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NAG Toolbox: nag_lapack_zgecon (f07au)

Purpose

nag_lapack_zgecon (f07au) estimates the condition number of a complex matrix AA, where AA has been factorized by nag_lapack_zgetrf (f07ar).

Syntax

[rcond, info] = f07au(norm_p, a, anorm, 'n', n)
[rcond, info] = nag_lapack_zgecon(norm_p, a, anorm, 'n', n)

Description

nag_lapack_zgecon (f07au) estimates the condition number of a complex matrix AA, in either the 11-norm or the -norm:
κ1 (A) = A1 A11   or   κ (A) = A A1 .
κ1 (A) = A1 A-11   or   κ (A) = A A-1 .
Note that κ(A) = κ1(AH)κ(A)=κ1(AH).
Because the condition number is infinite if AA is singular, the function actually returns an estimate of the reciprocal of the condition number.

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 κ1(A)κ1(A) or κ(A)κ(A) is estimated.
norm = '1'norm='1' or 'O''O'
κ1(A)κ1(A) is estimated.
norm = 'I'norm='I'
κ(A)κ(A) is estimated.
Constraint: norm = '1'norm='1', 'O''O' or 'I''I'.
2:     a(lda, : :) – complex array
The first dimension of the array a must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,n)max(1,n)
The LULU factorization of AA, as returned by nag_lapack_zgetrf (f07ar).
3:     anorm – double scalar
If norm = '1'norm='1' or 'O''O', the 11-norm of the original matrix AA.
If norm = 'I'norm='I', the -norm of the original matrix AA.
anorm may be computed by calling nag_blas_zlange (f06ua) with the same value for the parameter norm_p.
anorm must be computed either before calling nag_lapack_zgetrf (f07ar) or else from a copy of the original matrix AA (see Section [Example]).
Constraint: anorm0.0anorm0.0.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array a The second dimension of the array a.
nn, the order of the matrix AA.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

lda work rwork

Output Parameters

1:     rcond – double scalar
An estimate of the reciprocal of the condition number of AA. rcond is set to zero if exact singularity is detected or the estimate underflows. If rcond is less than machine precision, AA is singular to working precision.
2:     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: norm_p, 2: n, 3: a, 4: lda, 5: anorm, 6: rcond, 7: work, 8: rwork, 9: 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 computed estimate rcond is never less than the true value ρρ, and in practice is nearly always less than 10ρ10ρ, although examples can be constructed where rcond is much larger.

Further Comments

A call to nag_lapack_zgecon (f07au) 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 8n28n2 real floating point operations but takes considerably longer than a call to nag_lapack_zgetrs (f07as) with one right-hand side, because extra care is taken to avoid overflow when AA is approximately singular.
The real analogue of this function is nag_lapack_dgecon (f07ag).

Example

function nag_lapack_zgecon_example
a = [ -1.34 + 2.55i,  0.28 + 3.17i,  -6.39 - 2.2i,  0.72 - 0.92i;
      -0.17 - 1.41i,  3.31 - 0.15i,  -0.15 + 1.34i,  1.29 + 1.38i;
      -3.29 - 2.39i,  -1.91 + 4.42i,  -0.14 - 1.35i,  1.72 + 1.35i;
      2.41 + 0.39i,  -0.56 + 1.47i, -0.83 - 0.69i,  -1.96 + 0.67i];

norm_p = '1';
anorm = norm(a, 1);

% Factorise a
[a, ipiv, info] = nag_lapack_zgetrf(a);

% Estimate condition number
[rcond, info] = nag_lapack_zgecon(norm_p, a, anorm);

if rcond > nag_machine_precision
  fprintf('\nEstimate of condition number = %10.2e\n', 1/rcond);
else
  fprintf('\nA is singular to working precision\n');
end
 

Estimate of condition number =   1.50e+02

function f07au_example
a = [ -1.34 + 2.55i,  0.28 + 3.17i,  -6.39 - 2.2i,  0.72 - 0.92i;
      -0.17 - 1.41i,  3.31 - 0.15i,  -0.15 + 1.34i,  1.29 + 1.38i;
      -3.29 - 2.39i,  -1.91 + 4.42i,  -0.14 - 1.35i,  1.72 + 1.35i;
      2.41 + 0.39i,  -0.56 + 1.47i, -0.83 - 0.69i,  -1.96 + 0.67i];

norm_p = '1';
anorm = norm(a, 1);

% Factorise a
[a, ipiv, info] = f07ar(a);

% Estimate condition number
[rcond, info] = f07au(norm_p, a, anorm);

if rcond > x02aj
  fprintf('\nEstimate of condition number = %10.2e\n', 1/rcond);
else
  fprintf('\nA is singular to working precision\n');
end
 

Estimate of condition number =   1.50e+02


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

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