Developer Reference

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

?gttrs

Solves a system of linear equations with a tridiagonal coefficient matrix using the LU factorization computed by
?gttrf
.

Syntax

lapack_int
LAPACKE_sgttrs
(
int
matrix_layout
,
char
trans
,
lapack_int
n
,
lapack_int
nrhs
,
const
float
*
dl
,
const
float
*
d
,
const
float
*
du
,
const
float
*
du2
,
const
lapack_int
*
ipiv
,
float
*
b
,
lapack_int
ldb
);
lapack_int
LAPACKE_dgttrs
(
int
matrix_layout
,
char
trans
,
lapack_int
n
,
lapack_int
nrhs
,
const
double
*
dl
,
const
double
*
d
,
const
double
*
du
,
const
double
*
du2
,
const
lapack_int
*
ipiv
,
double
*
b
,
lapack_int
ldb
);
lapack_int
LAPACKE_cgttrs
(
int
matrix_layout
,
char
trans
,
lapack_int
n
,
lapack_int
nrhs
,
const
lapack_complex_float
*
dl
,
const
lapack_complex_float
*
d
,
const
lapack_complex_float
*
du
,
const
lapack_complex_float
*
du2
,
const
lapack_int
*
ipiv
,
lapack_complex_float
*
b
,
lapack_int
ldb
);
lapack_int
LAPACKE_zgttrs
(
int
matrix_layout
,
char
trans
,
lapack_int
n
,
lapack_int
nrhs
,
const
lapack_complex_double
*
dl
,
const
lapack_complex_double
*
d
,
const
lapack_complex_double
*
du
,
const
lapack_complex_double
*
du2
,
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 with multiple right hand sides:
A*X
=
B
if
trans
=
'N'
,
A
T
*X
=
B
if
trans
=
'T'
,
A
H
*X
=
B
if
trans
=
'C'
(for complex matrices only).
Before calling this routine, you must call
?gttrf
to compute the
LU
factorization of
A
.
Input Parameters
matrix_layout
Specifies whether matrix storage layout for array
b
is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
trans
Must be
'N'
or
'T'
or
'C'
.
Indicates the form of the equations:
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
.
n
The order of
A
;
n
0.
nrhs
The number of right-hand sides, that is, the number of columns in
B
;
nrhs
0
.
dl
,
d
,
du
,
du2
Arrays:
dl
(
n
-1)
,
d
(
n
)
,
du
(
n
-1)
,
du2
(
n
-2)
.
The array
dl
contains the
(
n
- 1)
multipliers that define the matrix
L
from the
LU
factorization of
A
.
The array
d
contains the
n
diagonal elements of the upper triangular matrix
U
from the
LU
factorization of
A
.
The array
du
contains the (
n
- 1) elements of the first superdiagonal of
U
.
The array
du2
contains the (
n
- 2) elements of the second superdiagonal of
U
.
b
Array of size max(1,
ldb
*
nrhs
) for column major layout and max(1,
n
*
ldb
) for row major layout. Contains the matrix
B
whose columns are the right-hand sides for the systems of equations.
ldb
The leading dimension of
b
;
ldb
max(1,
n
)
for column major layout and
ldb
nrhs
for row major layout
.
ipiv
Array, size (
n
). The
ipiv
array, as returned by
?gttrf
.
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 P|L||U|
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
7
n
(including
n
divisions) for real flavors and
34
n
(including
2
n
divisions) for complex flavors.
To estimate the condition number
κ
(
A
)
, call
?gtcon
.
To refine the solution and estimate the error, call
?gtrfs
.

Product and Performance Information

1

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