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NAME

        ZGBTRF  - compute an LU factorization of a complex m-by-n band matrix A
        using partial pivoting with row interchanges
 

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 ver‐
        sion 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.
        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 ele‐
        ments of U because of fill-in resulting from the row interchanges.
 

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