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NAG Toolbox: nag_lapack_ztbrfs (f07vv)

Purpose

nag_lapack_ztbrfs (f07vv) returns error bounds for the solution of a complex triangular band system of linear equations with multiple right-hand sides, AX = BAX=B, ATX = BATX=B or AHX = BAHX=B.

Syntax

[ferr, berr, info] = f07vv(uplo, trans, diag, kd, ab, b, x, 'n', n, 'nrhs_p', nrhs_p)
[ferr, berr, info] = nag_lapack_ztbrfs(uplo, trans, diag, kd, ab, b, x, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_ztbrfs (f07vv) returns the backward errors and estimated bounds on the forward errors for the solution of a complex triangular band system of linear equations with multiple right-hand sides AX = BAX=B, ATX = BATX=B or AHX = BAHX=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_ztbrfs (f07vv) 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'
The equations are of the form ATX = BATX=B.
trans = 'C'trans='C'
The equations are of the form AHX = BAHX=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:     kd – int64int32nag_int scalar
kdkd, the number of superdiagonals of the matrix AA if uplo = 'U'uplo='U', or the number of subdiagonals if uplo = 'L'uplo='L'.
Constraint: kd0kd0.
5:     ab(ldab, : :) – complex array
The first dimension of the array ab must be at least kd + 1kd+1
The second dimension of the array must be at least max (1,n)max(1,n)
The nn by nn triangular band matrix AA.
The matrix is stored in rows 11 to kd + 1kd+1, more precisely,
  • if uplo = 'U'uplo='U', the elements of the upper triangle of AA within the band must be stored with element AijAij in ab(kd + 1 + ij,j)​ for ​max (1,jkd)ijabkd+1+i-jj​ for ​max(1,j-kd)ij;
  • if uplo = 'L'uplo='L', the elements of the lower triangle of AA within the band must be stored with element AijAij in ab(1 + ij,j)​ for ​jimin (n,j + kd).ab1+i-jj​ for ​jimin(n,j+kd).
If diag = 'U'diag='U', the diagonal elements of AA are assumed to be 11, and are not referenced.
6:     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.
7:     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_ztbtrs (f07vs).

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The second dimension of the array ab.
nn, the order of the matrix AA.
Constraint: n0n0.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the arrays b, x.
rr, the number of right-hand sides.
Constraint: nrhs0nrhs0.

Input Parameters Omitted from the MATLAB Interface

ldab ldb ldx work rwork

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: kd, 6: nrhs_p, 7: ab, 8: ldab, 9: b, 10: ldb, 11: x, 12: ldx, 13: ferr, 14: berr, 15: work, 16: rwork, 17: 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_ztbrfs (f07vv), for each right-hand side, involves solving a number of systems of linear equations of the form Ax = bAx=b or AHx = bAHx=b; the number is usually 55 and never more than 1111. Each solution involves approximately 8nk8nk real floating point operations (assuming nknk).
The real analogue of this function is nag_lapack_dtbrfs (f07vh).

Example

function nag_lapack_ztbrfs_example
uplo = 'L';
trans = 'N';
diag = 'N';
kd = int64(2);
ab = [ -1.94 + 4.43i,  4.12 - 4.27i,  0.43 - 2.66i,  0.44 + 0.1i;
      -3.39 + 3.44i,  -1.84 + 5.53i,  1.74 - 0.04i,  0 + 0i;
      1.62 + 3.68i,  -2.77 - 1.93i,  0 + 0i,  0 + 0i];
b = [ -8.86 - 3.88i,  -24.09 - 5.27i;
      -15.57 - 23.41i,  -57.97 + 8.14i;
      -7.63 + 22.78i,  19.09 - 29.51i;
      -14.74 - 2.4i,  19.17 + 21.33i];
[x, info] = nag_lapack_ztbtrs(uplo, trans, diag, kd, ab, b);
[ferr, berr, info] = nag_lapack_ztbrfs(uplo, trans, diag, kd, ab, b, x)
 

ferr =

   1.0e-13 *

    0.1803
    0.2199


berr =

   1.0e-16 *

    0.0904
    0.6571


info =

                    0


function f07vv_example
uplo = 'L';
trans = 'N';
diag = 'N';
kd = int64(2);
ab = [ -1.94 + 4.43i,  4.12 - 4.27i,  0.43 - 2.66i,  0.44 + 0.1i;
      -3.39 + 3.44i,  -1.84 + 5.53i,  1.74 - 0.04i,  0 + 0i;
      1.62 + 3.68i,  -2.77 - 1.93i,  0 + 0i,  0 + 0i];
b = [ -8.86 - 3.88i,  -24.09 - 5.27i;
      -15.57 - 23.41i,  -57.97 + 8.14i;
      -7.63 + 22.78i,  19.09 - 29.51i;
      -14.74 - 2.4i,  19.17 + 21.33i];
[x, info] = f07vs(uplo, trans, diag, kd, ab, b);
[ferr, berr, info] = f07vv(uplo, trans, diag, kd, ab, b, x)
 

ferr =

   1.0e-13 *

    0.1803
    0.2199


berr =

   1.0e-16 *

    0.0904
    0.6571


info =

                    0



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