Annotation of rpl/lapack/lapack/zpoequb.f, revision 1.14
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: *
1.12 bertrand 115: *> \date December 2016
1.6 bertrand 116: *
117: *> \ingroup complex16POcomputational
118: *
119: * =====================================================================
1.1 bertrand 120: SUBROUTINE ZPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO )
121: *
1.12 bertrand 122: * -- LAPACK computational routine (version 3.7.0) --
1.6 bertrand 123: * -- LAPACK is a software package provided by Univ. of Tennessee, --
124: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.12 bertrand 125: * December 2016
1.1 bertrand 126: *
127: * .. Scalar Arguments ..
128: INTEGER INFO, LDA, N
129: DOUBLE PRECISION AMAX, SCOND
130: * ..
131: * .. Array Arguments ..
132: COMPLEX*16 A( LDA, * )
133: DOUBLE PRECISION S( * )
134: * ..
135: *
136: * =====================================================================
137: *
138: * .. Parameters ..
139: DOUBLE PRECISION ZERO, ONE
140: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
141: * ..
142: * .. Local Scalars ..
143: INTEGER I
144: DOUBLE PRECISION SMIN, BASE, TMP
145: * ..
146: * .. External Functions ..
147: DOUBLE PRECISION DLAMCH
148: EXTERNAL DLAMCH
149: * ..
150: * .. External Subroutines ..
151: EXTERNAL XERBLA
152: * ..
153: * .. Intrinsic Functions ..
154: INTRINSIC MAX, MIN, SQRT, LOG, INT, REAL, DIMAG
155: * ..
156: * .. Executable Statements ..
157: *
158: * Test the input parameters.
159: *
160: * Positive definite only performs 1 pass of equilibration.
161: *
162: INFO = 0
163: IF( N.LT.0 ) THEN
164: INFO = -1
165: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
166: INFO = -3
167: END IF
168: IF( INFO.NE.0 ) THEN
169: CALL XERBLA( 'ZPOEQUB', -INFO )
170: RETURN
171: END IF
172: *
173: * Quick return if possible.
174: *
175: IF( N.EQ.0 ) THEN
176: SCOND = ONE
177: AMAX = ZERO
178: RETURN
179: END IF
180:
181: BASE = DLAMCH( 'B' )
182: TMP = -0.5D+0 / LOG ( BASE )
183: *
184: * Find the minimum and maximum diagonal elements.
185: *
186: S( 1 ) = A( 1, 1 )
187: SMIN = S( 1 )
188: AMAX = S( 1 )
189: DO 10 I = 2, N
190: S( I ) = A( I, I )
191: SMIN = MIN( SMIN, S( I ) )
192: AMAX = MAX( AMAX, S( I ) )
193: 10 CONTINUE
194: *
195: IF( SMIN.LE.ZERO ) THEN
196: *
197: * Find the first non-positive diagonal element and return.
198: *
199: DO 20 I = 1, N
200: IF( S( I ).LE.ZERO ) THEN
201: INFO = I
202: RETURN
203: END IF
204: 20 CONTINUE
205: ELSE
206: *
207: * Set the scale factors to the reciprocals
208: * of the diagonal elements.
209: *
210: DO 30 I = 1, N
211: S( I ) = BASE ** INT( TMP * LOG( S( I ) ) )
212: 30 CONTINUE
213: *
214: * Compute SCOND = min(S(I)) / max(S(I)).
215: *
216: SCOND = SQRT( SMIN ) / SQRT( AMAX )
217: END IF
218: *
219: RETURN
220: *
221: * End of ZPOEQUB
222: *
223: END
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