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Annotation of rpl/lapack/lapack/dpoequb.f, revision 1.3

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