SYNOPSIS

SUBROUTINE ZGBTRF(

M, N, KL, KU, AB, LDAB, IPIV, INFO )

    

INTEGER INFO, KL, KU, LDAB, M, N

    

INTEGER IPIV( * )

    

COMPLEX*16 AB( LDAB, * )

PURPOSE

ZGBTRF computes an LU factorization of a complex m-by-n band matrix A using partial pivoting with row interchanges. This is the blocked version of the algorithm, calling Level 3 BLAS.

ARGUMENTS

M (input) INTEGER

The number of rows of the matrix A. M >= 0.

N (input) INTEGER

The number of columns 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.

AB (input/output) COMPLEX*16 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(m,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 (min(M,N))

The pivot indices; for 1 <= i <= min(M,N), row i of the matrix was interchanged with row IPIV(i).

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 division by zero will occur if it is used to solve a system of equations.

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.