naginterfaces.library.sparse.direct_​real_​gen_​cond

naginterfaces.library.sparse.direct_real_gen_cond(norm, n, il, lval, iu, uval, anorm)

direct_real_gen_cond computes an estimate of the reciprocal of the condition number of a sparse matrix given an $LU$ factorization of the matrix computed by direct_real_gen_lu().

For full information please refer to the NAG Library document for f11mg

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f11/f11mgf.html

Parameters

normstr, length 1

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

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

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

$norm=\text{'I'}$

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

nint

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

ilint, array-like, shape $(:)$

Records the sparsity pattern of matrix $L$ as computed by direct_real_gen_lu().

lvalfloat, array-like, shape $(:)$

Records the nonzero values of matrix $L$ and some nonzero values of matrix $U$ as computed by direct_real_gen_lu().

iuint, array-like, shape $(:)$

Records the sparsity pattern of matrix $U$ as computed by direct_real_gen_lu().

uvalfloat, array-like, shape $(:)$

Records some nonzero values of matrix $U$ as computed by direct_real_gen_lu().

anormfloat

If $norm=\text{'1'}$ or $\text{'O'}$, the $1$-norm of the matrix $A$.

If $norm=\text{'I'}$, the $\infty$-norm of the matrix $A$.

$anorm$ may be computed by calling direct_real_gen_norm() with the same value for the argument $norm$.

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, $anorm=⟨value⟩$.

Constraint: $anorm\ge 0.0$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $1$)

On entry, $norm=⟨value⟩$.

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

Notes

direct_real_gen_cond estimates the condition number of a real sparse matrix $A$, in either the $1$-norm or the $\infty$-norm:

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

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 should be preceded by a call to direct_real_gen_norm() to compute ${\parallel A\parallel }_1$ or ${\parallel A\parallel }_{\infty }$, and a call to direct_real_gen_lu() to compute the $LU$ factorization of $A$. The function then estimates ${\Vert A^{-1}\Vert }_1$ or ${\Vert A^{-1}\Vert }_{\infty }$ and computes the reciprocal of the condition number.