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

## Description

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 reciprocal condition number, see the details.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```kappa(z, ...) ## Default S3 method: kappa(z, exact = FALSE, norm = NULL, method = c("qr", "direct"), ...) ## S3 method for class 'lm' kappa(z, ...) ## S3 method for class '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 One- 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`.

## Author(s)

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

## Source

The LAPACK routines `DTRCON` and `ZTRCON` and the LINPACK routine `DTRCO`.

LAPACK and LINPACK are from http://www.netlib.org/lapack and http://www.netlib.org/linpack and their guides are listed in the references.

## 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.

`norm`; `svd` for the singular value decomposition and `qr` for the QR one.
 ```1 2 3 4 5 6 7 8 9``` ```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%) ```