Annotation of rpl/lapack/lapack/dpoequb.f, revision 1.8
1.5 bertrand 1: *> \brief \b DPOEQUB
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DPOEQUB + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpoequb.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpoequb.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpoequb.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DPOEQUB( 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: *> \date November 2011
109: *
110: *> \ingroup doublePOcomputational
111: *
112: * =====================================================================
1.1 bertrand 113: SUBROUTINE DPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO )
114: *
1.5 bertrand 115: * -- LAPACK computational routine (version 3.4.0) --
116: * -- LAPACK is a software package provided by Univ. of Tennessee, --
117: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
118: * November 2011
1.1 bertrand 119: *
120: * .. Scalar Arguments ..
121: INTEGER INFO, LDA, N
122: DOUBLE PRECISION AMAX, SCOND
123: * ..
124: * .. Array Arguments ..
125: DOUBLE PRECISION A( LDA, * ), S( * )
126: * ..
127: *
128: * =====================================================================
129: *
130: * .. Parameters ..
131: DOUBLE PRECISION ZERO, ONE
132: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
133: * ..
134: * .. Local Scalars ..
135: INTEGER I
136: DOUBLE PRECISION SMIN, BASE, TMP
137: * ..
138: * .. External Functions ..
139: DOUBLE PRECISION DLAMCH
140: EXTERNAL DLAMCH
141: * ..
142: * .. External Subroutines ..
143: EXTERNAL XERBLA
144: * ..
145: * .. Intrinsic Functions ..
146: INTRINSIC MAX, MIN, SQRT, LOG, INT
147: * ..
148: * .. Executable Statements ..
149: *
150: * Test the input parameters.
151: *
152: * Positive definite only performs 1 pass of equilibration.
153: *
154: INFO = 0
155: IF( N.LT.0 ) THEN
156: INFO = -1
157: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
158: INFO = -3
159: END IF
160: IF( INFO.NE.0 ) THEN
161: CALL XERBLA( 'DPOEQUB', -INFO )
162: RETURN
163: END IF
164: *
165: * Quick return if possible.
166: *
167: IF( N.EQ.0 ) THEN
168: SCOND = ONE
169: AMAX = ZERO
170: RETURN
171: END IF
172:
173: BASE = DLAMCH( 'B' )
174: TMP = -0.5D+0 / LOG ( BASE )
175: *
176: * Find the minimum and maximum diagonal elements.
177: *
178: S( 1 ) = A( 1, 1 )
179: SMIN = S( 1 )
180: AMAX = S( 1 )
181: DO 10 I = 2, N
182: S( I ) = A( I, I )
183: SMIN = MIN( SMIN, S( I ) )
184: AMAX = MAX( AMAX, S( I ) )
185: 10 CONTINUE
186: *
187: IF( SMIN.LE.ZERO ) THEN
188: *
189: * Find the first non-positive diagonal element and return.
190: *
191: DO 20 I = 1, N
192: IF( S( I ).LE.ZERO ) THEN
193: INFO = I
194: RETURN
195: END IF
196: 20 CONTINUE
197: ELSE
198: *
199: * Set the scale factors to the reciprocals
200: * of the diagonal elements.
201: *
202: DO 30 I = 1, N
203: S( I ) = BASE ** INT( TMP * LOG( S( I ) ) )
204: 30 CONTINUE
205: *
206: * Compute SCOND = min(S(I)) / max(S(I)).
207: *
208: SCOND = SQRT( SMIN ) / SQRT( AMAX )
209: END IF
210: *
211: RETURN
212: *
213: * End of DPOEQUB
214: *
215: END
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