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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