

   EEssttiimmaattee tthhee CCoonnddiittiioonn NNuummbbeerr ooff aa MMaattrriixx,, QQRR DDeeccoommppoossiittiioonn
   oorr FFiitt

        kappa(z, ...)
        kappa.lm     (z, ...)
        kappa.default(z, exact = FALSE)
        kappa.qr     (z, ...)
        kappa.tri    (z, exact = FALSE, ...)

   AArrgguummeennttss::

          z: A matrix or a the result of `qr' or a fit from a
             class inheriting from `"lm"'.

      exact: Should the result be exact?

   DDeessccrriippttiioonn::

        An estimate of the condition number of a matrix or of
        the R matrix of a QR decomposition, perhaps of a linear
        fit.  The condition number is defined as the ratio of
        the largest to the smallest non-zero singular value of
        the matrix.

   DDeettaaiillss::

        If `exact = FALSE' (the default) the condition number
        is estimated by a cheap approximation. Following S,
        this uses the LINPACK routine `dtrco.f'.  However, in R
        (or S) the exact calculation is also likely to be quick
        enough.

   VVaalluuee::

        The condition number, kappa, or an approximation if
        `exact=FALSE'.

   AAuutthhoorr((ss))::

        B.D. Ripley

   SSeeee AAllssoo::

        `svd' for the singular value decomposition and `qr' for
        the QR one.

   EExxaammpplleess::

        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%)

