00001 SUBROUTINE ZPOEQU( N, A, LDA, S, SCOND, AMAX, INFO ) 00002 * 00003 * -- LAPACK routine (version 3.2) -- 00004 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00005 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00006 * November 2006 00007 * 00008 * .. Scalar Arguments .. 00009 INTEGER INFO, LDA, N 00010 DOUBLE PRECISION AMAX, SCOND 00011 * .. 00012 * .. Array Arguments .. 00013 DOUBLE PRECISION S( * ) 00014 COMPLEX*16 A( LDA, * ) 00015 * .. 00016 * 00017 * Purpose 00018 * ======= 00019 * 00020 * ZPOEQU computes row and column scalings intended to equilibrate a 00021 * Hermitian positive definite matrix A and reduce its condition number 00022 * (with respect to the two-norm). S contains the scale factors, 00023 * S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with 00024 * elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This 00025 * choice of S puts the condition number of B within a factor N of the 00026 * smallest possible condition number over all possible diagonal 00027 * scalings. 00028 * 00029 * Arguments 00030 * ========= 00031 * 00032 * N (input) INTEGER 00033 * The order of the matrix A. N >= 0. 00034 * 00035 * A (input) COMPLEX*16 array, dimension (LDA,N) 00036 * The N-by-N Hermitian positive definite matrix whose scaling 00037 * factors are to be computed. Only the diagonal elements of A 00038 * are referenced. 00039 * 00040 * LDA (input) INTEGER 00041 * The leading dimension of the array A. LDA >= max(1,N). 00042 * 00043 * S (output) DOUBLE PRECISION array, dimension (N) 00044 * If INFO = 0, S contains the scale factors for A. 00045 * 00046 * SCOND (output) DOUBLE PRECISION 00047 * If INFO = 0, S contains the ratio of the smallest S(i) to 00048 * the largest S(i). If SCOND >= 0.1 and AMAX is neither too 00049 * large nor too small, it is not worth scaling by S. 00050 * 00051 * AMAX (output) DOUBLE PRECISION 00052 * Absolute value of largest matrix element. If AMAX is very 00053 * close to overflow or very close to underflow, the matrix 00054 * should be scaled. 00055 * 00056 * INFO (output) INTEGER 00057 * = 0: successful exit 00058 * < 0: if INFO = -i, the i-th argument had an illegal value 00059 * > 0: if INFO = i, the i-th diagonal element is nonpositive. 00060 * 00061 * ===================================================================== 00062 * 00063 * .. Parameters .. 00064 DOUBLE PRECISION ZERO, ONE 00065 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00066 * .. 00067 * .. Local Scalars .. 00068 INTEGER I 00069 DOUBLE PRECISION SMIN 00070 * .. 00071 * .. External Subroutines .. 00072 EXTERNAL XERBLA 00073 * .. 00074 * .. Intrinsic Functions .. 00075 INTRINSIC DBLE, MAX, MIN, SQRT 00076 * .. 00077 * .. Executable Statements .. 00078 * 00079 * Test the input parameters. 00080 * 00081 INFO = 0 00082 IF( N.LT.0 ) THEN 00083 INFO = -1 00084 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00085 INFO = -3 00086 END IF 00087 IF( INFO.NE.0 ) THEN 00088 CALL XERBLA( 'ZPOEQU', -INFO ) 00089 RETURN 00090 END IF 00091 * 00092 * Quick return if possible 00093 * 00094 IF( N.EQ.0 ) THEN 00095 SCOND = ONE 00096 AMAX = ZERO 00097 RETURN 00098 END IF 00099 * 00100 * Find the minimum and maximum diagonal elements. 00101 * 00102 S( 1 ) = DBLE( A( 1, 1 ) ) 00103 SMIN = S( 1 ) 00104 AMAX = S( 1 ) 00105 DO 10 I = 2, N 00106 S( I ) = DBLE( A( I, I ) ) 00107 SMIN = MIN( SMIN, S( I ) ) 00108 AMAX = MAX( AMAX, S( I ) ) 00109 10 CONTINUE 00110 * 00111 IF( SMIN.LE.ZERO ) THEN 00112 * 00113 * Find the first non-positive diagonal element and return. 00114 * 00115 DO 20 I = 1, N 00116 IF( S( I ).LE.ZERO ) THEN 00117 INFO = I 00118 RETURN 00119 END IF 00120 20 CONTINUE 00121 ELSE 00122 * 00123 * Set the scale factors to the reciprocals 00124 * of the diagonal elements. 00125 * 00126 DO 30 I = 1, N 00127 S( I ) = ONE / SQRT( S( I ) ) 00128 30 CONTINUE 00129 * 00130 * Compute SCOND = min(S(I)) / max(S(I)) 00131 * 00132 SCOND = SQRT( SMIN ) / SQRT( AMAX ) 00133 END IF 00134 RETURN 00135 * 00136 * End of ZPOEQU 00137 * 00138 END
1.7.2