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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_ztpcon (f07uu)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_lapack_ztpcon (f07uu) estimates the condition number of a complex triangular matrix, using packed storage.


[rcond, info] = f07uu(norm_p, uplo, diag, n, ap)
[rcond, info] = nag_lapack_ztpcon(norm_p, uplo, diag, n, ap)


nag_lapack_ztpcon (f07uu) estimates the condition number of a complex triangular matrix A, in either the 1-norm or the -norm, using packed storage:
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.


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


Compulsory Input Parameters

1:     norm_p – string (length ≥ 1)
Indicates whether κ1A or κA is estimated.
norm_p='1' or 'O'
κ1A is estimated.
κA is estimated.
Constraint: norm_p='1', 'O' or 'I'.
2:     uplo – string (length ≥ 1)
Specifies whether A is upper or lower triangular.
A is upper triangular.
A is lower triangular.
Constraint: uplo='U' or 'L'.
3:     diag – string (length ≥ 1)
Indicates whether A is a nonunit or unit triangular matrix.
A is a nonunit triangular matrix.
A is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1.
Constraint: diag='N' or 'U'.
4:     n int64int32nag_int scalar
n, the order of the matrix A.
Constraint: n0.
5:     ap: – complex array
The dimension of the array ap must be at least max1,n×n+1/2
The n by n triangular matrix A, packed by columns.
More precisely,
  • if uplo='U', the upper triangle of A must be stored with element Aij in api+jj-1/2 for ij;
  • if uplo='L', the lower triangle of A must be stored with element Aij in api+2n-jj-1/2 for ij.
If diag='U', the diagonal elements of A are assumed to be 1, and are not referenced; the same storage scheme is used whether diag='N' or ‘U’.

Optional Input Parameters


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 the estimate underflows. If rcond is less than machine precision, 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

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


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_ztpcon (f07uu) involves solving a number of systems of linear equations of the form Ax=b or AHx=b; the number is usually 5 and never more than 11. Each solution involves approximately 4n2 real floating-point operations but takes considerably longer than a call to nag_lapack_ztptrs (f07us) with one right-hand side, because extra care is taken to avoid overflow when A is approximately singular.
The real analogue of this function is nag_lapack_dtpcon (f07ug).


This example estimates the condition number in the 1-norm of the matrix A, where
A= 4.78+4.56i 0.00+0.00i 0.00+0.00i 0.00+0.00i 2.00-0.30i -4.11+1.25i 0.00+0.00i 0.00+0.00i 2.89-1.34i 2.36-4.25i 4.15+0.80i 0.00+0.00i -1.89+1.15i 0.04-3.69i -0.02+0.46i 0.33-0.26i ,  
using packed storage. The true condition number in the 1-norm is 70.27.
function f07uu_example

fprintf('f07uu example results\n\n');

% Condition number of A, where A is Complex Lower triangular and packed.
n = int64(4);
ap = [ 4.78 + 4.56i;  2.00 - 0.30i;  2.89 - 1.34i; -1.89 + 1.15i;
                     -4.11 + 1.25i;  2.36 - 4.25i;  0.04 - 3.69i;
                                     4.15 + 0.80i; -0.02 + 0.46i;
                                                    0.33 - 0.26i];

norm_p = '1';
uplo   = 'L';
diag   = 'N';

% Resciprocal condition estimate
[rcond, info] = f07uu( ...
                       norm_p, uplo, diag, n, ap);

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

f07uu example results

Estimate of condition number =  3.74e+01

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Chapter Contents
Chapter Introduction
NAG Toolbox

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