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
SUBROUTINE DGBTRF( M, N, KL, KU, AB, LDAB, IPIV, INFO )
INTEGER INFO, KL, KU, LDAB, M, N
INTEGER IPIV( * )
DOUBLE PRECISION AB( LDAB, * )
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.
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
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)