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# zgbtf2

```
NAME
ZGBTF2 - compute an LU factorization of a complex m-by-n
band matrix A using partial pivoting with row interchanges

SYNOPSIS
SUBROUTINE ZGBTF2( M, N, KL, KU, AB, LDAB, IPIV, INFO )

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

INTEGER        IPIV( * )

COMPLEX*16     AB( LDAB, * )

PURPOSE
ZGBTF2 computes an LU factorization of a complex m-by-n band
matrix A using partial pivoting with row interchanges.

This is the unblocked version of the algorithm, calling
Level 2 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 super-
diagonals 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 fac-
torization 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; ele-
ments 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.
```