LAPACK  3.10.1
LAPACK: Linear Algebra PACKage
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
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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