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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, AX = BAX=B. 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 AX = BAX=B. The function handles each right-hand side vector (stored as a column of the matrix BB) independently, so we describe the function of nag_lapack_dporfs (f07fh) in terms of a single right-hand side bb and solution xx.
Given a computed solution xx, the function computes the component-wise backward error ββ. This is the size of the smallest relative perturbation in each element of AA and bb such that xx is the exact solution of a perturbed system
(A + δA)x = b + δb
|δaij|β|aij|   and   |δbi|β|bi| .
(A+δA)x=b+δb |δaij|β|aij|   and   |δbi|β|bi| .
Then the function estimates a bound for the component-wise forward error in the computed solution, defined by:
max |xii| / max |xi|
i i
maxi|xi-x^i|/maxi|xi|
where x^ 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:     uplo – string (length ≥ 1)
Specifies whether the upper or lower triangular part of AA is stored and how AA is to be factorized.
uplo = 'U'uplo='U'
The upper triangular part of AA is stored and AA is factorized as UTUUTU, where UU is upper triangular.
uplo = 'L'uplo='L'
The lower triangular part of AA is stored and AA is factorized as LLTLLT, where LL is lower triangular.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
2:     a(lda, : :) – double array
The first dimension of the array a must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,n)max(1,n)
The nn by nn original symmetric positive definite matrix AA as supplied to nag_lapack_dpotrf (f07fd).
3:     af(ldaf, : :) – double array
The first dimension of the array af must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,n)max(1,n)
The Cholesky factor of AA, as returned by nag_lapack_dpotrf (f07fd).
4:     b(ldb, : :) – double array
The first dimension of the array b must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,nrhs)max(1,nrhs)
The nn by rr right-hand side matrix BB.
5:     x(ldx, : :) – double array
The first dimension of the array x must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,nrhs)max(1,nrhs)
The nn by rr solution matrix XX, as returned by nag_lapack_dpotrs (f07fe).

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the arrays a, af, b, x The second dimension of the arrays a, af.
nn, the order of the matrix AA.
Constraint: n0n0.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the arrays b, x. (An error is raised if these dimensions are not equal.)
rr, the number of right-hand sides.
Constraint: nrhs0nrhs0.

Input Parameters Omitted from the MATLAB Interface

lda ldaf ldb ldx work iwork

Output Parameters

1:     x(ldx, : :) – double array
The first dimension of the array x will be max (1,n)max(1,n)
The second dimension of the array will be max (1,nrhs)max(1,nrhs)
ldxmax (1,n)ldxmax(1,n).
The improved solution matrix XX.
2:     ferr(nrhs_p) – double array
ferr(j)ferrj contains an estimated error bound for the jjth solution vector, that is, the jjth column of XX, for j = 1,2,,rj=1,2,,r.
3:     berr(nrhs_p) – double array
berr(j)berrj contains the component-wise backward error bound ββ for the jjth solution vector, that is, the jjth column of XX, for j = 1,2,,rj=1,2,,r.
4:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

  info = iinfo=-i
If info = iinfo=-i, parameter ii had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: nrhs_p, 4: a, 5: lda, 6: af, 7: ldaf, 8: b, 9: ldb, 10: x, 11: ldx, 12: ferr, 13: berr, 14: work, 15: iwork, 16: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

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 4n24n2 floating point operations. Each step of iterative refinement involves an additional 6n26n2 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 Ax = bAx=b; the number is usually 44 or 55 and never more than 1111. Each solution involves approximately 2n22n2 operations.
The complex analogue of this function is nag_lapack_zporfs (f07fv).

Example

function nag_lapack_dporfs_example
uplo = 'L';
a = [4.16, 0, 0, 0;
     -3.12, 5.03, 0, 0;
     0.56, -0.83, 0.76, 0;
     -0.1, 1.18, 0.34, 1.18];
b = [8.7, 8.3;
     -13.35, 2.13;
     1.89, 1.61;
     -4.14, 5];
[af, info] = nag_lapack_dpotrf(uplo, a);
[x, info]  = nag_lapack_dpotrs(uplo, af, b);
[xOut, ferr, berr, info] = nag_lapack_dporfs(uplo, a, af, b, x)
 

xOut =

    1.0000    4.0000
   -1.0000    3.0000
    2.0000    2.0000
   -3.0000    1.0000


ferr =

   1.0e-13 *

    0.2334
    0.2253


berr =

   1.0e-15 *

    0.1021
    0.0500


info =

                    0


function f07fh_example
uplo = 'L';
a = [4.16, 0, 0, 0;
     -3.12, 5.03, 0, 0;
     0.56, -0.83, 0.76, 0;
     -0.1, 1.18, 0.34, 1.18];
b = [8.7, 8.3;
     -13.35, 2.13;
     1.89, 1.61;
     -4.14, 5];
[af, info] = f07fd(uplo, a);
[x, info]  = f07fe(uplo, af, b);
[xOut, ferr, berr, info] = f07fh(uplo, a, af, b, x)
 

xOut =

    1.0000    4.0000
   -1.0000    3.0000
    2.0000    2.0000
   -3.0000    1.0000


ferr =

   1.0e-13 *

    0.2334
    0.2253


berr =

   1.0e-15 *

    0.1021
    0.0500


info =

                    0



PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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