naginterfaces.library.lapacklin.ztpcon

naginterfaces.library.lapacklin.ztpcon(norm, uplo, diag, n, ap)

ztpcon estimates the condition number of a complex triangular matrix, using packed storage.

For full information please refer to the NAG Library document for f07uu

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f07/f07uuf.html

Parameters

normstr, length 1

Indicates whether ${\kappa }_1\left(A\right)$ or ${\kappa }_{\infty }\left(A\right)$ is estimated.

$norm=\text{'1'}$ or $\text{'O'}$

${\kappa }_1\left(A\right)$ is estimated.

$norm=\text{'I'}$

${\kappa }_{\infty }\left(A\right)$ is estimated.

uplostr, length 1

Specifies whether $A$ is upper or lower triangular.

$uplo=\text{'U'}$

$A$ is upper triangular.

$uplo=\text{'L'}$

$A$ is lower triangular.

diagstr, length 1

Indicates whether $A$ is a nonunit or unit triangular matrix.

$diag=\text{'N'}$

$A$ is a nonunit triangular matrix.

$diag=\text{'U'}$

$A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$.

nint

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

apcomplex, array-like, shape $(n\times (n+1)/2)$

The $n\times n$ triangular matrix $A$, packed by columns.

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 $norm$.

Constraint: $norm=\text{'1'}$, $\text{'O'}$ or $\text{'I'}$.

(errno $-2$)

On entry, error in parameter $uplo$.

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

(errno $-3$)

On entry, error in parameter $diag$.

Constraint: $diag=\text{'N'}$ or $\text{'U'}$.

(errno $-4$)

On entry, error in parameter $n$.

Constraint: $n\ge 0$.

Notes

ztpcon estimates the condition number of a complex triangular matrix $A$, in either the $1$-norm or the $\infty$-norm, using packed storage:

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

Note that ${\kappa }_{\infty }\left(A\right)={\kappa }_1\left(A^T\right)$.

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 ${\parallel A\parallel }_1$ or ${\parallel A\parallel }_{\infty }$ exactly, and uses Higham’s implementation of Hager’s method (see Higham (1988)) to estimate ${\Vert A^{-1}\Vert }_1$ or ${\Vert A^{-1}\Vert }_{\infty }$.

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