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NAG Toolbox: nag_lapack_dtprfs (f07uh)

Purpose

nag_lapack_dtprfs (f07uh) returns error bounds for the solution of a real triangular system of linear equations with multiple right-hand sides, AX = BAX=B or ATX = BATX=B, using packed storage.

Syntax

[ferr, berr, info] = f07uh(uplo, trans, diag, ap, b, x, 'n', n, 'nrhs_p', nrhs_p)
[ferr, berr, info] = nag_lapack_dtprfs(uplo, trans, diag, ap, b, x, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dtprfs (f07uh) returns the backward errors and estimated bounds on the forward errors for the solution of a real triangular system of linear equations with multiple right-hand sides AX = BAX=B or ATX = BATX=B, using packed storage. 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_dtprfs (f07uh) 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 AA is upper or lower triangular.
uplo = 'U'uplo='U'
AA is upper triangular.
uplo = 'L'uplo='L'
AA is lower triangular.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
2:     trans – string (length ≥ 1)
Indicates the form of the equations.
trans = 'N'trans='N'
The equations are of the form AX = BAX=B.
trans = 'T'trans='T' or 'C''C'
The equations are of the form ATX = BATX=B.
Constraint: trans = 'N'trans='N', 'T''T' or 'C''C'.
3:     diag – string (length ≥ 1)
Indicates whether AA is a nonunit or unit triangular matrix.
diag = 'N'diag='N'
AA is a nonunit triangular matrix.
diag = 'U'diag='U'
AA is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 11.
Constraint: diag = 'N'diag='N' or 'U''U'.
4:     ap( : :) – double array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)max(1,n×(n+1)/2).
The nn by nn triangular matrix AA, packed by columns.
More precisely,
  • if uplo = 'U'uplo='U', the upper triangle of AA must be stored with element AijAij in ap(i + j(j1) / 2)api+j(j-1)/2 for ijij;
  • if uplo = 'L'uplo='L', the lower triangle of AA must be stored with element AijAij in ap(i + (2nj)(j1) / 2)api+(2n-j)(j-1)/2 for ijij.
If diag = 'U'diag='U', the diagonal elements of AA are assumed to be 11, and are not referenced; the same storage scheme is used whether diag = 'N'diag='N' or ‘U’.
5:     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.
6:     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_dtptrs (f07ue).

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the arrays b, x.
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

ldb ldx work iwork

Output Parameters

1:     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.
2:     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.
3:     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: trans, 3: diag, 4: n, 5: nrhs_p, 6: ap, 7: b, 8: ldb, 9: x, 10: ldx, 11: ferr, 12: berr, 13: work, 14: iwork, 15: 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

A call to nag_lapack_dtprfs (f07uh), for each right-hand side, involves solving a number of systems of linear equations of the form Ax = bAx=b or ATx = bATx=b; the number is usually 44 or 55 and never more than 1111. Each solution involves approximately n2n2 floating point operations.
The complex analogue of this function is nag_lapack_ztprfs (f07uv).

Example

function nag_lapack_dtprfs_example
uplo = 'L';
trans = 'N';
diag = 'N';
ap = [4.3;
     -3.96;
     0.4;
     -0.27;
     -4.87;
     0.31;
     0.07;
     -8.02;
     -5.95;
     0.12];
b = [-12.9, -21.5;
     16.75, 14.93;
     -17.55, 6.33;
     -11.04, 8.09];
x = [-3, -5;
     -1, 1;
     2, -1;
     1, 6];
[ferr, berr, info] = nag_lapack_dtprfs(uplo, trans, diag, ap, b, x)
 

ferr =

   1.0e-13 *

    0.8824
    0.2637


berr =

   1.0e-16 *

    0.7420
         0


info =

                    0


function f07uh_example
uplo = 'L';
trans = 'N';
diag = 'N';
ap = [4.3;
     -3.96;
     0.4;
     -0.27;
     -4.87;
     0.31;
     0.07;
     -8.02;
     -5.95;
     0.12];
b = [-12.9, -21.5;
     16.75, 14.93;
     -17.55, 6.33;
     -11.04, 8.09];
x = [-3, -5;
     -1, 1;
     2, -1;
     1, 6];
[ferr, berr, info] = f07uh(uplo, trans, diag, ap, b, x)
 

ferr =

   1.0e-13 *

    0.8824
    0.2637


berr =

   1.0e-16 *

    0.7420
         0


info =

                    0



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

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