Matrix (version 0.9975-2)

rcond: Estimate the Reciprocal Condition Number

Description

Estimate the reciprocal of the condition number of a matrix.

This is a generic function with several methods, as seen by showMethods(rcond).

Usage

rcond(x, type, ...)

Arguments

x

an Robject that inherits from the Matrix class.

type

Character indicating the type of norm to be used in the estimate. The default is "O" for the 1-norm. The other possible value is "I" for the infinity norm.

...

further arguments passed to or from other methods.

Value

• An estimate of the reciprocal condition number of x.

BACKGROUND

The condition number of a matrix is the product of the matrix and the norm of its inverse (or pseudo-inverse if the matrix is not square). Since it can take on values between 1 and infinity, inclusive, it can be viewed as a measure of how close a matrix is to being rank deficient. It can also be viewed as a factor by which errors in solving linear systems with this matrix as coefficient matrix could be magnified.

Condition numbers are usually estimated, since exact computation is costly in terms of floating-point operations. An (over) estimate of reciprocal condition number is given, since by doing so overflow is avoided. Matrices are well-conditioned if the reciprocal condition number is near 1 and ill-conditioned if it is near zero.

References

Golub, G., and Van Loan, C. F. (1989). Matrix Computations, 2nd edition, Johns Hopkins, Baltimore.

Examples

x <- Matrix(rnorm(9), 3, 3)
rcond(x)
rcond(Hilbert(9))  # should be about 9.1e-13