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

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

1

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