Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

  • 2021.1
  • 12/04/2020
  • Public Content
Contents

?gbtrs

Solves a system of linear equations with an LU-factored band coefficient matrix, with multiple right-hand sides.

Syntax

lapack_int
LAPACKE_sgbtrs
(
int
matrix_layout
,
char
trans
,
lapack_int
n
,
lapack_int
kl
,
lapack_int
ku
,
lapack_int
nrhs
,
const
float
*
ab
,
lapack_int
ldab
,
const
lapack_int
*
ipiv
,
float
*
b
,
lapack_int
ldb
);
lapack_int
LAPACKE_dgbtrs
(
int
matrix_layout
,
char
trans
,
lapack_int
n
,
lapack_int
kl
,
lapack_int
ku
,
lapack_int
nrhs
,
const
double
*
ab
,
lapack_int
ldab
,
const
lapack_int
*
ipiv
,
double
*
b
,
lapack_int
ldb
);
lapack_int
LAPACKE_cgbtrs
(
int
matrix_layout
,
char
trans
,
lapack_int
n
,
lapack_int
kl
,
lapack_int
ku
,
lapack_int
nrhs
,
const
lapack_complex_float
*
ab
,
lapack_int
ldab
,
const
lapack_int
*
ipiv
,
lapack_complex_float
*
b
,
lapack_int
ldb
);
lapack_int
LAPACKE_zgbtrs
(
int
matrix_layout
,
char
trans
,
lapack_int
n
,
lapack_int
kl
,
lapack_int
ku
,
lapack_int
nrhs
,
const
lapack_complex_double
*
ab
,
lapack_int
ldab
,
const
lapack_int
*
ipiv
,
lapack_complex_double
*
b
,
lapack_int
ldb
);
Include Files
  • mkl.h
Description
The routine solves for
X
 the following systems of linear equations:
A*X
 =
B
if
trans
=
'N'
,
A
T
*X
 =
B
if
trans
=
'T'
,
A
H
*X
 =
B
if
trans
=
'C'
 (for complex matrices only).
Here
A
 is an
LU
-factored general band matrix of order
n
 with
kl
 non-zero subdiagonals and
ku
 nonzero superdiagonals. Before calling this routine, call
?gbtrf
 to compute the
LU
 factorization of
A
.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
trans
Must be
'N'
 or
'T'
 or
'C'
.
n
The order of
A
; the number of rows in
B
;
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.
nrhs
The number of right-hand sides;
nrhs
≥
 0.
ab
Array
ab
 size max(1,
ldab
*
n
)
The array
ab
 contains elements of the LU factors of the matrix
A
 as returned by gbtrf.
b
Array
b
 size max(1,
ldb
*
nrhs
) for column major layout and max(1,
ldb
*
n
) for row major layout.
The array
b
 contains the matrix
B
 whose columns are the right-hand sides for the systems of equations.
ldab
The leading dimension of the array
ab
;
ldab
≥
 2*
kl
 +
ku
 +1.
ldb
The leading dimension of
b
;
ldb
≥
 max(1,
n
) for column major layout and
ldb
≥
nrhs
 for row major layout
.
ipiv
Array, size at least
max(1,
n
)
. The
ipiv
 array, as returned by
?gbtrf
.
Output Parameters
b
Overwritten by the solution matrix
X
.
Return Values
This function returns a value
info
.
If
info
=0
, the execution is successful.
If
info
 =
-i
, parameter
i
 had an illegal value.
Application Notes
For each right-hand side
b
, the computed solution is the exact solution of a perturbed system of equations
(
A
 +
E
)
x
 =
b
, 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.
If
x
0
 is the true solution, the computed solution
x
 satisfies this error bound:
Equation
where
cond(
A
,
x
)
= || |
A
-1
||
A
| |
x
| ||
∞
 / ||
x
||
∞
≤
 ||
A
-1
||
∞
 ||
A
||
∞
 =
κ
∞
(
A
).
Note that
cond(
A
,
x
)
 can be much smaller than
κ
∞
(
A
)
; the condition number of
A
T
 and
A
H
 might or might not be equal to
κ
∞
(
A
)
.
The approximate number of floating-point operations for one right-hand side vector is 2
n
(
ku
 + 2
kl
) for real flavors. The number of operations for complex flavors is 4 times greater. All these estimates assume that
kl
 and
ku
 are much less than min(
m
,
n
).
To estimate the condition number
κ
∞
(
A
)
, call
?gbcon
.
To refine the solution and estimate the error, call
?gbrfs
.

Product and Performance Information

1

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