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 $Q R$ 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

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 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".

Value

The condition number, $k a p p a$ , or an approximation if exact = FALSE.

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.

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.

Examples

Run this code

# 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%)
# }

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