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, ...)
## S3 method for class 'lm':
kappa(z, ...)
## Default S3 method:
kappa(z, exact = FALSE, ...)
## S3 method for class 'qr':
kappa(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.

`kappa.tri`

is an internal function called by `kappa.qr`

.

### Value

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

.

### Author(s)

The design was inspired by (but differs considerably from)
the S function of the same name described in Chambers (1992).

### References

Chambers, J. M. (1992)
*Linear models.*
Chapter 4 of *Statistical Models in S*
eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole.

### 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%)

[Package

*base* version 2.5.0

Index]