ZGBTRF(3)      LAPACK routine of NEC Numeric Library Collection      ZGBTRF(3)



NAME
       ZGBTRF

SYNOPSIS
       SUBROUTINE ZGBTRF (M, N, KL, KU, AB, LDAB, IPIV, INFO)



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

           N         (input)
                     N is INTEGER
                     The number of columns of the matrix A.  N >= 0.

           KL        (input)
                     KL is INTEGER
                     The number of subdiagonals within the band of A.  KL >= 0.

           KU        (input)
                     KU is INTEGER
                     The number of superdiagonals within the band of A.  KU >= 0.

           AB        (input/output)
                     AB is 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)
                     LDAB is INTEGER
                     The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.

           IPIV      (output)
                     IPIV is 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)
                     INFO is 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.



LAPACK routine                  31 October 2017                      ZGBTRF(3)