Matrix (version 1.0-4)

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, norm, ...)
## S3 method for class 'sparseMatrix,character':
rcond(x, norm, useInv=FALSE, \dots)

Arguments

x

an Robject that inherits from the Matrix class.

norm

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

useInv

logical (or "Matrix" containing solve(x)). If not false, compute the reciprocal condition number as $1/(\|x\| \cdot \|x^{-1}\|)$, where $x^{-1}$ is the inverse of $x$, solve(x)

...

further arguments passed to or from other methods.

Value

• An estimate of the reciprocal condition number of x.

BACKGROUND

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).

More generally, the condition number is defined (also for non-square matrices $A$) as $$\kappa(A) = \frac{\max_{\|v\| = 1} \|A v\|}{\min_{\|v\| = 1} \|A v\|}.$$ Whenever x is not a square matrix, in our method definitions, this is typically computed via rcond(qr.R(qr(X)), ...) where X is x or t(x).

The condition number takes on values between 1 and infinity, inclusive, and can be viewed as a factor by which errors in solving linear systems with this matrix as coefficient matrix could be magnified.

rcond() computes the reciprocal condition number $1/\kappa$ with values in $[0,1]$ and can be viewed as a scaled measure of how close a matrix is to being rank deficient (aka singular).

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.

See Also

norm, kappa() from package base computes an approximate condition number of a traditional matrix, even non-square ones, with respect to the $p=2$ (Euclidean) norm. solve.

condest, a newer approximate estimate of the (1-norm) condition number, particularly efficient for large sparse matrices.

Examples

x <- Matrix(rnorm(9), 3, 3)
rcond(x)
## typically "the same" (with more computational effort):
1 / (norm(x) * norm(solve(x)))
rcond(Hilbert(9))  # should be about 9.1e-13

## For non-square matrices:
rcond(x1 <- cbind(1,1:10))# 0.05278
rcond(x2 <- cbind(x1, 2:11))# practically 0, since x2 does not have full rank

## sparse
(S1 <- Matrix(rbind(0:1,0, diag(3:-2))))
rcond(S1)
m1 <- as(S1, "denseMatrix")
all.equal(rcond(S1), rcond(m1))

## wide and sparse
rcond(Matrix(cbind(0, diag(2:-1))))

## Large sparse example ----------
m <- Matrix(c(3,0:2), 2,2)
M <- bdiag(kronecker(Diagonal(2), m), kronecker(m,m))
36*(iM <- solve(M)) # still sparse
MM <- kronecker(Diagonal(10), kronecker(Diagonal(5),kronecker(m,M)))
dim(M3 <- kronecker(bdiag(M,M),MM)) # 12'800 ^ 2
if(interactive()) ## takes about 2 seconds
  system.time(r <- rcond(M3))
## whereas this is *fast* even though it computes  solve(M3)
system.time(r. <- rcond(M3, useInv=TRUE))
if(interactive()) ## the values are not the same
  c(r, r.)  # 0.05555 0.013888
stopifnot(all.equal(r., 1/72, tol=1e-12))