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

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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.

See Also

norm; svd for the singular value decomposition and qr for the QR one.

Examples

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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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