Annotation of rpl/lapack/lapack/zpoequb.f, revision 1.2
1.1 bertrand 1: SUBROUTINE ZPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO )
2: *
3: * -- LAPACK routine (version 3.2) --
4: * -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
5: * -- Jason Riedy of Univ. of California Berkeley. --
6: * -- November 2008 --
7: *
8: * -- LAPACK is a software package provided by Univ. of Tennessee, --
9: * -- Univ. of California Berkeley and NAG Ltd. --
10: *
11: IMPLICIT NONE
12: * ..
13: * .. Scalar Arguments ..
14: INTEGER INFO, LDA, N
15: DOUBLE PRECISION AMAX, SCOND
16: * ..
17: * .. Array Arguments ..
18: COMPLEX*16 A( LDA, * )
19: DOUBLE PRECISION S( * )
20: * ..
21: *
22: * Purpose
23: * =======
24: *
25: * ZPOEQUB computes row and column scalings intended to equilibrate a
26: * symmetric positive definite matrix A and reduce its condition number
27: * (with respect to the two-norm). S contains the scale factors,
28: * S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
29: * elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
30: * choice of S puts the condition number of B within a factor N of the
31: * smallest possible condition number over all possible diagonal
32: * scalings.
33: *
34: * Arguments
35: * =========
36: *
37: * N (input) INTEGER
38: * The order of the matrix A. N >= 0.
39: *
40: * A (input) COMPLEX*16 array, dimension (LDA,N)
41: * The N-by-N symmetric positive definite matrix whose scaling
42: * factors are to be computed. Only the diagonal elements of A
43: * are referenced.
44: *
45: * LDA (input) INTEGER
46: * The leading dimension of the array A. LDA >= max(1,N).
47: *
48: * S (output) DOUBLE PRECISION array, dimension (N)
49: * If INFO = 0, S contains the scale factors for A.
50: *
51: * SCOND (output) DOUBLE PRECISION
52: * If INFO = 0, S contains the ratio of the smallest S(i) to
53: * the largest S(i). If SCOND >= 0.1 and AMAX is neither too
54: * large nor too small, it is not worth scaling by S.
55: *
56: * AMAX (output) DOUBLE PRECISION
57: * Absolute value of largest matrix element. If AMAX is very
58: * close to overflow or very close to underflow, the matrix
59: * should be scaled.
60: *
61: * INFO (output) INTEGER
62: * = 0: successful exit
63: * < 0: if INFO = -i, the i-th argument had an illegal value
64: * > 0: if INFO = i, the i-th diagonal element is nonpositive.
65: *
66: * =====================================================================
67: *
68: * .. Parameters ..
69: DOUBLE PRECISION ZERO, ONE
70: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
71: * ..
72: * .. Local Scalars ..
73: INTEGER I
74: DOUBLE PRECISION SMIN, BASE, TMP
75: COMPLEX*16 ZDUM
76: * ..
77: * .. External Functions ..
78: DOUBLE PRECISION DLAMCH
79: EXTERNAL DLAMCH
80: * ..
81: * .. External Subroutines ..
82: EXTERNAL XERBLA
83: * ..
84: * .. Intrinsic Functions ..
85: INTRINSIC MAX, MIN, SQRT, LOG, INT, REAL, DIMAG
86: * ..
87: * .. Statement Functions ..
88: DOUBLE PRECISION CABS1
89: * ..
90: * .. Statement Function Definitions ..
91: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
92: * ..
93: * .. Executable Statements ..
94: *
95: * Test the input parameters.
96: *
97: * Positive definite only performs 1 pass of equilibration.
98: *
99: INFO = 0
100: IF( N.LT.0 ) THEN
101: INFO = -1
102: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
103: INFO = -3
104: END IF
105: IF( INFO.NE.0 ) THEN
106: CALL XERBLA( 'ZPOEQUB', -INFO )
107: RETURN
108: END IF
109: *
110: * Quick return if possible.
111: *
112: IF( N.EQ.0 ) THEN
113: SCOND = ONE
114: AMAX = ZERO
115: RETURN
116: END IF
117:
118: BASE = DLAMCH( 'B' )
119: TMP = -0.5D+0 / LOG ( BASE )
120: *
121: * Find the minimum and maximum diagonal elements.
122: *
123: S( 1 ) = A( 1, 1 )
124: SMIN = S( 1 )
125: AMAX = S( 1 )
126: DO 10 I = 2, N
127: S( I ) = A( I, I )
128: SMIN = MIN( SMIN, S( I ) )
129: AMAX = MAX( AMAX, S( I ) )
130: 10 CONTINUE
131: *
132: IF( SMIN.LE.ZERO ) THEN
133: *
134: * Find the first non-positive diagonal element and return.
135: *
136: DO 20 I = 1, N
137: IF( S( I ).LE.ZERO ) THEN
138: INFO = I
139: RETURN
140: END IF
141: 20 CONTINUE
142: ELSE
143: *
144: * Set the scale factors to the reciprocals
145: * of the diagonal elements.
146: *
147: DO 30 I = 1, N
148: S( I ) = BASE ** INT( TMP * LOG( S( I ) ) )
149: 30 CONTINUE
150: *
151: * Compute SCOND = min(S(I)) / max(S(I)).
152: *
153: SCOND = SQRT( SMIN ) / SQRT( AMAX )
154: END IF
155: *
156: RETURN
157: *
158: * End of ZPOEQUB
159: *
160: END
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