Annotation of rpl/lapack/lapack/dpbequ.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DPBEQU( UPLO, N, KD, AB, LDAB, 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, KD, LDAB, N
! 11: DOUBLE PRECISION AMAX, SCOND
! 12: * ..
! 13: * .. Array Arguments ..
! 14: DOUBLE PRECISION AB( LDAB, * ), S( * )
! 15: * ..
! 16: *
! 17: * Purpose
! 18: * =======
! 19: *
! 20: * DPBEQU computes row and column scalings intended to equilibrate a
! 21: * symmetric positive definite band matrix A and reduce its condition
! 22: * number (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: * UPLO (input) CHARACTER*1
! 33: * = 'U': Upper triangular of A is stored;
! 34: * = 'L': Lower triangular of A is stored.
! 35: *
! 36: * N (input) INTEGER
! 37: * The order of the matrix A. N >= 0.
! 38: *
! 39: * KD (input) INTEGER
! 40: * The number of superdiagonals of the matrix A if UPLO = 'U',
! 41: * or the number of subdiagonals if UPLO = 'L'. KD >= 0.
! 42: *
! 43: * AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
! 44: * The upper or lower triangle of the symmetric band matrix A,
! 45: * stored in the first KD+1 rows of the array. The j-th column
! 46: * of A is stored in the j-th column of the array AB as follows:
! 47: * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
! 48: * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
! 49: *
! 50: * LDAB (input) INTEGER
! 51: * The leading dimension of the array A. LDAB >= KD+1.
! 52: *
! 53: * S (output) DOUBLE PRECISION array, dimension (N)
! 54: * If INFO = 0, S contains the scale factors for A.
! 55: *
! 56: * SCOND (output) DOUBLE PRECISION
! 57: * If INFO = 0, S contains the ratio of the smallest S(i) to
! 58: * the largest S(i). If SCOND >= 0.1 and AMAX is neither too
! 59: * large nor too small, it is not worth scaling by S.
! 60: *
! 61: * AMAX (output) DOUBLE PRECISION
! 62: * Absolute value of largest matrix element. If AMAX is very
! 63: * close to overflow or very close to underflow, the matrix
! 64: * should be scaled.
! 65: *
! 66: * INFO (output) INTEGER
! 67: * = 0: successful exit
! 68: * < 0: if INFO = -i, the i-th argument had an illegal value.
! 69: * > 0: if INFO = i, the i-th diagonal element is nonpositive.
! 70: *
! 71: * =====================================================================
! 72: *
! 73: * .. Parameters ..
! 74: DOUBLE PRECISION ZERO, ONE
! 75: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 76: * ..
! 77: * .. Local Scalars ..
! 78: LOGICAL UPPER
! 79: INTEGER I, J
! 80: DOUBLE PRECISION SMIN
! 81: * ..
! 82: * .. External Functions ..
! 83: LOGICAL LSAME
! 84: EXTERNAL LSAME
! 85: * ..
! 86: * .. External Subroutines ..
! 87: EXTERNAL XERBLA
! 88: * ..
! 89: * .. Intrinsic Functions ..
! 90: INTRINSIC MAX, MIN, SQRT
! 91: * ..
! 92: * .. Executable Statements ..
! 93: *
! 94: * Test the input parameters.
! 95: *
! 96: INFO = 0
! 97: UPPER = LSAME( UPLO, 'U' )
! 98: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
! 99: INFO = -1
! 100: ELSE IF( N.LT.0 ) THEN
! 101: INFO = -2
! 102: ELSE IF( KD.LT.0 ) THEN
! 103: INFO = -3
! 104: ELSE IF( LDAB.LT.KD+1 ) THEN
! 105: INFO = -5
! 106: END IF
! 107: IF( INFO.NE.0 ) THEN
! 108: CALL XERBLA( 'DPBEQU', -INFO )
! 109: RETURN
! 110: END IF
! 111: *
! 112: * Quick return if possible
! 113: *
! 114: IF( N.EQ.0 ) THEN
! 115: SCOND = ONE
! 116: AMAX = ZERO
! 117: RETURN
! 118: END IF
! 119: *
! 120: IF( UPPER ) THEN
! 121: J = KD + 1
! 122: ELSE
! 123: J = 1
! 124: END IF
! 125: *
! 126: * Initialize SMIN and AMAX.
! 127: *
! 128: S( 1 ) = AB( J, 1 )
! 129: SMIN = S( 1 )
! 130: AMAX = S( 1 )
! 131: *
! 132: * Find the minimum and maximum diagonal elements.
! 133: *
! 134: DO 10 I = 2, N
! 135: S( I ) = AB( J, I )
! 136: SMIN = MIN( SMIN, S( I ) )
! 137: AMAX = MAX( AMAX, S( I ) )
! 138: 10 CONTINUE
! 139: *
! 140: IF( SMIN.LE.ZERO ) THEN
! 141: *
! 142: * Find the first non-positive diagonal element and return.
! 143: *
! 144: DO 20 I = 1, N
! 145: IF( S( I ).LE.ZERO ) THEN
! 146: INFO = I
! 147: RETURN
! 148: END IF
! 149: 20 CONTINUE
! 150: ELSE
! 151: *
! 152: * Set the scale factors to the reciprocals
! 153: * of the diagonal elements.
! 154: *
! 155: DO 30 I = 1, N
! 156: S( I ) = ONE / SQRT( S( I ) )
! 157: 30 CONTINUE
! 158: *
! 159: * Compute SCOND = min(S(I)) / max(S(I))
! 160: *
! 161: SCOND = SQRT( SMIN ) / SQRT( AMAX )
! 162: END IF
! 163: RETURN
! 164: *
! 165: * End of DPBEQU
! 166: *
! 167: END
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