PDF version (NAG web site
, 64-bit version, 64-bit version)
NAG Toolbox: nag_lapack_dporfs (f07fh)
Purpose
nag_lapack_dporfs (f07fh) returns error bounds for the solution of a real symmetric positive definite system of linear equations with multiple right-hand sides, . It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.
Syntax
[
x,
ferr,
berr,
info] = f07fh(
uplo,
a,
af,
b,
x, 'n',
n, 'nrhs_p',
nrhs_p)
[
x,
ferr,
berr,
info] = nag_lapack_dporfs(
uplo,
a,
af,
b,
x, 'n',
n, 'nrhs_p',
nrhs_p)
Description
nag_lapack_dporfs (f07fh) returns the backward errors and estimated bounds on the forward errors for the solution of a real symmetric positive definite system of linear equations with multiple right-hand sides . The function handles each right-hand side vector (stored as a column of the matrix ) independently, so we describe the function of nag_lapack_dporfs (f07fh) in terms of a single right-hand side and solution .
Given a computed solution
, the function computes the
component-wise backward error
. This is the size of the smallest relative perturbation in each element of
and
such that
is the exact solution of a perturbed system
Then the function estimates a bound for the
component-wise forward error in the computed solution, defined by:
where
is the true solution.
For details of the method, see the
F07 Chapter Introduction.
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 whether the upper or lower triangular part of
is stored and how
is to be factorized.
- The upper triangular part of is stored and is factorized as , where is upper triangular.
- The lower triangular part of is stored and is factorized as , 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
.
The
by
original symmetric positive definite matrix
as supplied to
nag_lapack_dpotrf (f07fd).
- 3:
– double array
-
The first dimension of the array
af must be at least
.
The second dimension of the array
af must be at least
.
The Cholesky factor of
, as returned by
nag_lapack_dpotrf (f07fd).
- 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 .
- 5:
– double array
-
The first dimension of the array
x must be at least
.
The second dimension of the array
x must be at least
.
The
by
solution matrix
, as returned by
nag_lapack_dpotrs (f07fe).
Optional Input Parameters
- 1:
– int64int32nag_int scalar
-
Default:
the first dimension of the arrays
a,
af,
b,
x and the second dimension of the arrays
a,
af.
, the order of the matrix .
Constraint:
.
- 2:
– int64int32nag_int scalar
-
Default:
the second dimension of the arrays
b,
x. (An error is raised if these dimensions are not equal.)
, the number of right-hand sides.
Constraint:
.
Output Parameters
- 1:
– double array
-
The first dimension of the array
x will be
.
The second dimension of the array
x will be
.
The improved solution matrix .
- 2:
– double array
-
contains an estimated error bound for the th solution vector, that is, the th column of , for .
- 3:
– double array
-
contains the component-wise backward error bound for the th solution vector, that is, the th column of , for .
- 4:
– 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
The bounds returned in
ferr are not rigorous, because they are estimated, not computed exactly; but in practice they almost always overestimate the actual error.
Further Comments
For each right-hand side, computation of the backward error involves a minimum of floating-point operations. Each step of iterative refinement involves an additional operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required.
Estimating the forward error involves solving a number of systems of linear equations of the form ; the number is usually or and never more than . Each solution involves approximately operations.
The complex analogue of this function is
nag_lapack_zporfs (f07fv).
Example
This example solves the system of equations
using iterative refinement and to compute the forward and backward error bounds, where
Here
is symmetric positive definite and must first be factorized by
nag_lapack_dpotrf (f07fd).
Open in the MATLAB editor:
f07fh_example
function f07fh_example
fprintf('f07fh example results\n\n');
uplo = 'Lower';
a = [ 4.16, 0, 0, 0;
-3.12, 5.03, 0, 0;
0.56, -0.83, 0.76, 0;
-0.10, 1.18, 0.34, 1.18];
[L, info] = f07fd( ...
uplo, a);
b = [ 8.70, 8.30;
-13.35, 2.13;
1.89, 1.61;
-4.14, 5.00];
[x, info] = f07fe( ...
uplo, L, b);
[x, ferr, berr, info] = f07fh( ...
uplo, a, L, b, x);
[ifail] = x04ca( ...
'General', 'N', x, 'Solution(s)');
fprintf('\nBackward errors (machine-dependent)\n ')
fprintf('%11.1e', berr);
fprintf('\nEstimated forward error bounds (machine-dependent)\n ')
fprintf('%11.1e', ferr);
fprintf('\n');
f07fh example results
Solution(s)
1 2
1 1.0000 4.0000
2 -1.0000 3.0000
3 2.0000 2.0000
4 -3.0000 1.0000
Backward errors (machine-dependent)
1.0e-16 5.0e-17
Estimated forward error bounds (machine-dependent)
2.3e-14 2.3e-14
PDF version (NAG web site
, 64-bit version, 64-bit version)
© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015