Contents

# ?ptsvx

Uses factorization to compute the solution to the system of linear equations with a symmetric (Hermitian) positive definite tridiagonal coefficient matrix
A
, and provides error bounds on the solution.

## Syntax

Include Files
• mkl.h
Description
The routine uses the Cholesky factorization
A
=
L*D*L
T
(real)/
A
=
L*D*L
H
(complex) to compute the solution to a real or complex system of linear equations
A*X
=
B
, where
A
is a
n
-by-
n
symmetric or Hermitian positive definite tridiagonal matrix, 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
?ptsvx
performs the following steps:
1. If
fact
=
'N'
, the matrix
A
is factored as
A
=
L*D*L
T
(real flavors)/
A
=
L*D*L
H
(complex flavors), where
L
is a unit lower bidiagonal matrix and
D
is diagonal. The factorization can also be regarded as having the form
A
=
U
T
*D*U
(real flavors)/
A
=
U
H
*D*U
(complex flavors).
i
-by-
i
principal minor is not positive-definite, 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
is supplied on entry.
If
fact
=
'F'
: on entry,
df
and
ef
contain the factored form of
A
. Arrays
d
,
e
,
df
, and
ef
will not be modified.
If
fact
=
'N'
, the matrix
A
will be copied to
df
and
ef
, and factored.
n
The order of matrix
A
;
n
0.
nrhs
The number of right-hand sides, the number of columns in
B
;
nrhs
0
.
d
,
df
Arrays:
d
(size
n
),
df
(size
n
).
The array
d
contains the
n
diagonal elements of the tridiagonal matrix
A
.
The array
df
is an input argument if
fact
=
'F'
and on entry contains the
n
diagonal elements of the diagonal matrix
D
from the
L*D*L
T
(real)/
L*D*L
H
(complex) factorization of
A
.
e
,
ef
,
b
Arrays:
e
(size
n
-1),
ef
(size
n
-1),
b
, size max(
ldb
*
nrhs
) for column major layout and max(
ldb
*
n
) for row major layout
. The array
e
contains the
(
n
- 1)
subdiagonal elements of the tridiagonal matrix
A
.
The array
ef
is an input argument if
fact
=
'F'
and on entry contains the
(
n
- 1)
subdiagonal elements of the unit bidiagonal factor
L
from the
L*D*L
T
(real)/
L*D*L
H
(complex) factorization of
A
.
The array
b
contains the matrix
B
whose columns are the right-hand sides for the systems of equations.
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
.
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
to the system of equations.
df
,
ef
These arrays are output arguments if
fact
=
'N'
. See the description of
df
,
ef
in
Input Arguments
section.
rcond
An estimate of the reciprocal condition number of the matrix
A
after equilibration (if done). 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
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
, the leading minor of order
i
(and therefore the matrix
A
itself) is not positive-definite, so the factorization could not be completed, and 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.