Annotation of rpl/lapack/lapack/zpoequb.f, revision 1.15
1.6 bertrand 1: *> \brief \b ZPOEQUB
2: *
3: * =========== DOCUMENTATION ===========
4: *
1.12 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.6 bertrand 7: *
8: *> \htmlonly
1.12 bertrand 9: *> Download ZPOEQUB + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zpoequb.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zpoequb.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zpoequb.f">
1.6 bertrand 15: *> [TXT]</a>
1.12 bertrand 16: *> \endhtmlonly
1.6 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO )
1.12 bertrand 22: *
1.6 bertrand 23: * .. Scalar Arguments ..
24: * INTEGER INFO, LDA, N
25: * DOUBLE PRECISION AMAX, SCOND
26: * ..
27: * .. Array Arguments ..
28: * COMPLEX*16 A( LDA, * )
29: * DOUBLE PRECISION S( * )
30: * ..
1.12 bertrand 31: *
1.6 bertrand 32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> ZPOEQUB computes row and column scalings intended to equilibrate a
1.12 bertrand 39: *> Hermitian positive definite matrix A and reduce its condition number
1.6 bertrand 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.
1.12 bertrand 46: *>
47: *> This routine differs from ZPOEQU 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
51: *> between sqrt(radix) and 1/sqrt(radix).
1.6 bertrand 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*16 array, dimension (LDA,N)
1.12 bertrand 66: *> The N-by-N Hermitian positive definite matrix whose scaling
1.6 bertrand 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 DOUBLE PRECISION 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 DOUBLE PRECISION
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 DOUBLE PRECISION
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: *
1.12 bertrand 110: *> \author Univ. of Tennessee
111: *> \author Univ. of California Berkeley
112: *> \author Univ. of Colorado Denver
113: *> \author NAG Ltd.
1.6 bertrand 114: *
115: *> \ingroup complex16POcomputational
116: *
117: * =====================================================================
1.1 bertrand 118: SUBROUTINE ZPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO )
119: *
1.15 ! bertrand 120: * -- LAPACK computational routine --
1.6 bertrand 121: * -- LAPACK is a software package provided by Univ. of Tennessee, --
122: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.1 bertrand 123: *
124: * .. Scalar Arguments ..
125: INTEGER INFO, LDA, N
126: DOUBLE PRECISION AMAX, SCOND
127: * ..
128: * .. Array Arguments ..
129: COMPLEX*16 A( LDA, * )
130: DOUBLE PRECISION S( * )
131: * ..
132: *
133: * =====================================================================
134: *
135: * .. Parameters ..
136: DOUBLE PRECISION ZERO, ONE
137: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
138: * ..
139: * .. Local Scalars ..
140: INTEGER I
141: DOUBLE PRECISION SMIN, BASE, TMP
142: * ..
143: * .. External Functions ..
144: DOUBLE PRECISION DLAMCH
145: EXTERNAL DLAMCH
146: * ..
147: * .. External Subroutines ..
148: EXTERNAL XERBLA
149: * ..
150: * .. Intrinsic Functions ..
151: INTRINSIC MAX, MIN, SQRT, LOG, INT, REAL, DIMAG
152: * ..
153: * .. Executable Statements ..
154: *
155: * Test the input parameters.
156: *
157: * Positive definite only performs 1 pass of equilibration.
158: *
159: INFO = 0
160: IF( N.LT.0 ) THEN
161: INFO = -1
162: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
163: INFO = -3
164: END IF
165: IF( INFO.NE.0 ) THEN
166: CALL XERBLA( 'ZPOEQUB', -INFO )
167: RETURN
168: END IF
169: *
170: * Quick return if possible.
171: *
172: IF( N.EQ.0 ) THEN
173: SCOND = ONE
174: AMAX = ZERO
175: RETURN
176: END IF
177:
178: BASE = DLAMCH( 'B' )
179: TMP = -0.5D+0 / LOG ( BASE )
180: *
181: * Find the minimum and maximum diagonal elements.
182: *
1.15 ! bertrand 183: S( 1 ) = DBLE( A( 1, 1 ) )
1.1 bertrand 184: SMIN = S( 1 )
185: AMAX = S( 1 )
186: DO 10 I = 2, N
1.15 ! bertrand 187: S( I ) = DBLE( A( I, I ) )
1.1 bertrand 188: SMIN = MIN( SMIN, S( I ) )
189: AMAX = MAX( AMAX, S( I ) )
190: 10 CONTINUE
191: *
192: IF( SMIN.LE.ZERO ) THEN
193: *
194: * Find the first non-positive diagonal element and return.
195: *
196: DO 20 I = 1, N
197: IF( S( I ).LE.ZERO ) THEN
198: INFO = I
199: RETURN
200: END IF
201: 20 CONTINUE
202: ELSE
203: *
204: * Set the scale factors to the reciprocals
205: * of the diagonal elements.
206: *
207: DO 30 I = 1, N
208: S( I ) = BASE ** INT( TMP * LOG( S( I ) ) )
209: 30 CONTINUE
210: *
211: * Compute SCOND = min(S(I)) / max(S(I)).
212: *
213: SCOND = SQRT( SMIN ) / SQRT( AMAX )
214: END IF
215: *
216: RETURN
217: *
218: * End of ZPOEQUB
219: *
220: END
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