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sgbsv

 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, * )

 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 factorization: U is stored as an
              upper triangular band matrix with KL+KU superdiago-
              nals in rows 1 to KL+KU+1, and the multipliers used
              during the factorization 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 illegal
              value
              > 0:  if INFO = i, U(i,i) is exactly zero.  The fac-
              torization has been completed, but the factor U is
              exactly singular, and the solution has not been com-
              puted.

 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.