rcond-methods {Matrix} | R Documentation |
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, ...)
## S4 method for signature 'sparseMatrix,character'
rcond(x, norm, useInv=FALSE, ...)
Arguments
x |
an R object that inherits from the |
norm |
character string indicating the type of norm to be used in
the estimate. The default is |
useInv |
logical (or This may be an efficient alternative (only) in situations where
Note that the result may differ depending on |
... |
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 ) as
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
with values in
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
(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 if you have >= 8 GB RAM
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
## for all 4 norms available for sparseMatrix :
cbind(rr <- sapply(c("1","I","F","M"),
function(N) rcond(M3, norm=N, useInv=TRUE)))