sgbequb()

subroutine sgbequb	(	integer	m,
		integer	n,
		integer	kl,
		integer	ku,
		real, dimension( ldab, * )	ab,
		integer	ldab,
		real, dimension( * )	r,
		real, dimension( * )	c,
		real	rowcnd,
		real	colcnd,
		real	amax,
		integer	info )

SGBEQUB

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Purpose:

!>
!> SGBEQUB computes row and column scalings intended to equilibrate an
!> M-by-N 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 an absolute value of at most
!> the radix.
!>
!> R(i) and C(j) are restricted to be a power of the radix 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.
!>
!> This routine differs from SGEEQU by restricting the scaling factors
!> to a power of the radix.  Barring over- and underflow, scaling by
!> these factors introduces no additional rounding errors.  However, the
!> scaled entries' magnitudes are no longer approximately 1 but lie
!> between sqrt(radix) and 1/sqrt(radix).
!> 

Parameters

[in]	M	!> M is INTEGER !> The number of rows of the matrix A. M >= 0. !>
[in]	N	!> N is INTEGER !> The number of columns of the matrix A. N >= 0. !>
[in]	KL	!> KL is INTEGER !> The number of subdiagonals within the band of A. KL >= 0. !>
[in]	KU	!> KU is INTEGER !> The number of superdiagonals within the band of A. KU >= 0. !>
[in]	AB	!> AB is REAL array, dimension (LDAB,N) !> On entry, the matrix A in band storage, 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(N,j+kl) !>
[in]	LDAB	!> LDAB is INTEGER !> The leading dimension of the array A. LDAB >= max(1,M). !>
[out]	R	!> R is REAL array, dimension (M) !> If INFO = 0 or INFO > M, R contains the row scale factors !> for A. !>
[out]	C	!> C is REAL array, dimension (N) !> If INFO = 0, C contains the column scale factors for A. !>
[out]	ROWCND	!> ROWCND is REAL !> 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. !>
[out]	COLCND	!> COLCND is REAL !> 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. !>
[out]	AMAX	!> AMAX is REAL !> Absolute value of largest matrix element. If AMAX is very !> close to overflow or very close to underflow, the matrix !> should be scaled. !>
[out]	INFO	!> INFO is 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 !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 156 of file sgbequb.f.

*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, KL, KU, LDAB, M, N
      REAL               AMAX, COLCND, ROWCND
*     ..
*     .. Array Arguments ..
      REAL               AB( LDAB, * ), C( * ), R( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ONE, ZERO
      parameter( one = 1.0e+0, zero = 0.0e+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J, KD
      REAL               BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX, LOGRDX
*     ..
*     .. External Functions ..
      REAL               SLAMCH
      EXTERNAL           slamch
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, min, log
*     ..
*     .. 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( 'SGBEQUB', -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.  Assume SMLNUM is a power of the radix.
*
      smlnum = slamch( 'S' )
      bignum = one / smlnum
      radix = slamch( 'B' )
      logrdx = log(radix)
*
*     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
      DO i = 1, m
         IF( r( i ).GT.zero ) THEN
            r( i ) = radix**int( log( r( i ) ) / logrdx )
         END IF
      END DO
*
*     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.
*
      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
         IF( c( j ).GT.zero ) THEN
            c( j ) = radix**int( log( c( j ) ) / logrdx )
         END IF
   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 SGBEQUB
*

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