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NAG Toolbox

NAG Toolbox: nag_lapack_dtrcon (f07tg)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dtrcon (f07tg) estimates the condition number of a real triangular matrix.

Syntax

[rcond, info] = f07tg(norm_p, uplo, diag, a, 'n', n)
[rcond, info] = nag_lapack_dtrcon(norm_p, uplo, diag, a, 'n', n)

Description

nag_lapack_dtrcon (f07tg) estimates the condition number of a real triangular matrix A, in either the 1-norm or the -norm:
κ1 A = A1 A-11   or   κ A = A A-1 .  
Note that κA=κ1AT.
Because the condition number is infinite if A is singular, the function actually returns an estimate of the reciprocal of the condition number.
The function computes A1 or A exactly, and uses Higham's implementation of Hager's method (see Higham (1988)) to estimate A-11 or A-1.

References

Higham N J (1988) FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation ACM Trans. Math. Software 14 381–396

Parameters

Compulsory Input Parameters

1:     norm_p – string (length ≥ 1)
Indicates whether κ1A or κA is estimated.
norm_p='1' or 'O'
κ1A is estimated.
norm_p='I'
κA is estimated.
Constraint: norm_p='1', 'O' or 'I'.
2:     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'.
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.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array a and the second dimension of the array a.
n, the order of the matrix A.
Constraint: n0.

Output Parameters

1:     rcond – double scalar
An estimate of the reciprocal of the condition number of A. rcond is set to zero if exact singularity is detected or if the estimate underflows. If rcond is less than machine precision, then A is singular to working precision.
2:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.

Accuracy

The computed estimate rcond is never less than the true value ρ, and in practice is nearly always less than 10ρ, although examples can be constructed where rcond is much larger.

Further Comments

A call to nag_lapack_dtrcon (f07tg) involves solving a number of systems of linear equations of the form Ax=b or ATx=b; the number is usually 4 or 5 and never more than 11. Each solution involves approximately n2 floating-point operations but takes considerably longer than a call to nag_lapack_dtrtrs (f07te) with one right-hand side, because extra care is taken to avoid overflow when A is approximately singular.
The complex analogue of this function is nag_lapack_ztrcon (f07tu).

Example

This example estimates the condition number in the 1-norm of the matrix A, 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 .  
The true condition number in the 1-norm is 116.41.
function f07tg_example


fprintf('f07tg example results\n\n');

% Estimate condition number of A, 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];

norm_p = '1';
uplo = 'L';
diag = 'N';
[rcond, info] = f07tg( ...
                       norm_p, uplo, diag, a);

fprintf('Estimate of condition number = %9.2e\n', 1/rcond);


f07tg example results

Estimate of condition number =  1.16e+02

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