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Annotation of rpl/lapack/lapack/dpoequb.f, revision 1.1

1.1     ! bertrand    1:       SUBROUTINE DPOEQUB( 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:       DOUBLE PRECISION   A( LDA, * ), S( * )
        !            19: *     ..
        !            20: *
        !            21: *  Purpose
        !            22: *  =======
        !            23: *
        !            24: *  DPOEQU computes row and column scalings intended to equilibrate a
        !            25: *  symmetric positive definite matrix A and reduce its condition number
        !            26: *  (with respect to the two-norm).  S contains the scale factors,
        !            27: *  S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
        !            28: *  elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
        !            29: *  choice of S puts the condition number of B within a factor N of the
        !            30: *  smallest possible condition number over all possible diagonal
        !            31: *  scalings.
        !            32: *
        !            33: *  Arguments
        !            34: *  =========
        !            35: *
        !            36: *  N       (input) INTEGER
        !            37: *          The order of the matrix A.  N >= 0.
        !            38: *
        !            39: *  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
        !            40: *          The N-by-N symmetric positive definite matrix whose scaling
        !            41: *          factors are to be computed.  Only the diagonal elements of A
        !            42: *          are referenced.
        !            43: *
        !            44: *  LDA     (input) INTEGER
        !            45: *          The leading dimension of the array A.  LDA >= max(1,N).
        !            46: *
        !            47: *  S       (output) DOUBLE PRECISION array, dimension (N)
        !            48: *          If INFO = 0, S contains the scale factors for A.
        !            49: *
        !            50: *  SCOND   (output) DOUBLE PRECISION
        !            51: *          If INFO = 0, S contains the ratio of the smallest S(i) to
        !            52: *          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
        !            53: *          large nor too small, it is not worth scaling by S.
        !            54: *
        !            55: *  AMAX    (output) DOUBLE PRECISION
        !            56: *          Absolute value of largest matrix element.  If AMAX is very
        !            57: *          close to overflow or very close to underflow, the matrix
        !            58: *          should be scaled.
        !            59: *
        !            60: *  INFO    (output) INTEGER
        !            61: *          = 0:  successful exit
        !            62: *          < 0:  if INFO = -i, the i-th argument had an illegal value
        !            63: *          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
        !            64: *
        !            65: *  =====================================================================
        !            66: *
        !            67: *     .. Parameters ..
        !            68:       DOUBLE PRECISION   ZERO, ONE
        !            69:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
        !            70: *     ..
        !            71: *     .. Local Scalars ..
        !            72:       INTEGER            I
        !            73:       DOUBLE PRECISION   SMIN, BASE, TMP
        !            74: *     ..
        !            75: *     .. External Functions ..
        !            76:       DOUBLE PRECISION   DLAMCH
        !            77:       EXTERNAL           DLAMCH
        !            78: *     ..
        !            79: *     .. External Subroutines ..
        !            80:       EXTERNAL           XERBLA
        !            81: *     ..
        !            82: *     .. Intrinsic Functions ..
        !            83:       INTRINSIC          MAX, MIN, SQRT, LOG, INT
        !            84: *     ..
        !            85: *     .. Executable Statements ..
        !            86: *
        !            87: *     Test the input parameters.
        !            88: *
        !            89: *     Positive definite only performs 1 pass of equilibration.
        !            90: *
        !            91:       INFO = 0
        !            92:       IF( N.LT.0 ) THEN
        !            93:          INFO = -1
        !            94:       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
        !            95:          INFO = -3
        !            96:       END IF
        !            97:       IF( INFO.NE.0 ) THEN
        !            98:          CALL XERBLA( 'DPOEQUB', -INFO )
        !            99:          RETURN
        !           100:       END IF
        !           101: *
        !           102: *     Quick return if possible.
        !           103: *
        !           104:       IF( N.EQ.0 ) THEN
        !           105:          SCOND = ONE
        !           106:          AMAX = ZERO
        !           107:          RETURN
        !           108:       END IF
        !           109: 
        !           110:       BASE = DLAMCH( 'B' )
        !           111:       TMP = -0.5D+0 / LOG ( BASE )
        !           112: *
        !           113: *     Find the minimum and maximum diagonal elements.
        !           114: *
        !           115:       S( 1 ) = A( 1, 1 )
        !           116:       SMIN = S( 1 )
        !           117:       AMAX = S( 1 )
        !           118:       DO 10 I = 2, N
        !           119:          S( I ) = A( I, I )
        !           120:          SMIN = MIN( SMIN, S( I ) )
        !           121:          AMAX = MAX( AMAX, S( I ) )
        !           122:    10 CONTINUE
        !           123: *
        !           124:       IF( SMIN.LE.ZERO ) THEN
        !           125: *
        !           126: *        Find the first non-positive diagonal element and return.
        !           127: *
        !           128:          DO 20 I = 1, N
        !           129:             IF( S( I ).LE.ZERO ) THEN
        !           130:                INFO = I
        !           131:                RETURN
        !           132:             END IF
        !           133:    20    CONTINUE
        !           134:       ELSE
        !           135: *
        !           136: *        Set the scale factors to the reciprocals
        !           137: *        of the diagonal elements.
        !           138: *
        !           139:          DO 30 I = 1, N
        !           140:             S( I ) = BASE ** INT( TMP * LOG( S( I ) ) )
        !           141:    30    CONTINUE
        !           142: *
        !           143: *        Compute SCOND = min(S(I)) / max(S(I)).
        !           144: *
        !           145:          SCOND = SQRT( SMIN ) / SQRT( AMAX )
        !           146:       END IF
        !           147: *
        !           148:       RETURN
        !           149: *
        !           150: *     End of DPOEQUB
        !           151: *
        !           152:       END