Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

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
≤
n
, then
U
i
,
i
 is exactly zero. The factorization has not been completed unless
i
 =
n
, but the factor
U
 is exactly singular, so the solution and error bounds could not be computed;
rcond
 = 0 is returned. If
info
 =
i
, and
i
 =
n
 + 1
, then
U
 is nonsingular, but
rcond
 is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of
rcond
 would suggest.

Product and Performance Information