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1: *> \brief \b ZPOEQUB
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
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">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZPOEQUB( 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: * COMPLEX*16 A( LDA, * )
29: * DOUBLE PRECISION S( * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> ZPOEQUB computes row and column scalings intended to equilibrate a
39: *> symmetric 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: *> \endverbatim
47: *
48: * Arguments:
49: * ==========
50: *
51: *> \param[in] N
52: *> \verbatim
53: *> N is INTEGER
54: *> The order of the matrix A. N >= 0.
55: *> \endverbatim
56: *>
57: *> \param[in] A
58: *> \verbatim
59: *> A is COMPLEX*16 array, dimension (LDA,N)
60: *> The N-by-N symmetric positive definite matrix whose scaling
61: *> factors are to be computed. Only the diagonal elements of A
62: *> are referenced.
63: *> \endverbatim
64: *>
65: *> \param[in] LDA
66: *> \verbatim
67: *> LDA is INTEGER
68: *> The leading dimension of the array A. LDA >= max(1,N).
69: *> \endverbatim
70: *>
71: *> \param[out] S
72: *> \verbatim
73: *> S is DOUBLE PRECISION array, dimension (N)
74: *> If INFO = 0, S contains the scale factors for A.
75: *> \endverbatim
76: *>
77: *> \param[out] SCOND
78: *> \verbatim
79: *> SCOND is DOUBLE PRECISION
80: *> If INFO = 0, S contains the ratio of the smallest S(i) to
81: *> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
82: *> large nor too small, it is not worth scaling by S.
83: *> \endverbatim
84: *>
85: *> \param[out] AMAX
86: *> \verbatim
87: *> AMAX is DOUBLE PRECISION
88: *> Absolute value of largest matrix element. If AMAX is very
89: *> close to overflow or very close to underflow, the matrix
90: *> should be scaled.
91: *> \endverbatim
92: *>
93: *> \param[out] INFO
94: *> \verbatim
95: *> INFO is INTEGER
96: *> = 0: successful exit
97: *> < 0: if INFO = -i, the i-th argument had an illegal value
98: *> > 0: if INFO = i, the i-th diagonal element is nonpositive.
99: *> \endverbatim
100: *
101: * Authors:
102: * ========
103: *
104: *> \author Univ. of Tennessee
105: *> \author Univ. of California Berkeley
106: *> \author Univ. of Colorado Denver
107: *> \author NAG Ltd.
108: *
109: *> \date November 2011
110: *
111: *> \ingroup complex16POcomputational
112: *
113: * =====================================================================
114: SUBROUTINE ZPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO )
115: *
116: * -- LAPACK computational routine (version 3.4.0) --
117: * -- LAPACK is a software package provided by Univ. of Tennessee, --
118: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
119: * November 2011
120: *
121: * .. Scalar Arguments ..
122: INTEGER INFO, LDA, N
123: DOUBLE PRECISION AMAX, SCOND
124: * ..
125: * .. Array Arguments ..
126: COMPLEX*16 A( LDA, * )
127: DOUBLE PRECISION S( * )
128: * ..
129: *
130: * =====================================================================
131: *
132: * .. Parameters ..
133: DOUBLE PRECISION ZERO, ONE
134: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
135: * ..
136: * .. Local Scalars ..
137: INTEGER I
138: DOUBLE PRECISION SMIN, BASE, TMP
139: * ..
140: * .. External Functions ..
141: DOUBLE PRECISION DLAMCH
142: EXTERNAL DLAMCH
143: * ..
144: * .. External Subroutines ..
145: EXTERNAL XERBLA
146: * ..
147: * .. Intrinsic Functions ..
148: INTRINSIC MAX, MIN, SQRT, LOG, INT, REAL, DIMAG
149: * ..
150: * .. Executable Statements ..
151: *
152: * Test the input parameters.
153: *
154: * Positive definite only performs 1 pass of equilibration.
155: *
156: INFO = 0
157: IF( N.LT.0 ) THEN
158: INFO = -1
159: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
160: INFO = -3
161: END IF
162: IF( INFO.NE.0 ) THEN
163: CALL XERBLA( 'ZPOEQUB', -INFO )
164: RETURN
165: END IF
166: *
167: * Quick return if possible.
168: *
169: IF( N.EQ.0 ) THEN
170: SCOND = ONE
171: AMAX = ZERO
172: RETURN
173: END IF
174:
175: BASE = DLAMCH( 'B' )
176: TMP = -0.5D+0 / LOG ( BASE )
177: *
178: * Find the minimum and maximum diagonal elements.
179: *
180: S( 1 ) = A( 1, 1 )
181: SMIN = S( 1 )
182: AMAX = S( 1 )
183: DO 10 I = 2, N
184: S( I ) = A( I, I )
185: SMIN = MIN( SMIN, S( I ) )
186: AMAX = MAX( AMAX, S( I ) )
187: 10 CONTINUE
188: *
189: IF( SMIN.LE.ZERO ) THEN
190: *
191: * Find the first non-positive diagonal element and return.
192: *
193: DO 20 I = 1, N
194: IF( S( I ).LE.ZERO ) THEN
195: INFO = I
196: RETURN
197: END IF
198: 20 CONTINUE
199: ELSE
200: *
201: * Set the scale factors to the reciprocals
202: * of the diagonal elements.
203: *
204: DO 30 I = 1, N
205: S( I ) = BASE ** INT( TMP * LOG( S( I ) ) )
206: 30 CONTINUE
207: *
208: * Compute SCOND = min(S(I)) / max(S(I)).
209: *
210: SCOND = SQRT( SMIN ) / SQRT( AMAX )
211: END IF
212: *
213: RETURN
214: *
215: * End of ZPOEQUB
216: *
217: END
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