naginterfaces.library.lapacklin.zpocon

naginterfaces.library.lapacklin.zpocon(uplo, n, a, anorm)

zpocon estimates the condition number of a complex Hermitian positive definite matrix $A$, where $A$ has been factorized by zpotrf().

For full information please refer to the NAG Library document for f07fu

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f07/f07fuf.html

Parameters

uplostr, length 1

Specifies how $A$ has been factorized.

$uplo=\text{'U'}$

$A=U^{H}U$, where $U$ is upper triangular.

$uplo=\text{'L'}$

$A=L{L}^H$, where $L$ is lower triangular.

nint

$n$, the order of the matrix $A$.

acomplex, array-like, shape $(n,n)$

The Cholesky factor of $A$, as returned by zpotrf().

anormfloat

The $1$-norm of the original matrix $A$, which may be computed by calling blas.zlanhe with its argument $\text{norm}=\text{'1'}$. $anorm$ must be computed either before calling zpotrf() or else from a copy of the original matrix $A$.

Returns

rcondfloat

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.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $uplo$.

Constraint: $uplo=\text{'U'}$ or $\text{'L'}$.

(errno $-2$)

On entry, error in parameter $n$.

Constraint: $n\ge 0$.

(errno $-5$)

On entry, error in parameter $anorm$.

Constraint: $anorm\ge 0.0$.

Notes

zpocon estimates the condition number (in the $1$-norm) of a complex Hermitian positive definite matrix $A$:

$${\kappa }_1\left(A\right)={\parallel A\parallel }_1{\Vert A^{-1}\Vert }_1\text{.}$$

Since $A$ is Hermitian, ${\kappa }_1\left(A\right)={\kappa }_{\infty }\left(A\right)={\parallel A\parallel }_{\infty }{\Vert A^{-1}\Vert }_{\infty }$.

Because ${\kappa }_1\left(A\right)$ is infinite if $A$ is singular, the function actually returns an estimate of the reciprocal of ${\kappa }_1\left(A\right)$.

The function should be preceded by a call to blas.zlanhe to compute ${\parallel A\parallel }_1$ and a call to zpotrf() to compute the Cholesky factorization of $A$. The function then uses Higham’s implementation of Hager’s method (see Higham (1988)) to estimate ${\Vert A^{-1}\Vert }_1$.

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