dgbequ.f

          SUBROUTINE DGBEQU( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
         $                   AMAX, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            INFO, KL, KU, LDAB, M, N
          DOUBLE PRECISION   AMAX, COLCND, ROWCND
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   AB( LDAB, * ), C( * ), R( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DGBEQU computes row and column scalings intended to equilibrate an
    *  M-by-N band matrix A and reduce its condition number.  R returns the
    *  row scale factors and C the column scale factors, chosen to try to
    *  make the largest element in each row and column of the matrix B with
    *  elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.
    *
    *  R(i) and C(j) are restricted to be between SMLNUM = smallest safe
    *  number and BIGNUM = largest safe number.  Use of these scaling
    *  factors is not guaranteed to reduce the condition number of A but
    *  works well in practice.
    *
    *  Arguments
    *  =========
    *
    *  M       (input) INTEGER
    *          The number of rows of the matrix A.  M >= 0.
    *
    *  N       (input) INTEGER
    *          The number of columns of the matrix A.  N >= 0.
    *
    *  KL      (input) INTEGER
    *          The number of subdiagonals within the band of A.  KL >= 0.
    *
    *  KU      (input) INTEGER
    *          The number of superdiagonals within the band of A.  KU >= 0.
    *
    *  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
    *          The band matrix A, stored in rows 1 to KL+KU+1.  The j-th
    *          column of A is stored in the j-th column of the array AB as
    *          follows:
    *          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
    *
    *  LDAB    (input) INTEGER
    *          The leading dimension of the array AB.  LDAB >= KL+KU+1.
    *
    *  R       (output) DOUBLE PRECISION array, dimension (M)
    *          If INFO = 0, or INFO > M, R contains the row scale factors
    *          for A.
    *
    *  C       (output) DOUBLE PRECISION array, dimension (N)
    *          If INFO = 0, C contains the column scale factors for A.
    *
    *  ROWCND  (output) DOUBLE PRECISION
    *          If INFO = 0 or INFO > M, ROWCND contains the ratio of the
    *          smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
    *          AMAX is neither too large nor too small, it is not worth
    *          scaling by R.
    *
    *  COLCND  (output) DOUBLE PRECISION
    *          If INFO = 0, COLCND contains the ratio of the smallest
    *          C(i) to the largest C(i).  If COLCND >= 0.1, it is not
    *          worth scaling by C.
    *
    *  AMAX    (output) DOUBLE PRECISION
    *          Absolute value of largest matrix element.  If AMAX is very
    *          close to overflow or very close to underflow, the matrix
    *          should be scaled.
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          < 0:  if INFO = -i, the i-th argument had an illegal value
    *          > 0:  if INFO = i, and i is
    *                <= M:  the i-th row of A is exactly zero
    *                >  M:  the (i-M)-th column of A is exactly zero
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ONE, ZERO
          PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
    *     ..
    *     .. Local Scalars ..
          INTEGER            I, J, KD
          DOUBLE PRECISION   BIGNUM, RCMAX, RCMIN, SMLNUM
    *     ..
    *     .. External Functions ..
          DOUBLE PRECISION   DLAMCH
          EXTERNAL           DLAMCH
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, MAX, MIN
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters
    *
          INFO = 0
          IF( M.LT.0 ) THEN
             INFO = -1
          ELSE IF( N.LT.0 ) THEN
             INFO = -2
          ELSE IF( KL.LT.0 ) THEN
             INFO = -3
          ELSE IF( KU.LT.0 ) THEN
             INFO = -4
          ELSE IF( LDAB.LT.KL+KU+1 ) THEN
             INFO = -6
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'DGBEQU', -INFO )
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( M.EQ.0 .OR. N.EQ.0 ) THEN
             ROWCND = ONE
             COLCND = ONE
             AMAX = ZERO
             RETURN
          END IF
    *
    *     Get machine constants.
    *
          SMLNUM = DLAMCH( 'S' )
          BIGNUM = ONE / SMLNUM
    *
    *     Compute row scale factors.
    *
          DO 10 I = 1, M
             R( I ) = ZERO
       10 CONTINUE
    *
    *     Find the maximum element in each row.
    *
          KD = KU + 1
          DO 30 J = 1, N
             DO 20 I = MAX( J-KU, 1 ), MIN( J+KL, M )
                R( I ) = MAX( R( I ), ABS( AB( KD+I-J, J ) ) )
       20    CONTINUE
       30 CONTINUE
    *
    *     Find the maximum and minimum scale factors.
    *
          RCMIN = BIGNUM
          RCMAX = ZERO
          DO 40 I = 1, M
             RCMAX = MAX( RCMAX, R( I ) )
             RCMIN = MIN( RCMIN, R( I ) )
       40 CONTINUE
          AMAX = RCMAX
    *
          IF( RCMIN.EQ.ZERO ) THEN
    *
    *        Find the first zero scale factor and return an error code.
    *
             DO 50 I = 1, M
                IF( R( I ).EQ.ZERO ) THEN
                   INFO = I
                   RETURN
                END IF
       50    CONTINUE
          ELSE
    *
    *        Invert the scale factors.
    *
             DO 60 I = 1, M
                R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
       60    CONTINUE
    *
    *        Compute ROWCND = min(R(I)) / max(R(I))
    *
             ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
          END IF
    *
    *     Compute column scale factors
    *
          DO 70 J = 1, N
             C( J ) = ZERO
       70 CONTINUE
    *
    *     Find the maximum element in each column,
    *     assuming the row scaling computed above.
    *
          KD = KU + 1
          DO 90 J = 1, N
             DO 80 I = MAX( J-KU, 1 ), MIN( J+KL, M )
                C( J ) = MAX( C( J ), ABS( AB( KD+I-J, J ) )*R( I ) )
       80    CONTINUE
       90 CONTINUE
    *
    *     Find the maximum and minimum scale factors.
    *
          RCMIN = BIGNUM
          RCMAX = ZERO
          DO 100 J = 1, N
             RCMIN = MIN( RCMIN, C( J ) )
             RCMAX = MAX( RCMAX, C( J ) )
      100 CONTINUE
    *
          IF( RCMIN.EQ.ZERO ) THEN
    *
    *        Find the first zero scale factor and return an error code.
    *
             DO 110 J = 1, N
                IF( C( J ).EQ.ZERO ) THEN
                   INFO = M + J
                   RETURN
                END IF
      110    CONTINUE
          ELSE
    *
    *        Invert the scale factors.
    *
             DO 120 J = 1, N
                C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
      120    CONTINUE
    *
    *        Compute COLCND = min(C(J)) / max(C(J))
    *
             COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
          END IF
    *
          RETURN
    *
    *     End of DGBEQU
    *
          END