NAG CL Interface
f07mec (dsytrs)
1
Purpose
f07mec solves a real symmetric indefinite system of linear equations with multiple right-hand sides,
where
has been factorized by
f07mdc.
2
Specification
void |
f07mec (Nag_OrderType order,
Nag_UploType uplo,
Integer n,
Integer nrhs,
const double a[],
Integer pda,
const Integer ipiv[],
double b[],
Integer pdb,
NagError *fail) |
|
The function may be called by the names: f07mec, nag_lapacklin_dsytrs or nag_dsytrs.
3
Description
f07mec is used to solve a real symmetric indefinite system of linear equations
, this function must be preceded by a call to
f07mdc which computes the Bunch–Kaufman factorization of
.
If , , where is a permutation matrix, is an upper triangular matrix and is a symmetric block diagonal matrix with by and by blocks; the solution is computed by solving and then .
If , , where is a lower triangular matrix; the solution is computed by solving and then .
4
References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5
Arguments
-
1:
– Nag_OrderType
Input
-
On entry: the
order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See
Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
-
2:
– Nag_UploType
Input
-
On entry: specifies how
has been factorized.
- , where is upper triangular.
- , where is lower triangular.
Constraint:
or .
-
3:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
4:
– Integer
Input
-
On entry: , the number of right-hand sides.
Constraint:
.
-
5:
– const double
Input
-
Note: the dimension,
dim, of the array
a
must be at least
.
On entry: details of the factorization of
, as returned by
f07mdc.
-
6:
– Integer
Input
On entry: the stride separating row or column elements (depending on the value of
order) of the matrix in the array
a.
Constraint:
.
-
7:
– const Integer
Input
-
Note: the dimension,
dim, of the array
ipiv
must be at least
.
On entry: details of the interchanges and the block structure of
, as returned by
f07mdc.
-
8:
– double
Input/Output
-
Note: the dimension,
dim, of the array
b
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the by right-hand side matrix .
On exit: the by solution matrix .
-
9:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
;
- if , .
-
10:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
7
Accuracy
For each right-hand side vector
, the computed solution
is the exact solution of a perturbed system of equations
, where
- if , ;
- if , ,
is a modest linear function of
, and
is the
machine precision.
If
is the true solution, then the computed solution
satisfies a forward error bound of the form
where
.
Note that can be much smaller than .
Forward and backward error bounds can be computed by calling
f07mhc, and an estimate for
(
) can be obtained by calling
f07mgc.
8
Parallelism and Performance
f07mec makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The total number of floating-point operations is approximately .
This function may be followed by a call to
f07mhc to refine the solution and return an error estimate.
The complex analogues of this function are
f07msc for Hermitian matrices and
f07nsc for symmetric matrices.
10
Example
This example solves the system of equations
, where
Here
is symmetric indefinite and must first be factorized by
f07mdc.
10.1
Program Text
10.2
Program Data
10.3
Program Results