SGBTF2 - compute an LU factorization of a real m-by-n band
matrix A using partial pivoting with row interchanges
SYNOPSIS
SUBROUTINE SGBTF2( M, N, KL, KU, AB, LDAB, IPIV, INFO )
INTEGER INFO, KL, KU, LDAB, M, N
INTEGER IPIV( * )
REAL AB( LDAB, * )
PURPOSE
SGBTF2 computes an LU factorization of a real 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) REAL 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 mul
tipliers used during the factorization are stored
in rows KL+KU+2 to 2*KL+KU+1. See below for fur
ther details.
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >=
2*KL+KU+1.
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 ille
gal 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 equa
tions.
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.
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