Annotation of rpl/lapack/lapack/dpoequb.f, revision 1.13
1.5 bertrand 1: *> \brief \b DPOEQUB
2: *
3: * =========== DOCUMENTATION ===========
4: *
1.11 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.5 bertrand 7: *
8: *> \htmlonly
1.11 bertrand 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">
1.5 bertrand 15: *> [TXT]</a>
1.11 bertrand 16: *> \endhtmlonly
1.5 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO )
1.11 bertrand 22: *
1.5 bertrand 23: * .. Scalar Arguments ..
24: * INTEGER INFO, LDA, N
25: * DOUBLE PRECISION AMAX, SCOND
26: * ..
27: * .. Array Arguments ..
28: * DOUBLE PRECISION A( LDA, * ), S( * )
29: * ..
1.11 bertrand 30: *
1.5 bertrand 31: *
32: *> \par Purpose:
33: * =============
34: *>
35: *> \verbatim
36: *>
1.11 bertrand 37: *> DPOEQUB computes row and column scalings intended to equilibrate a
1.5 bertrand 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.
1.11 bertrand 45: *>
46: *> This routine differs from DPOEQU 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
50: *> between sqrt(radix) and 1/sqrt(radix).
1.5 bertrand 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION
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 DOUBLE PRECISION
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: *
1.11 bertrand 109: *> \author Univ. of Tennessee
110: *> \author Univ. of California Berkeley
111: *> \author Univ. of Colorado Denver
112: *> \author NAG Ltd.
1.5 bertrand 113: *
1.11 bertrand 114: *> \date December 2016
1.5 bertrand 115: *
116: *> \ingroup doublePOcomputational
117: *
118: * =====================================================================
1.1 bertrand 119: SUBROUTINE DPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO )
120: *
1.11 bertrand 121: * -- LAPACK computational routine (version 3.7.0) --
1.5 bertrand 122: * -- LAPACK is a software package provided by Univ. of Tennessee, --
123: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.11 bertrand 124: * December 2016
1.1 bertrand 125: *
126: * .. Scalar Arguments ..
127: INTEGER INFO, LDA, N
128: DOUBLE PRECISION AMAX, SCOND
129: * ..
130: * .. Array Arguments ..
131: DOUBLE PRECISION A( LDA, * ), S( * )
132: * ..
133: *
134: * =====================================================================
135: *
136: * .. Parameters ..
137: DOUBLE PRECISION ZERO, ONE
138: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
139: * ..
140: * .. Local Scalars ..
141: INTEGER I
142: DOUBLE PRECISION SMIN, BASE, TMP
143: * ..
144: * .. External Functions ..
145: DOUBLE PRECISION DLAMCH
146: EXTERNAL DLAMCH
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( 'DPOEQUB', -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 = DLAMCH( 'B' )
180: TMP = -0.5D+0 / 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 DPOEQUB
220: *
221: END
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