Description Usage Arguments Details Value Note Author(s) References See Also Examples
chol
performs a Cholesky
decomposition of a symmetric positive definite sparse matrix x
of class spam
.
1 2 3 4 5 6 7 8 9 10 
x 
symmetric positive definite matrix of class 
pivot 
should the matrix be permuted, and if, with what algorithm, see ‘Details’ below. 
method 
Currently, only 
memory 
Parameters specific to the method, see ‘Details’ below. 
eps 
threshold to test symmetry. Defaults to

... 
further arguments passed to or from other methods. 
object 
an object from a previous call to 
chol
performs a Cholesky decomposition of a symmetric
positive definite sparse matrix x
of class
spam
. Currently, there is only the block sparse Cholesky
algorithm of Ng and Peyton (1993) implemented (method="NgPeyton"
).
To pivot/permute the matrix, you can choose between the multiple minimum
degree (pivot="MMD"
) or reverse CuthillMckee (pivot="RCM"
)
from George and Lui (1981). It is also possible to furnish a specific
permutation in which case pivot
is a vector. For compatibility
reasons, pivot
can also take a logical in which for FALSE
no permutation is done and for TRUE
is equivalent to
MMD
.
Often the sparsity structure is fixed and does not change, but the
entries do. In those cases, we can update the Cholesky factor with
update.spam.chol.NgPeyton
by suppling a Cholesky factor and the
updated matrix. Notice that the structure is effectively object <
update(object, x)
. The update feature without assignement has been disabled.
The option cholupdatesingular
determines how singular matrices
are handled by update
. The function hands back an error
("error"
), a warning ("warning"
) or the value NULL
("null"
).
The Cholesky decompositions requires parameters, linked to memory
allocation. If the default values are too small the Fortran routine
returns an error to R, which allocates more space and calls the Fortran
routine again. The user can also pass better estimates of the allocation
sizes to chol
with the argument memory=list(nnzR=...,
nnzcolindices=...)
. The minimal sizes for a fixed sparsity
structure can be obtained from a summary
call, see ‘Examples’.
The output of chol
can be used with forwardsolve
and
backsolve
to solve a system of linear equations.
Notice that the Cholesky factorization of the package SparseM
is also
based on the algorithm of Ng and Peyton (1993). Whereas the Cholesky
routine of the package Matrix
are based on
CHOLMOD
by Timothy A. Davis (C
code).
The function returns the Cholesky factor in an object of class
spam.chol.
method. Recall that the latter is the Cholesky
factor of a reordered matrix x
, see also ordering
.
Although the symmetric structure of x
is needed, only the upper
diagonal entries are used. By default, the code does check for
symmetry (contrarily to base:::chol
). However,
depending on the matrix size, this is a time consuming test.
A test is ignored if
options("spam.cholsymmetrycheck")
is set to FALSE
.
If a permutation is supplied with pivot
,
options("spam.cholpivotcheck")
determines if the permutation is
tested for validity (defaults to TRUE
).
Reinhard Furrer, based on Ng and Peyton (1993) Fortran routines
Ng, E. G. and Peyton, B. W. (1993) Block sparse Cholesky algorithms on advanced uniprocessor computers, SIAM J. Sci. Comput., 14, 1034–1056.
Gilbert, J. R., Ng, E. G. and Peyton, B. W. (1994) An efficient algorithm to compute row and column counts for sparse Cholesky factorization, SIAM J. Matrix Anal. Appl., 15, 1075–1091.
George, A. and Liu, J. (1981) Computer Solution of Large Sparse Positive Definite Systems, Prentice Hall.
det.spam
, solve.spam
,
forwardsolve.spam
, backsolve.spam
and ordering
.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  # generate multivariate normals:
set.seed(13)
n < 25 # dimension
N < 1000 # sample size
Sigma < .25^abs(outer(1:n,1:n,""))
Sigma < as.spam( Sigma, eps=1e4)
cholS < chol( Sigma)
# cholS is the upper triangular part of the permutated matrix Sigma
iord < ordering(cholS, inv=TRUE)
R < as.spam(cholS)
mvsample < ( array(rnorm(N*n),c(N,n)) %*% R)[,iord]
# It is often better to order the sample than the matrix
# R itself.
# 'mvsample' is of class 'spam'. We need to transform it to a
# regular matrix, as there is no method 'var' for 'spam' (should there?).
norm( var( as.matrix( mvsample))  Sigma, type='m')
norm( t(R) %*% R  Sigma)
# To speed up factorizations, memory allocations can be optimized:
opt < summary(cholS)
# here, some elements of Sigma may be changed...
cholS < chol( Sigma, memory=list(nnzR=opt$nnzR,nnzcolindices=opt$nnzc))

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