Estimate the Condition Number of a Matrix, QR Decomposition or Fit

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 Should the result be exact?

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.

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=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%)

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