LAPACK  3.8.0 LAPACK: Linear Algebra PACKage
spoequb.f
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1 *> \brief \b SPOEQUB
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SPOEQUB( 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 A( LDA, * ), S( * )
29 * ..
30 *
31 *
32 *> \par Purpose:
33 * =============
34 *>
35 *> \verbatim
36 *>
37 *> SPOEQUB 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 SPOEQU 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
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 REAL 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 REAL 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 REAL
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 REAL
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 *> \date December 2016
115 *
116 *> \ingroup realPOcomputational
117 *
118 * =====================================================================
119  SUBROUTINE spoequb( N, A, LDA, S, SCOND, AMAX, INFO )
120 *
121 * -- LAPACK computational routine (version 3.7.0) --
122 * -- LAPACK is a software package provided by Univ. of Tennessee, --
123 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
124 * December 2016
125 *
126 * .. Scalar Arguments ..
127  INTEGER INFO, LDA, N
128  REAL AMAX, SCOND
129 * ..
130 * .. Array Arguments ..
131  REAL A( lda, * ), S( * )
132 * ..
133 *
134 * =====================================================================
135 *
136 * .. Parameters ..
137  REAL ZERO, ONE
138  parameter( zero = 0.0e+0, one = 1.0e+0 )
139 * ..
140 * .. Local Scalars ..
141  INTEGER I
142  REAL SMIN, BASE, TMP
143 * ..
144 * .. External Functions ..
145  REAL SLAMCH
146  EXTERNAL slamch
147 * ..
148 * .. External Subroutines ..
149  EXTERNAL xerbla
150 * ..
151 * .. Intrinsic Functions ..
152  INTRINSIC max, min, sqrt, log, int
153 * ..
154 * .. Executable Statements ..
155 *
156 * Test the input parameters.
157 *
158 * Positive definite only performs 1 pass of equilibration.
159 *
160  info = 0
161  IF( n.LT.0 ) THEN
162  info = -1
163  ELSE IF( lda.LT.max( 1, n ) ) THEN
164  info = -3
165  END IF
166  IF( info.NE.0 ) THEN
167  CALL xerbla( 'SPOEQUB', -info )
168  RETURN
169  END IF
170 *
171 * Quick return if possible.
172 *
173  IF( n.EQ.0 ) THEN
174  scond = one
175  amax = zero
176  RETURN
177  END IF
178
179  base = slamch( 'B' )
180  tmp = -0.5 / log( base )
181 *
182 * Find the minimum and maximum diagonal elements.
183 *
184  s( 1 ) = a( 1, 1 )
185  smin = s( 1 )
186  amax = s( 1 )
187  DO 10 i = 2, n
188  s( i ) = a( i, i )
189  smin = min( smin, s( i ) )
190  amax = max( amax, s( i ) )
191  10 CONTINUE
192 *
193  IF( smin.LE.zero ) THEN
194 *
195 * Find the first non-positive diagonal element and return.
196 *
197  DO 20 i = 1, n
198  IF( s( i ).LE.zero ) THEN
199  info = i
200  RETURN
201  END IF
202  20 CONTINUE
203  ELSE
204 *
205 * Set the scale factors to the reciprocals
206 * of the diagonal elements.
207 *
208  DO 30 i = 1, n
209  s( i ) = base ** int( tmp * log( s( i ) ) )
210  30 CONTINUE
211 *
212 * Compute SCOND = min(S(I)) / max(S(I)).
213 *
214  scond = sqrt( smin ) / sqrt( amax )
215  END IF
216 *
217  RETURN
218 *
219 * End of SPOEQUB
220 *
221  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine spoequb(N, A, LDA, S, SCOND, AMAX, INFO)
SPOEQUB
Definition: spoequb.f:120