sgbequ(3S)

NAME
     SGBEQU - compute row and column scalings intended to equilibrate an M-
     by-N band matrix A and reduce its condition number

SYNOPSIS
     SUBROUTINE SGBEQU( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX,
                        INFO )

         INTEGER        INFO, KL, KU, LDAB, M, N

         REAL           AMAX, COLCND, ROWCND

         REAL           AB( LDAB, * ), C( * ), R( * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     SGBEQU 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) REAL 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) REAL array, dimension (M)
             If INFO = 0, or INFO > M, R contains the row scale factors for A.

     C       (output) REAL array, dimension (N)
             If INFO = 0, C contains the column scale factors for A.

     ROWCND  (output) 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.

     COLCND  (output) 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.

     AMAX    (output) REAL
             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

SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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