\Header{kappa}{Estimate the Condition Number of a Matrix, QR Decomposition or Fit}
\alias{kappa.default}{kappa}
\alias{kappa.lm}{kappa}
\alias{kappa.qr}{kappa}
\alias{kappa.tri}{kappa}
\keyword{math}{kappa}
\begin{Usage}
\begin{verbatim}
kappa(z, ...)
kappa.lm     (z, ...)
kappa.default(z, exact = FALSE)
kappa.qr     (z, ...) 
kappa.tri    (z, exact = FALSE, ...)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\texttt{z}] A matrix or a the result of \texttt{\Link{qr}} or a fit from a class
inheriting from \texttt{"lm"}.
\item[\texttt{exact}] Should the result be exact?
\end{ldescription}
\end{Arguments}
\begin{Description}
An estimate of the condition number of a matrix or of the \eqn{R}{} matrix of a
\eqn{QR}{} decomposition, perhaps of a linear fit.  The condition number is
defined as the ratio of the largest to the smallest \emph{non-zero}
singular value of the matrix.\end{Description}
\begin{Details}
If \texttt{exact = FALSE} (the default) the condition number is estimated
by a cheap approximation. Following S, this uses the LINPACK routine
\file{dtrco.f}.  However, in \R\ (or S) the exact calculation is also
likely to be quick enough.\end{Details}
\begin{Value}
The condition number, \eqn{kappa}{}, or an approximation if
\texttt{exact=FALSE}.\end{Value}
\begin{Author}
B.D. Ripley\end{Author}
\begin{SeeAlso}
\texttt{\Link{svd}} for the singular value decomposition and
\texttt{\Link{qr}} for the \eqn{QR}{} one.\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
kappa(x1 <- cbind(1,1:10))# 15.71
kappa(x1, exact=T)	  # 13.68
kappa(x2 <- cbind(x1,2:11))# high! [x2 is singular!]

hilbert <- function(n) { i <- 1:n; 1 / outer(i - 1, i, "+") }
sv9 <- svd(h9 <- hilbert(9))$ d
kappa(h9)# pretty high!
kappa(h9, exact=TRUE) == max(sv9) / min(sv9)
kappa(h9, exact=TRUE) / kappa(h9) # .677 (i.e. rel.error = 32%)
\end{ExampleCode}
\end{Examples}

