Developer Reference

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

lapack_int LAPACKE_sptsvx
(
int
matrix_layout
,
char
fact
,
lapack_int
n
,
lapack_int
nrhs
,
const float*
d
,
const float*
e
,
float*
df
,
float*
ef
,
const float*
b
,
lapack_int
ldb
,
float*
x
,
lapack_int
ldx
,
float*
rcond
,
float*
ferr
,
float*
berr
);
lapack_int LAPACKE_dptsvx
(
int
matrix_layout
,
char
fact
,
lapack_int
n
,
lapack_int
nrhs
,
const double*
d
,
const double*
e
,
double*
df
,
double*
ef
,
const double*
b
,
lapack_int
ldb
,
double*
x
,
lapack_int
ldx
,
double*
rcond
,
double*
ferr
,
double*
berr
);
lapack_int LAPACKE_cptsvx
(
int
matrix_layout
,
char
fact
,
lapack_int
n
,
lapack_int
nrhs
,
const float*
d
,
const lapack_complex_float*
e
,
float*
df
,
lapack_complex_float*
ef
,
const lapack_complex_float*
b
,
lapack_int
ldb
,
lapack_complex_float*
x
,
lapack_int
ldx
,
float*
rcond
,
float*
ferr
,
float*
berr
);
lapack_int LAPACKE_zptsvx
(
int
matrix_layout
,
char
fact
,
lapack_int
n
,
lapack_int
nrhs
,
const double*
d
,
const lapack_complex_double*
e
,
double*
df
,
lapack_complex_double*
ef
,
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 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).
  2. If the leading
    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
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
.
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
had an illegal value.
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

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