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NAG Toolbox: nag_lapack_dsytrs (f07me)
Purpose
nag_lapack_dsytrs (f07me) solves a real symmetric indefinite system of linear equations with multiple right-hand sides,
where
has been factorized by
nag_lapack_dsytrf (f07md).
Syntax
Description
nag_lapack_dsytrs (f07me) is used to solve a real symmetric indefinite system of linear equations
, this function must be preceded by a call to
nag_lapack_dsytrf (f07md) 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 .
References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Parameters
Compulsory Input Parameters
- 1:
– string (length ≥ 1)
-
Specifies how
has been factorized.
- , where is upper triangular.
- , where is lower triangular.
Constraint:
or .
- 2:
– double array
-
The first dimension of the array
a must be at least
.
The second dimension of the array
a must be at least
.
Details of the factorization of
, as returned by
nag_lapack_dsytrf (f07md).
- 3:
– int64int32nag_int array
-
The dimension of the array
ipiv
must be at least
Details of the interchanges and the block structure of
, as returned by
nag_lapack_dsytrf (f07md).
- 4:
– double array
-
The first dimension of the array
b must be at least
.
The second dimension of the array
b must be at least
.
The by right-hand side matrix .
Optional Input Parameters
- 1:
– int64int32nag_int scalar
-
Default:
the first dimension of the arrays
a,
b and the second dimension of the arrays
a,
ipiv.
, the order of the matrix .
Constraint:
.
- 2:
– int64int32nag_int scalar
-
Default:
the second dimension of the array
b.
, the number of right-hand sides.
Constraint:
.
Output Parameters
- 1:
– double array
-
The first dimension of the array
b will be
.
The second dimension of the array
b will be
.
The by solution matrix .
- 2:
– int64int32nag_int scalar
unless the function detects an error (see
Error Indicators and Warnings).
Error Indicators and Warnings
-
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
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 precisionIf
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
nag_lapack_dsyrfs (f07mh), and an estimate for
(
) can be obtained by calling
nag_lapack_dsycon (f07mg).
Further Comments
The total number of floating-point operations is approximately .
This function may be followed by a call to
nag_lapack_dsyrfs (f07mh) to refine the solution and return an error estimate.
The complex analogues of this function are
nag_lapack_zhetrs (f07ms) for Hermitian matrices and
nag_lapack_zsytrs (f07ns) for symmetric matrices.
Example
This example solves the system of equations
, where
Here
is symmetric indefinite and must first be factorized by
nag_lapack_dsytrf (f07md).
Open in the MATLAB editor:
f07me_example
function f07me_example
fprintf('f07me example results\n\n');
uplo = 'L';
a = [ 2.07, 0, 0, 0;
3.87, -0.21, 0, 0;
4.20, 1.87, 1.15, 0;
-1.15, 0.63, 2.06, -1.81];
b = [-9.50, 27.85;
-8.38, 9.90;
-6.07, 19.25;
-0.96, 3.93];
[af, ipiv, info] = f07md( ...
uplo, a);
[x, info] = f07me( ...
uplo, af, ipiv, b);
disp('Solution(s)');
disp(x);
f07me example results
Solution(s)
-4.0000 1.0000
-1.0000 4.0000
2.0000 3.0000
5.0000 2.0000
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