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NAME
     sgbsv - compute the solution to  a  real  system  of  linear
     equations  A  *  X  = B, where A is a band matrix of order N
     with KL subdiagonals and KU superdiagonals, and X and B  are
     N-by-NRHS matrices

SYNOPSIS
     SUBROUTINE SGBSV( N, KL, KU, NRHS, AB, LDAB, IPIV,  B,  LDB,
               INFO )

     INTEGER INFO, KL, KU, LDAB, LDB, N, NRHS

     INTEGER IPIV( * )

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



     #include <sunperf.h>

     void sgbsv(int n, int kl, int ku, int nrhs, float *sab,  int
               ldab,  int *ipivot, float *sb, int ldb, int *info)
               ;

PURPOSE
     SGBSV computes the solution to a real system of linear equa-
     tions A * X = B, where A is a band matrix of order N with KL
     subdiagonals and KU superdiagonals, and X and  B  are  N-by-
     NRHS matrices.

     The LU decomposition with partial pivoting  and  row  inter-
     changes  is used to factor A as A = L * U, where L is a pro-
     duct of permutation and unit lower triangular matrices  with
     KL subdiagonals, and U is upper triangular with KL+KU super-
     diagonals.  The factored form of A is then used to solve the
     system of equations A * X = B.


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

     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 matrix A in band  storage,  in  rows
               KL+1  to 2*KL+KU+1; rows 1 to KL of the array need
               not be set.  The j-th column of A is stored in the
               j-th   column   of   the   array  AB  as  follows:
               AB(KL+KU+1+i-j,j)   =    A(i,j)    for    max(1,j-
               KU)<=i<=min(N,j+KL)  On  exit, details of the fac-
               torization: U is stored  as  an  upper  triangular
               band matrix with KL+KU superdiagonals in rows 1 to
               KL+KU+1, and the multipliers used during the  fac-
               torization   are   stored   in   rows  KL+KU+2  to
               2*KL+KU+1.  See below for further details.

     LDAB      (input) INTEGER
               The leading dimension of the array  AB.   LDAB  >=
               2*KL+KU+1.

     IPIV      (output) INTEGER array, dimension (N)
               The pivot  indices  that  define  the  permutation
               matrix  P;  row  i  of the matrix was interchanged
               with row IPIV(i).

     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, U(i,i) is  exactly  zero.   The
               factorization has been completed, but the factor U
               is exactly singular, and the solution has not been
               computed.

FURTHER DETAILS
     The band storage scheme  is  illustrated  by  the  following
     example, when M = N = 6, KL = 2, KU = 1:

     On entry:                       On exit:

         *   *   *   +   +   +           *   *   *  u14 u25 u36
         *   *   +   +   +   +           *   *  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
        a21 a32 a43 a54 a65  *          m21 m32 m43 m54 m65  *
        a31 a42 a53 a64  *   *          m31 m42 m53 m64  *   *

     Array elements marked * are not used by  the  routine;  ele-
     ments marked + need not be set on entry, but are required by
     the routine to  store  elements  of  U  because  of  fill-in
     resulting from the row interchanges.

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