dgbsv(3F)

NAME
     DGBSV - 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 DGBSV( N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO )

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

         INTEGER       IPIV( * )

         DOUBLE        PRECISION AB( LDAB, * ), B( LDB, * )

PURPOSE
     DGBSV computes 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.

     The LU decomposition with partial pivoting and row interchanges is used
     to factor A as A = L * U, where L is a product of permutation and unit
     lower triangular matrices with KL subdiagonals, and U is upper triangular
     with KL+KU superdiagonals.  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) DOUBLE PRECISION 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
             superdiagonals 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) DOUBLE PRECISION 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 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; elements 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.