| rcond {Matrix} | R Documentation |
Estimate the reciprocal of the condition number of a matrix.
This is a generic function with several methods, as seen by
showMethods(rcond).
rcond(x, norm, ...) ## S4 method for signature 'sparseMatrix,character' rcond(x, norm, useInv=FALSE, ...)
x |
an R object that inherits from the |
norm |
Character 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. |
An estimate of the reciprocal condition number of x.
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) = (max_(||v|| = 1; || Av ||)) /(min_(||v|| = 1; || Av ||)).
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/κ 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.
Golub, G., and Van Loan, C. F. (1989). Matrix Computations, 2nd edition, Johns Hopkins, Baltimore.
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.
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