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NAME
     spbequ - compute row and column scalings intended to equili-
     brate a symmetric positive definite band matrix A and reduce
     its condition number (with respect to the two-norm)

SYNOPSIS
     SUBROUTINE SPBEQU( UPLO, N, KD, AB, LDAB,  S,  SCOND,  AMAX,
               INFO )

     CHARACTER UPLO

     INTEGER INFO, KD, LDAB, N

     REAL AMAX, SCOND

     REAL AB( LDAB, * ), S( * )



     #include <sunperf.h>

     void spbequ(char uplo, int n, int kd, float *sab, int  ldab,
               float *s, float *scond, float *amax, int *info) ;

PURPOSE
     SPBEQU computes row and column scalings intended to  equili-
     brate a symmetric positive definite band matrix A and reduce
     its condition number (with respect to the two-norm).  S con-
     tains  the  scale  factors, S(i) = 1/sqrt(A(i,i)), chosen so
     that  the  scaled  matrix   B   with   elements   B(i,j)   =
     S(i)*A(i,j)*S(j) has ones on the diagonal.  This choice of S
     puts the condition number of B within  a  factor  N  of  the
     smallest  possible condition number over all possible diago-
     nal scalings.


ARGUMENTS
     UPLO      (input) CHARACTER*1
               = 'U':  Upper triangular of A is stored;
               = 'L':  Lower triangular of A is stored.

     N         (input) INTEGER
               The order of the matrix A.  N >= 0.

     KD        (input) INTEGER
               The number of superdiagonals of the  matrix  A  if
               UPLO  = 'U', or the number of subdiagonals if UPLO
               = 'L'.  KD >= 0.

     AB        (input) REAL array, dimension (LDAB,N)
               The upper or lower triangle of the symmetric  band
               matrix  A,  stored  in  the first KD+1 rows of the
               array.  The j-th column of A is stored in the j-th
               column of the array AB as follows:  if UPLO = 'U',
               AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;  if
               UPLO   =   'L',   AB(1+i-j,j)      =   A(i,j)  for
               j<=i<=min(n,j+kd).

     LDAB      (input) INTEGER
               The leading dimension of the  array  A.   LDAB  >=
               KD+1.

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

     SCOND     (output) REAL
               If INFO = 0, S contains the ratio of the  smallest
               S(i)  to  the  largest  S(i).  If SCOND >= 0.1 and
               AMAX is neither too large nor too small, it is not
               worth scaling by S.

     AMAX      (output) REAL
               Absolute value of largest matrix element.  If AMAX
               is  very close to overflow or very close to under-
               flow, the matrix should be scaled.

     INFO      (output) INTEGER
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an ille-
               gal value.
               > 0:  if INFO = i, the i-th  diagonal  element  is
               nonpositive.

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