169 smin = s( 1 )
170 amax = s( 1 )
171 DO 10 i = 2, n
172 s( i ) = real( a( i, i ) )
173 smin = min( smin, s( i ) )
174 amax = max( amax, s( i ) )
175 10 CONTINUE
176*
177 IF( smin.LE.zero ) THEN
178*
179* Find the first non-positive diagonal element and return.
180*
181 DO 20 i = 1, n
182 IF( s( i ).LE.zero ) THEN
183 info = i
184 RETURN
185 END IF
186 20 CONTINUE
187 ELSE
188*
189* Set the scale factors to the reciprocals
190* of the diagonal elements.
191*
192 DO 30 i = 1, n
193 s( i ) = one / sqrt( s( i ) )
194 30 CONTINUE
195*
196* Compute SCOND = min(S(I)) / max(S(I))
197*
198 scond = sqrt( smin ) / sqrt( amax )
199 END IF
200 RETURN
201*
202* End of CPOEQU
203*
204 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine cpoequ(N, A, LDA, S, SCOND, AMAX, INFO)
CPOEQU
Definition: cpoequ.f:113