Developer Reference

  • 0.10
  • 10/21/2020
  • Public Content
Contents

?gtsvx

Computes the solution to the real or complex system of linear equations with a tridiagonal coefficient matrix A and multiple right-hand sides, and provides error bounds on the solution.

Syntax

lapack_int LAPACKE_sgtsvx
(
int
matrix_layout
,
char
fact
,
char
trans
,
lapack_int
n
,
lapack_int
nrhs
,
const float*
dl
,
const float*
d
,
const float*
du
,
float*
dlf
,
float*
df
,
float*
duf
,
float*
du2
,
lapack_int*
ipiv
,
const float*
b
,
lapack_int
ldb
,
float*
x
,
lapack_int
ldx
,
float*
rcond
,
float*
ferr
,
float*
berr
);
lapack_int LAPACKE_dgtsvx
(
int
matrix_layout
,
char
fact
,
char
trans
,
lapack_int
n
,
lapack_int
nrhs
,
const double*
dl
,
const double*
d
,
const double*
du
,
double*
dlf
,
double*
df
,
double*
duf
,
double*
du2
,
lapack_int*
ipiv
,
const double*
b
,
lapack_int
ldb
,
double*
x
,
lapack_int
ldx
,
double*
rcond
,
double*
ferr
,
double*
berr
);
lapack_int LAPACKE_cgtsvx
(
int
matrix_layout
,
char
fact
,
char
trans
,
lapack_int
n
,
lapack_int
nrhs
,
const lapack_complex_float*
dl
,
const lapack_complex_float*
d
,
const lapack_complex_float*
du
,
lapack_complex_float*
dlf
,
lapack_complex_float*
df
,
lapack_complex_float*
duf
,
lapack_complex_float*
du2
,
lapack_int*
ipiv
,
const lapack_complex_float*
b
,
lapack_int
ldb
,
lapack_complex_float*
x
,
lapack_int
ldx
,
float*
rcond
,
float*
ferr
,
float*
berr
);
lapack_int LAPACKE_zgtsvx
(
int
matrix_layout
,
char
fact
,
char
trans
,
lapack_int
n
,
lapack_int
nrhs
,
const lapack_complex_double*
dl
,
const lapack_complex_double*
d
,
const lapack_complex_double*
du
,
lapack_complex_double*
dlf
,
lapack_complex_double*
df
,
lapack_complex_double*
duf
,
lapack_complex_double*
du2
,
lapack_int*
ipiv
,
const lapack_complex_double*
b
,
lapack_int
ldb
,
lapack_complex_double*
x
,
lapack_int
ldx
,
double*
rcond
,
double*
ferr
,
double*
berr
);
Include Files
  • mkl.h
Description
The routine uses the
LU
factorization to compute the solution to a real or complex system of linear equations
A*X
=
B
,
A
T
*X
=
B
, or
A
H
*X
=
B
, where
A
is a tridiagonal matrix of order
n
, the columns of matrix
B
are individual right-hand sides, and the columns of
X
are the corresponding solutions.
Error bounds on the solution and a condition estimate are also provided.
The routine
?gtsvx
performs the following steps:
  1. If
    fact
    =
    'N'
    , the
    LU
    decomposition is used to factor the matrix
    A
    as
    A
    =
    L*U
    , where
    L
    is a product of permutation and unit lower bidiagonal matrices and
    U
    is an upper triangular matrix with nonzeroes in only the main diagonal and first two superdiagonals.
  2. If some
    U
    i
    ,
    i
    = 0, so that
    U
    is exactly singular, then the routine returns with
    info
    =
    i
    . Otherwise, the factored form of
    A
    is used to estimate the condition number of the matrix
    A
    . If the reciprocal of the condition number is less than machine precision,
    info
    =
    n
    + 1
    is returned as a warning, but the routine still goes on to solve for
    X
    and compute error bounds as described below.
  3. The system of equations is solved for
    X
    using the factored form of
    A
    .
  4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
fact
Must be
'F'
or
'N'
.
Specifies whether or not the factored form of the matrix
A
has been supplied on entry.
If
fact
=
'F'
: on entry,
dlf
,
df
,
duf
,
du2
, and
ipiv
contain the factored form of
A
; arrays
dl
,
d
,
du
,
dlf
,
df
,
duf
,
du2
, and
ipiv
will not be modified.
If
fact
=
'N'
, the matrix
A
will be copied to
dlf
,
df
, and
duf
and factored.
trans
Must be
'N'
,
'T'
, or
'C'
.
Specifies the form of the system of equations:
If
trans
=
'N'
, the system has the form
A
*
X
=
B
(No transpose).
If
trans
=
'T'
, the system has the form
A
T
*X
=
B
(Transpose).
If
trans
=
'C'
, the system has the form
A
H
*X
=
B
(Conjugate transpose).
n
The number of linear equations, the order of the matrix
A
;
n
0.
nrhs
The number of right hand sides, the number of columns of the matrices
B
and
X
;
nrhs
0.
dl
,
d
,
du
,
dlf
,
df
,
duf
,
du2
,
b
Arrays:
dl
, size (
n
-1), contains the subdiagonal elements of
A
.
d
, size (
n
), contains the diagonal elements of
A
.
du
, size (
n
-1), contains the superdiagonal elements of
A
.
dlf
, size (
n
-1). If
fact
=
'F'
, then
dlf
is an input argument and on entry contains the (
n
-1) multipliers that define the matrix
L
from the
LU
factorization of
A
as computed by
?gttrf
.
df
, size (
n
). If
fact
=
'F'
, then
df
is an input argument and on entry contains the
n
diagonal elements of the upper triangular matrix
U
from the
LU
factorization of
A
.
duf
, size (
n
-1). If
fact
=
'F'
, then
duf
is an input argument and on entry contains the (
n
-1) elements of the first superdiagonal of
U
.
du2
, size (
n
-2). If
fact
=
'F'
, then
du2
is an input argument and on entry contains the (
n
-2) elements of the second superdiagonal of
U
.
b
, size max(
ldb
*
nrhs
) for column major layout and max(
ldb
*
n
) for row major layout,
contains the right-hand side matrix
B
.
ldb
The leading dimension of
b
;
ldb
max(1,
n
) for column major layout and
ldb
nrhs
for row major layout
.
ldx
The leading dimension of
x
;
ldx
max(1,
n
) for column major layout and
ldx
nrhs
for row major layout
.
ipiv
Array, size at least
max(1,
n
)
. If
fact
=
'F'
, then
ipiv
is an input argument and on entry contains the pivot indices, as returned by
?gttrf
.
Output Parameters
x
Array, size
max(1,
ldx
*
nrhs
) for column major layout and max(1,
ldx
*
n
) for row major layout
.
If
info
= 0
or
info
=
n
+1
, the array
x
contains the solution matrix
X
.
dlf
If
fact
=
'N'
, then
dlf
is an output argument and on exit contains the
(
n
-1)
multipliers that define the matrix
L
from the
LU
factorization of A.
df
If
fact
=
'N'
, then
df
is an output argument and on exit contains the
n
diagonal elements of the upper triangular matrix
U
from the
LU
factorization of
A
.
duf
If
fact
=
'N'
, then
duf
is an output argument and on exit contains the
(
n
-1)
elements of the first superdiagonal of
U
.
du2
If
fact
=
'N'
, then
du2
is an output argument and on exit contains the
(
n
-2)
elements of the second superdiagonal of
U
.
ipiv
The array
ipiv
is an output argument if
fact
=
'N'
and, on exit, contains the pivot indices from the factorization
A
=
L*U
; row
i
of the matrix was interchanged with row
ipiv
[
i
-1]. The value of
ipiv
[
i
-1] will always be
i
or
i
+1;
ipiv
[
i
-1]=
i
indicates a row interchange was not required.
rcond
An estimate of the reciprocal condition number of the matrix
A
. If
rcond
is less than the machine precision (in particular, if
rcond
=0), the matrix is singular to working precision. This condition is indicated by a return code of
info
>0.
ferr
Array, size at least
max(1,
nrhs
)
. Contains the estimated forward error bound for each solution vector
x
j
(the
j
-th column of the solution matrix
X
). If
xtrue
is the true solution corresponding to
x
j
,
ferr
[
j
-1]
is an estimated upper bound for the magnitude of the largest element in
x
j
-
xtrue
divided by the magnitude of the largest element in
x
j
. The estimate is as reliable as the estimate for
rcond
, and is almost always a slight overestimate of the true error.
berr
Array, size at least
max(1,
nrhs
)
. Contains the component-wise relative backward error for each solution vector
x
j
, that is, the smallest relative change in any element of
A
or
B
that makes
x
j
an exact solution.
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
, and
i