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
b
;
ldb
max(1,
n
) for column major layout and
ldb
nrhs
for row major layout
.
ldx
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
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

Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804