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Annotation of rpl/lapack/lapack/zpoequ.f, revision 1.7

1.1       bertrand    1:       SUBROUTINE ZPOEQU( N, A, LDA, S, SCOND, AMAX, INFO )
                      2: *
                      3: *  -- LAPACK routine (version 3.2) --
                      4: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
                      5: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
                      6: *     November 2006
                      7: *
                      8: *     .. Scalar Arguments ..
                      9:       INTEGER            INFO, LDA, N
                     10:       DOUBLE PRECISION   AMAX, SCOND
                     11: *     ..
                     12: *     .. Array Arguments ..
                     13:       DOUBLE PRECISION   S( * )
                     14:       COMPLEX*16         A( LDA, * )
                     15: *     ..
                     16: *
                     17: *  Purpose
                     18: *  =======
                     19: *
                     20: *  ZPOEQU computes row and column scalings intended to equilibrate a
                     21: *  Hermitian positive definite matrix A and reduce its condition number
                     22: *  (with respect to the two-norm).  S contains the scale factors,
                     23: *  S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
                     24: *  elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
                     25: *  choice of S puts the condition number of B within a factor N of the
                     26: *  smallest possible condition number over all possible diagonal
                     27: *  scalings.
                     28: *
                     29: *  Arguments
                     30: *  =========
                     31: *
                     32: *  N       (input) INTEGER
                     33: *          The order of the matrix A.  N >= 0.
                     34: *
                     35: *  A       (input) COMPLEX*16 array, dimension (LDA,N)
                     36: *          The N-by-N Hermitian positive definite matrix whose scaling
                     37: *          factors are to be computed.  Only the diagonal elements of A
                     38: *          are referenced.
                     39: *
                     40: *  LDA     (input) INTEGER
                     41: *          The leading dimension of the array A.  LDA >= max(1,N).
                     42: *
                     43: *  S       (output) DOUBLE PRECISION array, dimension (N)
                     44: *          If INFO = 0, S contains the scale factors for A.
                     45: *
                     46: *  SCOND   (output) DOUBLE PRECISION
                     47: *          If INFO = 0, S contains the ratio of the smallest S(i) to
                     48: *          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
                     49: *          large nor too small, it is not worth scaling by S.
                     50: *
                     51: *  AMAX    (output) DOUBLE PRECISION
                     52: *          Absolute value of largest matrix element.  If AMAX is very
                     53: *          close to overflow or very close to underflow, the matrix
                     54: *          should be scaled.
                     55: *
                     56: *  INFO    (output) INTEGER
                     57: *          = 0:  successful exit
                     58: *          < 0:  if INFO = -i, the i-th argument had an illegal value
                     59: *          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
                     60: *
                     61: *  =====================================================================
                     62: *
                     63: *     .. Parameters ..
                     64:       DOUBLE PRECISION   ZERO, ONE
                     65:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
                     66: *     ..
                     67: *     .. Local Scalars ..
                     68:       INTEGER            I
                     69:       DOUBLE PRECISION   SMIN
                     70: *     ..
                     71: *     .. External Subroutines ..
                     72:       EXTERNAL           XERBLA
                     73: *     ..
                     74: *     .. Intrinsic Functions ..
                     75:       INTRINSIC          DBLE, MAX, MIN, SQRT
                     76: *     ..
                     77: *     .. Executable Statements ..
                     78: *
                     79: *     Test the input parameters.
                     80: *
                     81:       INFO = 0
                     82:       IF( N.LT.0 ) THEN
                     83:          INFO = -1
                     84:       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
                     85:          INFO = -3
                     86:       END IF
                     87:       IF( INFO.NE.0 ) THEN
                     88:          CALL XERBLA( 'ZPOEQU', -INFO )
                     89:          RETURN
                     90:       END IF
                     91: *
                     92: *     Quick return if possible
                     93: *
                     94:       IF( N.EQ.0 ) THEN
                     95:          SCOND = ONE
                     96:          AMAX = ZERO
                     97:          RETURN
                     98:       END IF
                     99: *
                    100: *     Find the minimum and maximum diagonal elements.
                    101: *
                    102:       S( 1 ) = DBLE( A( 1, 1 ) )
                    103:       SMIN = S( 1 )
                    104:       AMAX = S( 1 )
                    105:       DO 10 I = 2, N
                    106:          S( I ) = DBLE( A( I, I ) )
                    107:          SMIN = MIN( SMIN, S( I ) )
                    108:          AMAX = MAX( AMAX, S( I ) )
                    109:    10 CONTINUE
                    110: *
                    111:       IF( SMIN.LE.ZERO ) THEN
                    112: *
                    113: *        Find the first non-positive diagonal element and return.
                    114: *
                    115:          DO 20 I = 1, N
                    116:             IF( S( I ).LE.ZERO ) THEN
                    117:                INFO = I
                    118:                RETURN
                    119:             END IF
                    120:    20    CONTINUE
                    121:       ELSE
                    122: *
                    123: *        Set the scale factors to the reciprocals
                    124: *        of the diagonal elements.
                    125: *
                    126:          DO 30 I = 1, N
                    127:             S( I ) = ONE / SQRT( S( I ) )
                    128:    30    CONTINUE
                    129: *
                    130: *        Compute SCOND = min(S(I)) / max(S(I))
                    131: *
                    132:          SCOND = SQRT( SMIN ) / SQRT( AMAX )
                    133:       END IF
                    134:       RETURN
                    135: *
                    136: *     End of ZPOEQU
                    137: *
                    138:       END