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