LAPACK 3.3.1
Linear Algebra PACKage

zpoequb.f

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00001       SUBROUTINE ZPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO )
00002 *
00003 *     -- LAPACK routine (version 3.3.1)                                 --
00004 *     -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
00005 *     -- Jason Riedy of Univ. of California Berkeley.                 --
00006 *     -- November 2008                                                --
00007 *
00008 *     -- LAPACK is a software package provided by Univ. of Tennessee, --
00009 *     -- Univ. of California Berkeley and NAG Ltd.                    --
00010 *
00011       IMPLICIT NONE
00012 *     ..
00013 *     .. Scalar Arguments ..
00014       INTEGER            INFO, LDA, N
00015       DOUBLE PRECISION   AMAX, SCOND
00016 *     ..
00017 *     .. Array Arguments ..
00018       COMPLEX*16         A( LDA, * )
00019       DOUBLE PRECISION   S( * )
00020 *     ..
00021 *
00022 *  Purpose
00023 *  =======
00024 *
00025 *  ZPOEQUB computes row and column scalings intended to equilibrate a
00026 *  symmetric positive definite matrix A and reduce its condition number
00027 *  (with respect to the two-norm).  S contains the scale factors,
00028 *  S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
00029 *  elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
00030 *  choice of S puts the condition number of B within a factor N of the
00031 *  smallest possible condition number over all possible diagonal
00032 *  scalings.
00033 *
00034 *  Arguments
00035 *  =========
00036 *
00037 *  N       (input) INTEGER
00038 *          The order of the matrix A.  N >= 0.
00039 *
00040 *  A       (input) COMPLEX*16 array, dimension (LDA,N)
00041 *          The N-by-N symmetric positive definite matrix whose scaling
00042 *          factors are to be computed.  Only the diagonal elements of A
00043 *          are referenced.
00044 *
00045 *  LDA     (input) INTEGER
00046 *          The leading dimension of the array A.  LDA >= max(1,N).
00047 *
00048 *  S       (output) DOUBLE PRECISION array, dimension (N)
00049 *          If INFO = 0, S contains the scale factors for A.
00050 *
00051 *  SCOND   (output) DOUBLE PRECISION
00052 *          If INFO = 0, S contains the ratio of the smallest S(i) to
00053 *          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
00054 *          large nor too small, it is not worth scaling by S.
00055 *
00056 *  AMAX    (output) DOUBLE PRECISION
00057 *          Absolute value of largest matrix element.  If AMAX is very
00058 *          close to overflow or very close to underflow, the matrix
00059 *          should be scaled.
00060 *
00061 *  INFO    (output) INTEGER
00062 *          = 0:  successful exit
00063 *          < 0:  if INFO = -i, the i-th argument had an illegal value
00064 *          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
00065 *
00066 *  =====================================================================
00067 *
00068 *     .. Parameters ..
00069       DOUBLE PRECISION   ZERO, ONE
00070       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
00071 *     ..
00072 *     .. Local Scalars ..
00073       INTEGER            I
00074       DOUBLE PRECISION   SMIN, BASE, TMP
00075       COMPLEX*16         ZDUM
00076 *     ..
00077 *     .. External Functions ..
00078       DOUBLE PRECISION   DLAMCH
00079       EXTERNAL           DLAMCH
00080 *     ..
00081 *     .. External Subroutines ..
00082       EXTERNAL           XERBLA
00083 *     ..
00084 *     .. Intrinsic Functions ..
00085       INTRINSIC          MAX, MIN, SQRT, LOG, INT, REAL, DIMAG
00086 *     ..
00087 *     .. Executable Statements ..
00088 *
00089 *     Test the input parameters.
00090 *
00091 *     Positive definite only performs 1 pass of equilibration.
00092 *
00093       INFO = 0
00094       IF( N.LT.0 ) THEN
00095          INFO = -1
00096       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00097          INFO = -3
00098       END IF
00099       IF( INFO.NE.0 ) THEN
00100          CALL XERBLA( 'ZPOEQUB', -INFO )
00101          RETURN
00102       END IF
00103 *
00104 *     Quick return if possible.
00105 *
00106       IF( N.EQ.0 ) THEN
00107          SCOND = ONE
00108          AMAX = ZERO
00109          RETURN
00110       END IF
00111 
00112       BASE = DLAMCH( 'B' )
00113       TMP = -0.5D+0 / LOG ( BASE )
00114 *
00115 *     Find the minimum and maximum diagonal elements.
00116 *
00117       S( 1 ) = A( 1, 1 )
00118       SMIN = S( 1 )
00119       AMAX = S( 1 )
00120       DO 10 I = 2, N
00121          S( I ) = A( I, I )
00122          SMIN = MIN( SMIN, S( I ) )
00123          AMAX = MAX( AMAX, S( I ) )
00124    10 CONTINUE
00125 *
00126       IF( SMIN.LE.ZERO ) THEN
00127 *
00128 *        Find the first non-positive diagonal element and return.
00129 *
00130          DO 20 I = 1, N
00131             IF( S( I ).LE.ZERO ) THEN
00132                INFO = I
00133                RETURN
00134             END IF
00135    20    CONTINUE
00136       ELSE
00137 *
00138 *        Set the scale factors to the reciprocals
00139 *        of the diagonal elements.
00140 *
00141          DO 30 I = 1, N
00142             S( I ) = BASE ** INT( TMP * LOG( S( I ) ) )
00143    30    CONTINUE
00144 *
00145 *        Compute SCOND = min(S(I)) / max(S(I)).
00146 *
00147          SCOND = SQRT( SMIN ) / SQRT( AMAX )
00148       END IF
00149 *
00150       RETURN
00151 *
00152 *     End of ZPOEQUB
00153 *
00154       END
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