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NAG Toolbox: nag_lapack_dtrtrs (f07te)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dtrtrs (f07te) solves a real triangular system of linear equations with multiple right-hand sides, AX=B or ATX=B.

Syntax

[b, info] = f07te(uplo, trans, diag, a, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_dtrtrs(uplo, trans, diag, a, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dtrtrs (f07te) solves a real triangular system of linear equations AX=B or ATX=B.

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (1989) The accuracy of solutions to triangular systems SIAM J. Numer. Anal. 26 1252–1265

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies whether A is upper or lower triangular.
uplo='U'
A is upper triangular.
uplo='L'
A is lower triangular.
Constraint: uplo='U' or 'L'.
2:     trans – string (length ≥ 1)
Indicates the form of the equations.
trans='N'
The equations are of the form AX=B.
trans='T' or 'C'
The equations are of the form ATX=B.
Constraint: trans='N', 'T' or 'C'.
3:     diag – string (length ≥ 1)
Indicates whether A is a nonunit or unit triangular matrix.
diag='N'
A is a nonunit triangular matrix.
diag='U'
A is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1.
Constraint: diag='N' or 'U'.
4:     alda: – double array
The first dimension of the array a must be at least max1,n.
The second dimension of the array a must be at least max1,n.
The n by n triangular matrix A.
  • If uplo='U', a is upper triangular and the elements of the array below the diagonal are not referenced.
  • If uplo='L', a is lower triangular and the elements of the array above the diagonal are not referenced.
  • If diag='U', the diagonal elements of a are assumed to be 1, and are not referenced.
5:     bldb: – double array
The first dimension of the array b must be at least max1,n.
The second dimension of the array b must be at least max1,nrhs_p.
The n by r right-hand side matrix B.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the arrays a, b and the second dimension of the array a.
n, the order of the matrix A.
Constraint: n0.
2:     nrhs_p int64int32nag_int scalar
Default: the second dimension of the array b.
r, the number of right-hand sides.
Constraint: nrhs_p0.

Output Parameters

1:     bldb: – double array
The first dimension of the array b will be max1,n.
The second dimension of the array b will be max1,nrhs_p.
The n by r solution matrix X.
2:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
W  info>0
Element _ of the diagonal is exactly zero. A is singular and the solution has not been computed.

Accuracy

The solutions of triangular systems of equations are usually computed to high accuracy. See Higham (1989).
For each right-hand side vector b, the computed solution x is the exact solution of a perturbed system of equations A+Ex=b, where
EcnεA ,  
cn is a modest linear function of n, and ε is the machine precision.
If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form
x-x^ x cncondA,xε ,   provided   cncondA,xε<1 ,  
where condA,x=A-1Ax/x.
Note that condA,xcondA=A-1AκA; condA,x can be much smaller than condA and it is also possible for condAT to be much larger (or smaller) than condA.
Forward and backward error bounds can be computed by calling nag_lapack_dtrrfs (f07th), and an estimate for κA can be obtained by calling nag_lapack_dtrcon (f07tg) with norm_p='I'.

Further Comments

The total number of floating-point operations is approximately n2r.
The complex analogue of this function is nag_lapack_ztrtrs (f07ts).

Example

This example solves the system of equations AX=B, where
A= 4.30 0.00 0.00 0.00 -3.96 -4.87 0.00 0.00 0.40 0.31 -8.02 0.00 -0.27 0.07 -5.95 0.12   and   B= -12.90 -21.50 16.75 14.93 -17.55 6.33 -11.04 8.09 .  
function f07te_example


fprintf('f07te example results\n\n');

% Solve AX=B where A is Lower triangular
a = [ 4.30,  0,     0,    0;
     -3.96, -4.87,  0,    0;
      0.40,  0.31, -8.02, 0;
     -0.27,  0.07, -5.95, 0.12];
b = [-12.90, -21.50;
      16.75,  14.93;
     -17.55,   6.33;
     -11.04,   8.09];

uplo = 'L';
trans = 'N';
diag = 'N';

% Solve
[x, info] = f07te( ...
                   uplo, trans, diag, a, b);

% Display solution
[ifail] = x04ca( ...
                 'Gen', diag, x, 'Solution(s)');


f07te example results

 Solution(s)
             1          2
 1     -3.0000    -5.0000
 2     -1.0000     1.0000
 3      2.0000    -1.0000
 4      1.0000     6.0000

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Chapter Contents
Chapter Introduction
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