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NAG Toolbox: nag_lapack_zporfs (f07fv)

Purpose

nag_lapack_zporfs (f07fv) returns error bounds for the solution of a complex Hermitian 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] = f07fv(uplo, a, af, b, x, 'n', n, 'nrhs_p', nrhs_p)
[x, ferr, berr, info] = nag_lapack_zporfs(uplo, a, af, b, x, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_zporfs (f07fv) returns the backward errors and estimated bounds on the forward errors for the solution of a complex Hermitian 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_zporfs (f07fv) 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 UHUUHU, where UU is upper triangular.
uplo = 'L'uplo='L'
The lower triangular part of AA is stored and AA is factorized as LLHLLH, where LL is lower triangular.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
2:     a(lda, : :) – complex 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 Hermitian positive definite matrix AA as supplied to nag_lapack_zpotrf (f07fr).
3:     af(ldaf, : :) – complex 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_zpotrf (f07fr).
4:     b(ldb, : :) – complex 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, : :) – complex 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_zpotrs (f07fs).

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 rwork

Output Parameters

1:     x(ldx, : :) – complex 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: rwork, 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 16n216n2 real floating point operations. Each step of iterative refinement involves an additional 24n224n2 real 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 55 and never more than 1111. Each solution involves approximately 8n28n2 real operations.
The real analogue of this function is nag_lapack_dporfs (f07fh).

Example

function nag_lapack_zporfs_example
uplo = 'L';
a = [complex(3.23),  0 + 0i,  0 + 0i,  0 + 0i;
      1.51 + 1.92i,  3.58 + 0i,  0 + 0i,  0 + 0i;
      1.9 - 0.84i,  -0.23 - 1.11i,  4.09 + 0i,  0 + 0i;
      0.42 - 2.5i,  -1.18 - 1.37i,  2.33 + 0.14i,  4.29 + 0i];
b = [ 3.93 - 6.14i,  1.48 + 6.58i;
      6.17 + 9.42i,  4.65 - 4.75i;
      -7.17 - 21.83i,  -4.91 + 2.29i;
      1.99 - 14.38i,  7.64 - 10.79i];
[af, info] = nag_lapack_zpotrf(uplo, a);
[x, info] = nag_lapack_zpotrs(uplo, af, b);
[xOut, ferr, berr, info] = nag_lapack_zporfs(uplo, a, af, b, x)
 

xOut =

   1.0000 - 1.0000i  -1.0000 + 2.0000i
  -0.0000 + 3.0000i   3.0000 - 4.0000i
  -4.0000 - 5.0000i  -2.0000 + 3.0000i
   2.0000 + 1.0000i   4.0000 - 5.0000i


ferr =

   1.0e-13 *

    0.6021
    0.7231


berr =

   1.0e-15 *

    0.0698
    0.1053


info =

                    0


function f07fv_example
uplo = 'L';
a = [complex(3.23),  0 + 0i,  0 + 0i,  0 + 0i;
      1.51 + 1.92i,  3.58 + 0i,  0 + 0i,  0 + 0i;
      1.9 - 0.84i,  -0.23 - 1.11i,  4.09 + 0i,  0 + 0i;
      0.42 - 2.5i,  -1.18 - 1.37i,  2.33 + 0.14i,  4.29 + 0i];
b = [ 3.93 - 6.14i,  1.48 + 6.58i;
      6.17 + 9.42i,  4.65 - 4.75i;
      -7.17 - 21.83i,  -4.91 + 2.29i;
      1.99 - 14.38i,  7.64 - 10.79i];
[af, info] = f07fr(uplo, a);
[x, info] = f07fs(uplo, af, b);
[xOut, ferr, berr, info] = f07fv(uplo, a, af, b, x)
 

xOut =

   1.0000 - 1.0000i  -1.0000 + 2.0000i
  -0.0000 + 3.0000i   3.0000 - 4.0000i
  -4.0000 - 5.0000i  -2.0000 + 3.0000i
   2.0000 + 1.0000i   4.0000 - 5.0000i


ferr =

   1.0e-13 *

    0.6021
    0.7231


berr =

   1.0e-15 *

    0.0698
    0.1053


info =

                    0



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

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