LAPACK 3.3.1
Linear Algebra PACKage

dpoequb.f

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00001       SUBROUTINE DPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO )
00002 *
00003 *     -- LAPACK routine (version 3.2)                                 --
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       DOUBLE PRECISION   A( LDA, * ), S( * )
00019 *     ..
00020 *
00021 *  Purpose
00022 *  =======
00023 *
00024 *  DPOEQU computes row and column scalings intended to equilibrate a
00025 *  symmetric positive definite matrix A and reduce its condition number
00026 *  (with respect to the two-norm).  S contains the scale factors,
00027 *  S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
00028 *  elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
00029 *  choice of S puts the condition number of B within a factor N of the
00030 *  smallest possible condition number over all possible diagonal
00031 *  scalings.
00032 *
00033 *  Arguments
00034 *  =========
00035 *
00036 *  N       (input) INTEGER
00037 *          The order of the matrix A.  N >= 0.
00038 *
00039 *  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
00040 *          The N-by-N symmetric positive definite matrix whose scaling
00041 *          factors are to be computed.  Only the diagonal elements of A
00042 *          are referenced.
00043 *
00044 *  LDA     (input) INTEGER
00045 *          The leading dimension of the array A.  LDA >= max(1,N).
00046 *
00047 *  S       (output) DOUBLE PRECISION array, dimension (N)
00048 *          If INFO = 0, S contains the scale factors for A.
00049 *
00050 *  SCOND   (output) DOUBLE PRECISION
00051 *          If INFO = 0, S contains the ratio of the smallest S(i) to
00052 *          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
00053 *          large nor too small, it is not worth scaling by S.
00054 *
00055 *  AMAX    (output) DOUBLE PRECISION
00056 *          Absolute value of largest matrix element.  If AMAX is very
00057 *          close to overflow or very close to underflow, the matrix
00058 *          should be scaled.
00059 *
00060 *  INFO    (output) INTEGER
00061 *          = 0:  successful exit
00062 *          < 0:  if INFO = -i, the i-th argument had an illegal value
00063 *          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
00064 *
00065 *  =====================================================================
00066 *
00067 *     .. Parameters ..
00068       DOUBLE PRECISION   ZERO, ONE
00069       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
00070 *     ..
00071 *     .. Local Scalars ..
00072       INTEGER            I
00073       DOUBLE PRECISION   SMIN, BASE, TMP
00074 *     ..
00075 *     .. External Functions ..
00076       DOUBLE PRECISION   DLAMCH
00077       EXTERNAL           DLAMCH
00078 *     ..
00079 *     .. External Subroutines ..
00080       EXTERNAL           XERBLA
00081 *     ..
00082 *     .. Intrinsic Functions ..
00083       INTRINSIC          MAX, MIN, SQRT, LOG, INT
00084 *     ..
00085 *     .. Executable Statements ..
00086 *
00087 *     Test the input parameters.
00088 *
00089 *     Positive definite only performs 1 pass of equilibration.
00090 *
00091       INFO = 0
00092       IF( N.LT.0 ) THEN
00093          INFO = -1
00094       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00095          INFO = -3
00096       END IF
00097       IF( INFO.NE.0 ) THEN
00098          CALL XERBLA( 'DPOEQUB', -INFO )
00099          RETURN
00100       END IF
00101 *
00102 *     Quick return if possible.
00103 *
00104       IF( N.EQ.0 ) THEN
00105          SCOND = ONE
00106          AMAX = ZERO
00107          RETURN
00108       END IF
00109 
00110       BASE = DLAMCH( 'B' )
00111       TMP = -0.5D+0 / LOG ( BASE )
00112 *
00113 *     Find the minimum and maximum diagonal elements.
00114 *
00115       S( 1 ) = A( 1, 1 )
00116       SMIN = S( 1 )
00117       AMAX = S( 1 )
00118       DO 10 I = 2, N
00119          S( I ) = A( I, I )
00120          SMIN = MIN( SMIN, S( I ) )
00121          AMAX = MAX( AMAX, S( I ) )
00122    10 CONTINUE
00123 *
00124       IF( SMIN.LE.ZERO ) THEN
00125 *
00126 *        Find the first non-positive diagonal element and return.
00127 *
00128          DO 20 I = 1, N
00129             IF( S( I ).LE.ZERO ) THEN
00130                INFO = I
00131                RETURN
00132             END IF
00133    20    CONTINUE
00134       ELSE
00135 *
00136 *        Set the scale factors to the reciprocals
00137 *        of the diagonal elements.
00138 *
00139          DO 30 I = 1, N
00140             S( I ) = BASE ** INT( TMP * LOG( S( I ) ) )
00141    30    CONTINUE
00142 *
00143 *        Compute SCOND = min(S(I)) / max(S(I)).
00144 *
00145          SCOND = SQRT( SMIN ) / SQRT( AMAX )
00146       END IF
00147 *
00148       RETURN
00149 *
00150 *     End of DPOEQUB
00151 *
00152       END
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