kappa                  package:base                  R Documentation

Estimate the Condition Number

Description:

     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.

Usage:

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

Arguments:

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

   exact: logical. Should the result be exact?

     ...: further arguments passed to or from other methods.

Details:

     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.

Value:

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

Author(s):

     B.D. Ripley

See Also:

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

Examples:

     kappa(x1 <- cbind(1,1:10))# 15.71
     kappa(x1, exact = TRUE)        # 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%)

