Extract the (inverse) permutation used by the Cholesky decomposition
ordering( x, inv=FALSE)
- object of class
spam.chol.method returned by the function
- Return the permutation (default) or inverse thereof.
Recall that calculating a Cholesky factor from a sparse matrix consists of finding a permutation first, then calculating the factors of the permuted matrix. The ordering is important when working with the factors themselves.
The ordering from a full/regular matrix is
Note that there exists many different algorithms to find orderings.
See the examples, they speak more than 10 lines.
# Construct a pd matrix S to work with (size n) n <- 100 # dimension S <- .25^abs(outer(1:n,1:n,"-")) S <- as.spam( S, eps=1e-4) I <- diag(n) # Identity matrix cholS <- chol( S) ord <- ordering(cholS) iord <- ordering(cholS, inv=TRUE) R <- as.spam( cholS ) # R'R = P S P', with P=I[ord,], # a permutation matrix (rows permuted). RtR <- t(R) %*% R # the following are equivalent: as.spam( RtR - S[ord,ord] ) as.spam( RtR[iord,iord] - S ) as.spam( t(R[,iord]) %*% R[,iord] - S ) # trivially: as.spam( t(I[iord,]) - I[ord,]) # (P^-1)' = P as.spam( t(I[ord,]) - I[,ord]) # as.spam( I[iord,] - I[,ord]) as.spam( I[ord,]%*%S%*%I[,ord] - S[ord,ord] ) # pre and post multiplication with P and P' is ordering
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