Annotation of rpl/lapack/lapack/dppequ.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DPPEQU( UPLO, N, AP, 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: CHARACTER UPLO
! 10: INTEGER INFO, N
! 11: DOUBLE PRECISION AMAX, SCOND
! 12: * ..
! 13: * .. Array Arguments ..
! 14: DOUBLE PRECISION AP( * ), S( * )
! 15: * ..
! 16: *
! 17: * Purpose
! 18: * =======
! 19: *
! 20: * DPPEQU computes row and column scalings intended to equilibrate a
! 21: * symmetric positive definite matrix A in packed storage and reduce
! 22: * its condition number (with respect to the two-norm). S contains the
! 23: * scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix
! 24: * B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
! 25: * This choice of S puts the condition number of B within a factor N of
! 26: * the smallest possible condition number over all possible diagonal
! 27: * scalings.
! 28: *
! 29: * Arguments
! 30: * =========
! 31: *
! 32: * UPLO (input) CHARACTER*1
! 33: * = 'U': Upper triangle of A is stored;
! 34: * = 'L': Lower triangle of A is stored.
! 35: *
! 36: * N (input) INTEGER
! 37: * The order of the matrix A. N >= 0.
! 38: *
! 39: * AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
! 40: * The upper or lower triangle of the symmetric matrix A, packed
! 41: * columnwise in a linear array. The j-th column of A is stored
! 42: * in the array AP as follows:
! 43: * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
! 44: * if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
! 45: *
! 46: * S (output) DOUBLE PRECISION array, dimension (N)
! 47: * If INFO = 0, S contains the scale factors for A.
! 48: *
! 49: * SCOND (output) DOUBLE PRECISION
! 50: * If INFO = 0, S contains the ratio of the smallest S(i) to
! 51: * the largest S(i). If SCOND >= 0.1 and AMAX is neither too
! 52: * large nor too small, it is not worth scaling by S.
! 53: *
! 54: * AMAX (output) DOUBLE PRECISION
! 55: * Absolute value of largest matrix element. If AMAX is very
! 56: * close to overflow or very close to underflow, the matrix
! 57: * should be scaled.
! 58: *
! 59: * INFO (output) INTEGER
! 60: * = 0: successful exit
! 61: * < 0: if INFO = -i, the i-th argument had an illegal value
! 62: * > 0: if INFO = i, the i-th diagonal element is nonpositive.
! 63: *
! 64: * =====================================================================
! 65: *
! 66: * .. Parameters ..
! 67: DOUBLE PRECISION ONE, ZERO
! 68: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
! 69: * ..
! 70: * .. Local Scalars ..
! 71: LOGICAL UPPER
! 72: INTEGER I, JJ
! 73: DOUBLE PRECISION SMIN
! 74: * ..
! 75: * .. External Functions ..
! 76: LOGICAL LSAME
! 77: EXTERNAL LSAME
! 78: * ..
! 79: * .. External Subroutines ..
! 80: EXTERNAL XERBLA
! 81: * ..
! 82: * .. Intrinsic Functions ..
! 83: INTRINSIC MAX, MIN, SQRT
! 84: * ..
! 85: * .. Executable Statements ..
! 86: *
! 87: * Test the input parameters.
! 88: *
! 89: INFO = 0
! 90: UPPER = LSAME( UPLO, 'U' )
! 91: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
! 92: INFO = -1
! 93: ELSE IF( N.LT.0 ) THEN
! 94: INFO = -2
! 95: END IF
! 96: IF( INFO.NE.0 ) THEN
! 97: CALL XERBLA( 'DPPEQU', -INFO )
! 98: RETURN
! 99: END IF
! 100: *
! 101: * Quick return if possible
! 102: *
! 103: IF( N.EQ.0 ) THEN
! 104: SCOND = ONE
! 105: AMAX = ZERO
! 106: RETURN
! 107: END IF
! 108: *
! 109: * Initialize SMIN and AMAX.
! 110: *
! 111: S( 1 ) = AP( 1 )
! 112: SMIN = S( 1 )
! 113: AMAX = S( 1 )
! 114: *
! 115: IF( UPPER ) THEN
! 116: *
! 117: * UPLO = 'U': Upper triangle of A is stored.
! 118: * Find the minimum and maximum diagonal elements.
! 119: *
! 120: JJ = 1
! 121: DO 10 I = 2, N
! 122: JJ = JJ + I
! 123: S( I ) = AP( JJ )
! 124: SMIN = MIN( SMIN, S( I ) )
! 125: AMAX = MAX( AMAX, S( I ) )
! 126: 10 CONTINUE
! 127: *
! 128: ELSE
! 129: *
! 130: * UPLO = 'L': Lower triangle of A is stored.
! 131: * Find the minimum and maximum diagonal elements.
! 132: *
! 133: JJ = 1
! 134: DO 20 I = 2, N
! 135: JJ = JJ + N - I + 2
! 136: S( I ) = AP( JJ )
! 137: SMIN = MIN( SMIN, S( I ) )
! 138: AMAX = MAX( AMAX, S( I ) )
! 139: 20 CONTINUE
! 140: END IF
! 141: *
! 142: IF( SMIN.LE.ZERO ) THEN
! 143: *
! 144: * Find the first non-positive diagonal element and return.
! 145: *
! 146: DO 30 I = 1, N
! 147: IF( S( I ).LE.ZERO ) THEN
! 148: INFO = I
! 149: RETURN
! 150: END IF
! 151: 30 CONTINUE
! 152: ELSE
! 153: *
! 154: * Set the scale factors to the reciprocals
! 155: * of the diagonal elements.
! 156: *
! 157: DO 40 I = 1, N
! 158: S( I ) = ONE / SQRT( S( I ) )
! 159: 40 CONTINUE
! 160: *
! 161: * Compute SCOND = min(S(I)) / max(S(I))
! 162: *
! 163: SCOND = SQRT( SMIN ) / SQRT( AMAX )
! 164: END IF
! 165: RETURN
! 166: *
! 167: * End of DPPEQU
! 168: *
! 169: END
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