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NAME

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

SYNOPSIS

        SUBROUTINE ZGBEQU( M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,  AMAX,
                           INFO )
 
            INTEGER        INFO, KL, KU, LDAB, M, N
 
            DOUBLE         PRECISION AMAX, COLCND, ROWCND
 
            DOUBLE         PRECISION C( * ), R( * )
 
            COMPLEX*16     AB( LDAB, * )
 

PURPOSE

        ZGBEQU  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 num‐
        ber 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) COMPLEX*16 array, dimension (LDAB,N)
                The band matrix A, stored in rows 1 to KL+KU+1.  The j-th  col‐
                umn  of  A is stored in the j-th column of the array AB as fol‐
                lows: 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
 

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