dgbtrf man page on Scientific

```DGBTRF(1)		 LAPACK routine (version 3.2)		     DGBTRF(1)

NAME
DGBTRF  -  computes  an LU factorization of a real m-by-n band matrix A
using partial pivoting with row interchanges

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

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

INTEGER	  IPIV( * )

DOUBLE	  PRECISION AB( LDAB, * )

PURPOSE
DGBTRF computes an LU factorization of a	 real  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) DOUBLE PRECISION 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 superdiag‐
onals 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.

LAPACK routine (version 3.2)	 November 2008			     DGBTRF(1)
```
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