Developer Reference

Contents

?gbtrf

Computes the LU factorization of a general
m
-by-
n
band matrix.

Syntax

lapack_int
LAPACKE_sgbtrf
(
int
matrix_layout
lapack_int
m
lapack_int
n
lapack_int
kl
lapack_int
ku
float
*
ab
lapack_int
ldab
lapack_int
*
ipiv
);
lapack_int
LAPACKE_dgbtrf
(
int
matrix_layout
lapack_int
m
lapack_int
n
lapack_int
kl
lapack_int
ku
double
*
ab
lapack_int
ldab
lapack_int
*
ipiv
);
lapack_int
LAPACKE_cgbtrf
(
int
matrix_layout
lapack_int
m
lapack_int
n
lapack_int
kl
lapack_int
ku
lapack_complex_float
*
ab
lapack_int
ldab
lapack_int
*
ipiv
);
lapack_int
LAPACKE_zgbtrf
(
int
matrix_layout
lapack_int
m
lapack_int
n
lapack_int
kl
lapack_int
ku
lapack_complex_double
*
ab
lapack_int
ldab
lapack_int
*
ipiv
);
Include Files
  • mkl.h
Description
The routine forms the
LU
factorization of a general
m
-by-
n
band matrix
A
with
kl
non-zero subdiagonals and
ku
non-zero superdiagonals, that is,
A = P*L*U,
where
P
is a permutation matrix;
L
is lower triangular with unit diagonal elements and at most
kl
non-zero elements in each column;
U
is an upper triangular band matrix with
kl
+
ku
superdiagonals. The routine uses partial pivoting, with row interchanges (which creates the additional
kl
superdiagonals in
U
).
This routine supports the Progress Routine feature. See Progress Function for details.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
m
The number of rows in matrix
A
;
m
0.
n
The number of columns in matrix
A
;
n
0.
kl
The number of subdiagonals within the band of
A
;
kl
0.
ku
The number of superdiagonals within the band of
A
;
ku
0.
ab
Array, size
at least max(1,
ldab
*
n
) for column-major layout or max(1,
ldab
*
m
) for row-major layout
.
The array
ab
contains the matrix
A
in band storage as described in Band Storage.
ldab
The leading dimension of the array
ab
. (
ldab
2*
kl
+
ku
+ 1)
Output Parameters
ab
Overwritten with elements of
L
and
U
.
U
is stored as an upper triangular band matrix with
kl
+
ku
superdiagonals, and
L
is stored as a lower triangular band matrix with
kl
subdiagonals (diagonal unit values are not stored). Since the output array has more nonzero elements than the initial matrix
A
, there are limitations on the value of
ldab
and the placement of elements of
A
in array
ab
.
See Application Notes below for further details.
ipiv
Array, size at least
max(1,min(
m
,
n
))
. The pivot indices; for
1
i
min(
m
,
n
)
, row
i
was interchanged with row
ipiv
(
i
)
.
Return Values
This function returns a value
info
.
If
info
= 0
, the execution is successful.
If
info
=
-i
, parameter
i
had an illegal value.
If
info
=
i
,
u
i
i
is 0. The factorization has been completed, but
U
is exactly singular. Division by 0 will occur if you use the factor
U
for solving a system of linear equations.
Application Notes
The computed
L
and
U
are the exact factors of a perturbed matrix
A
+
E
, where
|E| ≤ c(kl+ku+1) ε P|L||U|
c
(
k
)
is a modest linear function of
k
, and
ε
is the machine precision.
The total number of floating-point operations for real flavors varies between approximately
2
n
(
ku
+1)
kl
and
2
n
(
kl
+
ku
+1)
kl
. The number of operations for complex flavors is four times greater. All these estimates assume that
kl
and
ku
are much less than
min(
m
,
n
)
.
As described in Band Storage, storage of a band matrix can be considered in two steps: packing band matrix elements into a matrix
AB
, then storing the elements in a linear array
ab
using a full storage scheme. The effect of the
?gbtrf
routine on matrix
AB
is illustrated by this example, for
m
=
n
= 6,
kl
= 2,
ku
= 1.
  • matrix_layout
    =
    LAPACK_COL_MAJOR
    On entry:
    On exit:
  • matrix_layout
    =
    LAPACK_ROW_MAJOR
    On entry:
    On exit:
Elements marked
*
are not used; 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.
After calling this routine with
m
=
n
, you can call the following routines:
to solve
A*X
=
B
or
A
T
*X
=
B
or
A
H
*X
=
B
to estimate the condition number of
A
.

Product and Performance Information

1

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.