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NAME
     spbsv - compute the solution to  a  real  system  of  linear
     equations  A * X = B,

SYNOPSIS
     SUBROUTINE SPBSV( UPLO, N, KD, NRHS, AB, LDAB, B, LDB,  INFO
               )

     CHARACTER UPLO

     INTEGER INFO, KD, LDAB, LDB, N, NRHS

     REAL AB( LDAB, * ), B( LDB, * )



     #include <sunperf.h>

     void spbsv(char uplo, int n, int kd, int nrhs,  float  *sab,
               int ldab, float *sb, int ldb, int *info) ;

PURPOSE
     SPBSV computes the solution to a real system of linear equa-
     tions
        A * X = B, where A is an N-by-N symmetric positive defin-
     ite band matrix and X and B are N-by-NRHS matrices.

     The Cholesky decomposition is used to factor A as
        A = U**T * U,  if UPLO = 'U', or
        A = L * L**T,  if UPLO = 'L',
     where U is an upper triangular band matrix, and L is a lower
     triangular  band matrix, with the same number of superdiago-
     nals or subdiagonals as A.  The factored form of A  is  then
     used to solve the system of equations A * X = B.


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

     N         (input) INTEGER
               The number of linear equations, i.e., 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.

     NRHS      (input) INTEGER
               The number of right hand sides, i.e.,  the  number
               of columns of the matrix B.  NRHS >= 0.

     AB        (input/output) REAL array, dimension (LDAB,N)
               On entry, the upper or lower triangle of the  sym-
               metric  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).   See  below  for  further
               details.

               On exit, if INFO = 0, the triangular factor U or L
               from  the Cholesky factorization A = U**T*U or A =
               L*L**T of the band matrix A, in the  same  storage
               format as A.

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

     B         (input/output) REAL array, dimension (LDB,NRHS)
               On entry, the N-by-NRHS right hand side matrix  B.
               On  exit,  if  INFO  =  0,  the N-by-NRHS solution
               matrix X.

     LDB       (input) INTEGER
               The leading dimension of  the  array  B.   LDB  >=
               max(1,N).

     INFO      (output) INTEGER
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an ille-
               gal value
               > 0:  if INFO = i, the leading minor of order i of
               A  is  not positive definite, so the factorization
               could not be completed, and the solution  has  not
               been computed.

FURTHER DETAILS
     The band storage scheme  is  illustrated  by  the  following
     example, when N = 6, KD = 2, and UPLO = 'U':

     On entry:                       On exit:

         *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
         *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
        a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66

     Similarly, if UPLO = 'L' the format of A is as follows:

     On entry:                       On exit:
        a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
        a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
        a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *

     Array elements marked * are not used by the routine.

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