kappa {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` |
Should the result be exact? |

### 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%)