SYNOPSIS

SUBROUTINE CGBSV(

N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO )

    

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

    

INTEGER IPIV( * )

    

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

PURPOSE

CGBSV computes the solution to a complex 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) COMPLEX 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) COMPLEX 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.