spam (version 2.11-1)

ordering: Extract the permutation

Description

Extract the (inverse) permutation used by the Cholesky decomposition

Usage

ordering( x, inv=FALSE)

Arguments

x

object of class spam.chol.method returned by the function chol.

inv

Return the permutation (default) or inverse thereof.

Author

Reinhard Furrer

Details

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 1:n.

Note that there exists many different algorithms to find orderings.

See the examples, they speak more than 10 lines.

See Also

chol.spam, solve.spam.

Examples

# 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],    eps=1e-15)
as.spam( RtR[iord,iord] - S,             eps=1e-15)
as.spam( t(R[,iord]) %*% R[,iord] - S, eps=1e-15)
# we use 'eps' to avoid issues close to machine precision

# 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