# kappa

##### Compute or Estimate the Condition Number of a Matrix

The condition number of a regular (square) matrix is the product of
the *norm* of the matrix and the norm of its inverse (or
pseudo-inverse), and hence depends on the kind of matrix-norm.

`kappa()`

computes by default (an estimate of) the 2-norm
condition number of a matrix or of the \(R\) matrix of a \(QR\)
decomposition, perhaps of a linear fit. The 2-norm condition number
can be shown to be the ratio of the largest to the smallest
*non-zero* singular value of the matrix.

`rcond()`

computes an approximation of the **r**eciprocal
**cond**ition number, see the details.

- Keywords
- math

##### Usage

```
kappa(z, …)
# S3 method for default
kappa(z, exact = FALSE,
norm = NULL, method = c("qr", "direct"), …)
# S3 method for lm
kappa(z, …)
# S3 method for qr
kappa(z, …)
```.kappa_tri(z, exact = FALSE, LINPACK = TRUE, norm = NULL, …)

rcond(x, norm = c("O","I","1"), triangular = FALSE, …)

##### Arguments

- z, x
A matrix or a the result of

`qr`

or a fit from a class inheriting from`"lm"`

.- exact
logical. Should the result be exact?

- norm
character string, specifying the matrix norm with respect to which the condition number is to be computed, see also

`norm`

. For`rcond`

, the default is`"O"`

, meaning the**O**ne- or 1-norm. The (currently only) other possible value is`"I"`

for the infinity norm.- method
a partially matched character string specifying the method to be used;

`"qr"`

is the default for back-compatibility, mainly.- triangular
logical. If true, the matrix used is just the lower triangular part of

`z`

.- LINPACK
logical. If true and

`z`

is not complex, the LINPACK routine`dtrco()`

is called; otherwise the relevant LAPACK routine is.- …
further arguments passed to or from other methods; for

`kappa.*()`

, notably`LINPACK`

when`norm`

is not`"2"`

.

##### Details

For `kappa()`

, if `exact = FALSE`

(the default) the 2-norm
condition number is estimated by a cheap approximation. However, the
exact calculation (via `svd`

) is also likely to be quick
enough.

Note that the 1- and Inf-norm condition numbers are much faster to
calculate, and `rcond()`

computes these * reciprocal*
condition numbers, also for complex matrices, using standard Lapack
routines.

`kappa`

and `rcond`

are different interfaces to
*partly* identical functionality.

`.kappa_tri`

is an internal function called by `kappa.qr`

and
`kappa.default`

.

Unsuccessful results from the underlying LAPACK code will result in an error giving a positive error code: these can only be interpreted by detailed study of the FORTRAN code.

##### Value

The condition number, \(kappa\), or an approximation if
`exact = FALSE`

.

##### References

Anderson. E. and ten others (1999)
*LAPACK Users' Guide*. Third Edition. SIAM.
Available on-line at
http://www.netlib.org/lapack/lug/lapack_lug.html.

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

Dongarra, J. J., Bunch, J. R., Moler, C. B. and Stewart, G. W. (1978)
*LINPACK Users Guide.* Philadelphia: SIAM Publications.

##### See Also

`norm`

;
`svd`

for the singular value decomposition and
`qr`

for the \(QR\) one.

##### Examples

`library(base)`

```
# NOT RUN {
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) # 0.677 (i.e., rel.error = 32%)
# }
```

*Documentation reproduced from package base, version 3.6.1, License: Part of R 3.6.1*

### Community examples

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