Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

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

?tbtrs

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

Syntax

lapack_int
LAPACKE_stbtrs
(
int
matrix_layout
,
char
uplo
,
char
trans
,
char
diag
,
lapack_int
n
,
lapack_int
kd
,
lapack_int
nrhs
,
const
float
*
ab
,
lapack_int
ldab
,
float
*
b
,
lapack_int
ldb
);
lapack_int
LAPACKE_dtbtrs
(
int
matrix_layout
,
char
uplo
,
char
trans
,
char
diag
,
lapack_int
n
,
lapack_int
kd
,
lapack_int
nrhs
,
const
double
*
ab
,
lapack_int
ldab
,
double
*
b
,
lapack_int
ldb
);
lapack_int
LAPACKE_ctbtrs
(
int
matrix_layout
,
char
uplo
,
char
trans
,
char
diag
,
lapack_int
n
,
lapack_int
kd
,
lapack_int
nrhs
,
const
lapack_complex_float
*
ab
,
lapack_int
ldab
,
lapack_complex_float
*
b
,
lapack_int
ldb
);
lapack_int
LAPACKE_ztbtrs
(
int
matrix_layout
,
char
uplo
,
char
trans
,
char
diag
,
lapack_int
n
,
lapack_int
kd
,
lapack_int
nrhs
,
const
lapack_complex_double
*
ab
,
lapack_int
ldab
,
lapack_complex_double
*
b
,
lapack_int
ldb
);
Include Files
  • mkl.h
Description
The routine solves for
X
 the following systems of linear equations with a band triangular matrix
A
, with multiple right-hand sides stored in
B
:
A*X
 =
B
if
trans
=
'N'
,
A
T
*X
 =
B
if
trans
=
'T'
,
A
H
*X
 =
B
if
trans
=
'C'
 (for complex matrices only).
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
uplo
Must be
'U'
 or
'L'
.
Indicates whether
A
 is upper or lower triangular:
If
uplo
 =
'U'
, then
A
 is upper triangular.
If
uplo
 =
'L'
, then
A
 is lower triangular.
trans
Must be
'N'
 or
'T'
 or
'C'
.
If
trans
 =
'N'
, then
A*X
 =
B
 is solved for
X
.
If
trans
 =
'T'
, then
A
T
*X
 =
B
 is solved for
X
.
If
trans
 =
'C'
, then
A
H
*X
 =
B
 is solved for
X
.
diag
Must be
'N'
 or
'U'
.
If
diag
 =
'N'
, then
A
 is not a unit triangular matrix.
If
diag
 =
'U'
, then
A
 is unit triangular: diagonal elements are assumed to be 1 and not referenced in the array
ab
.
n
The order of
A
; the number of rows in
B
;
n
≥
 0.
kd
The number of superdiagonals or subdiagonals in the matrix
A
;
kd
≥
 0.
nrhs
The number of right-hand sides;
nrhs
≥
 0.
ab
The array
ab
 contains the matrix
A
 in band storage form.
The size of
ab
 must be
max(1,
ldab
*
n
)
b
The array
b
 contains the matrix
B
 whose columns are the right-hand sides for the systems of equations.
The size of
b
 is max(1,
ldb
*
nrhs
) for column major layout and max(1,
ldb
*
n
) for row major layout.
ldab
The leading dimension of
ab
;
ldab
≥
kd
 + 1.
ldb
The leading dimension of
b
;
ldb
≥
 max(1,
n
) for column major layout and
ldb
≥
nrhs
 for row major layout
.
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(n)ε|A|
c
(
n
)
 is a modest linear function of
n
, 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
b
 is
2
n
*
kd
 for real flavors and
8
n
*
kd
 for complex flavors.
To estimate the condition number
κ
∞
(
A
)
, call
?tbcon
.
To estimate the error in the solution, call
?tbrfs
.

Product and Performance Information

1

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