LAPACK  3.8.0 LAPACK: Linear Algebra PACKage
cpoequb.f
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1 *> \brief \b CPOEQUB
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CPOEQUB( 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 * COMPLEX A( LDA, * )
29 * REAL S( * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> CPOEQUB 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 *>
47 *> This routine differs from CPOEQU by restricting the scaling factors
48 *> to a power of the radix. Barring over- and underflow, scaling by
49 *> these factors introduces no additional rounding errors. However, the
50 *> scaled diagonal entries are no longer approximately 1 but lie
52 *> \endverbatim
53 *
54 * Arguments:
55 * ==========
56 *
57 *> \param[in] N
58 *> \verbatim
59 *> N is INTEGER
60 *> The order of the matrix A. N >= 0.
61 *> \endverbatim
62 *>
63 *> \param[in] A
64 *> \verbatim
65 *> A is COMPLEX array, dimension (LDA,N)
66 *> The N-by-N Hermitian positive definite matrix whose scaling
67 *> factors are to be computed. Only the diagonal elements of A
68 *> are referenced.
69 *> \endverbatim
70 *>
71 *> \param[in] LDA
72 *> \verbatim
73 *> LDA is INTEGER
74 *> The leading dimension of the array A. LDA >= max(1,N).
75 *> \endverbatim
76 *>
77 *> \param[out] S
78 *> \verbatim
79 *> S is REAL array, dimension (N)
80 *> If INFO = 0, S contains the scale factors for A.
81 *> \endverbatim
82 *>
83 *> \param[out] SCOND
84 *> \verbatim
85 *> SCOND is REAL
86 *> If INFO = 0, S contains the ratio of the smallest S(i) to
87 *> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
88 *> large nor too small, it is not worth scaling by S.
89 *> \endverbatim
90 *>
91 *> \param[out] AMAX
92 *> \verbatim
93 *> AMAX is REAL
94 *> Absolute value of largest matrix element. If AMAX is very
95 *> close to overflow or very close to underflow, the matrix
96 *> should be scaled.
97 *> \endverbatim
98 *>
99 *> \param[out] INFO
100 *> \verbatim
101 *> INFO is INTEGER
102 *> = 0: successful exit
103 *> < 0: if INFO = -i, the i-th argument had an illegal value
104 *> > 0: if INFO = i, the i-th diagonal element is nonpositive.
105 *> \endverbatim
106 *
107 * Authors:
108 * ========
109 *
110 *> \author Univ. of Tennessee
111 *> \author Univ. of California Berkeley
112 *> \author Univ. of Colorado Denver
113 *> \author NAG Ltd.
114 *
115 *> \date December 2016
116 *
117 *> \ingroup complexPOcomputational
118 *
119 * =====================================================================
120  SUBROUTINE cpoequb( N, A, LDA, S, SCOND, AMAX, INFO )
121 *
122 * -- LAPACK computational routine (version 3.7.0) --
123 * -- LAPACK is a software package provided by Univ. of Tennessee, --
124 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
125 * December 2016
126 *
127 * .. Scalar Arguments ..
128  INTEGER INFO, LDA, N
129  REAL AMAX, SCOND
130 * ..
131 * .. Array Arguments ..
132  COMPLEX A( lda, * )
133  REAL S( * )
134 * ..
135 *
136 * =====================================================================
137 *
138 * .. Parameters ..
139  REAL ZERO, ONE
140  parameter( zero = 0.0e+0, one = 1.0e+0 )
141 * ..
142 * .. Local Scalars ..
143  INTEGER I
144  REAL SMIN, BASE, TMP
145 * ..
146 * .. External Functions ..
147  REAL SLAMCH
148  EXTERNAL slamch
149 * ..
150 * .. External Subroutines ..
151  EXTERNAL xerbla
152 * ..
153 * .. Intrinsic Functions ..
154  INTRINSIC max, min, sqrt, log, int
155 * ..
156 * .. Executable Statements ..
157 *
158 * Test the input parameters.
159 *
160 * Positive definite only performs 1 pass of equilibration.
161 *
162  info = 0
163  IF( n.LT.0 ) THEN
164  info = -1
165  ELSE IF( lda.LT.max( 1, n ) ) THEN
166  info = -3
167  END IF
168  IF( info.NE.0 ) THEN
169  CALL xerbla( 'CPOEQUB', -info )
170  RETURN
171  END IF
172 *
173 * Quick return if possible.
174 *
175  IF( n.EQ.0 ) THEN
176  scond = one
177  amax = zero
178  RETURN
179  END IF
180
181  base = slamch( 'B' )
182  tmp = -0.5 / log( base )
183 *
184 * Find the minimum and maximum diagonal elements.
185 *
186  s( 1 ) = a( 1, 1 )
187  smin = s( 1 )
188  amax = s( 1 )
189  DO 10 i = 2, n
190  s( i ) = a( i, i )
191  smin = min( smin, s( i ) )
192  amax = max( amax, s( i ) )
193  10 CONTINUE
194 *
195  IF( smin.LE.zero ) THEN
196 *
197 * Find the first non-positive diagonal element and return.
198 *
199  DO 20 i = 1, n
200  IF( s( i ).LE.zero ) THEN
201  info = i
202  RETURN
203  END IF
204  20 CONTINUE
205  ELSE
206 *
207 * Set the scale factors to the reciprocals
208 * of the diagonal elements.
209 *
210  DO 30 i = 1, n
211  s( i ) = base ** int( tmp * log( s( i ) ) )
212  30 CONTINUE
213 *
214 * Compute SCOND = min(S(I)) / max(S(I)).
215 *
216  scond = sqrt( smin ) / sqrt( amax )
217  END IF
218 *
219  RETURN
220 *
221 * End of CPOEQUB
222 *
223  END
subroutine cpoequb(N, A, LDA, S, SCOND, AMAX, INFO)
CPOEQUB
Definition: cpoequb.f:121
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62