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