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CHMfactor-class CHOLMOD-based Cholesky Factorizations

Description

The virtual class "CHMfactor" is a class of CHOLMOD-based Cholesky factorizations of symmetric, sparse, compressed, column-oriented matrices. Such a factorization is simplicial (virtual class "CHMsimpl") or supernodal (virtual class "CHMsuper"). Objects that inherit from these classes are either numeric factorizations (classes "dCHMsimpl" and "dCHMsuper") or symbolic factorizations (classes "nCHMsimpl" and "nCHMsuper").

Usage

isLDL(x)

## S4 method for signature 'CHMfactor'
update(object, parent, mult = 0, ...)
.updateCHMfactor(object, parent, mult)

## and many more methods, notably,
##   solve(a, b, system = c("A","LDLt","LD","DLt","L","Lt","D","P","Pt"), ...)
##   -----    see below

Arguments

x,object,a	a "CHMfactor" object (almost always the result of Cholesky()).
parent	a "dsCMatrix" or "dgCMatrix" matrix object with the same nonzero pattern as the matrix that generated object. If parent is symmetric, of class "dsCMatrix", then object should be a decomposition of a matrix with the same nonzero pattern as parent. If parent is not symmetric then object should be the decomposition of a matrix with the same nonzero pattern as tcrossprod(parent). Since Matrix version 1.0-8, other "sparseMatrix" matrices are coerced to dsparseMatrix and CsparseMatrix if needed.
mult	a numeric scalar (default 0). mult times the identity matrix is (implicitly) added to parent or tcrossprod(parent) before updating the decomposition object.
...	potentially further arguments to the methods.

Objects from the Class

Objects can be created by calls of the form new("dCHMsuper", ...) but are more commonly created via Cholesky(), applied to dsCMatrix or lsCMatrix objects.

For an introduction, it may be helpful to look at the expand() method and examples below.

Slots

of "CHMfactor" and all classes inheriting from it:

perm:

An integer vector giving the 0-based permutation of the rows and columns chosen to reduce fill-in and for post-ordering.

colcount:

Object of class "integer" ....

type:

Object of class "integer" ....

Slots of the non virtual classes “[dl]CHM(super|simpl)”:

p:

Object of class "integer" of pointers, one for each column, to the initial (zero-based) index of elements in the column. Only present in classes that contain "CHMsimpl".

i:

Object of class "integer" of length nnzero (number of non-zero elements). These are the row numbers for each non-zero element in the matrix. Only present in classes that contain "CHMsimpl".

x:

For the "d*" classes: "numeric" - the non-zero elements of the matrix.

Methods

isLDL

(x) returns a logical indicating if x is an LDL' decomposition or (when FALSE) an LL' one.

coerce

signature(from = "CHMfactor", to = "sparseMatrix") (or equivalently, to = "Matrix" or to = "triangularMatrix")

as(*, "sparseMatrix") returns the lower triangular factor L from the LL' form of the Cholesky factorization. Note that (currently) the factor from the LL' form is always returned, even if the "CHMfactor" object represents an LDL' decomposition. Furthermore, this is the factor after any fill-reducing permutation has been applied. See the expand method for obtaining both the permutation matrix, P, and the lower Cholesky factor, L.

coerce

signature(from = "CHMfactor", to = "pMatrix") returns the permutation matrix P, representing the fill-reducing permutation used in the decomposition.

expand

signature(x = "CHMfactor") returns a list with components P, the matrix representing the fill-reducing permutation, and L, the lower triangular Cholesky factor. The original positive-definite matrix A corresponds to the product A = P'LL'P. Because of fill-in during the decomposition the product may apparently have more non-zeros than the original matrix, even after applying drop0 to it. However, the extra "non-zeros" should be very small in magnitude.

image

signature(x = "CHMfactor"): Plot the image of the lower triangular factor, L, from the decomposition. This method is equivalent to image(as(x, "sparseMatrix")) so the comments in the above description of the coerce method apply here too.

solve

signature(a = "CHMfactor", b = "ddenseMatrix"), system= *: The solve methods for a "CHMfactor" object take an optional third argument system whose value can be one of the character strings "A", "LDLt", "LD", "DLt", "L", "Lt", "D", "P" or "Pt". This argument describes the system to be solved. The default, "A", is to solve Ax = b for x where A is the sparse, positive-definite matrix that was factored to produce a. Analogously, system = "L" returns the solution x, of Lx = b. Similarly, for all system codes but "P" and "Pt" where, e.g., x <- solve(a, b, system="P") is equivalent to x <- P %*% b.

See also solve-methods.

determinant

signature(x = "CHMfactor", logarithm = "logical") returns the determinant (or the logarithm of the determinant, if logarithm = TRUE, the default) of the factor L from the LL' decomposition (even if the decomposition represented by x is of the LDL' form (!)). This is the square root of the determinant (half the logarithm of the determinant when logarithm = TRUE) of the positive-definite matrix that was decomposed.

update

signature(object = "CHMfactor"), parent. The update method requires an additional argument parent, which is either a "dsCMatrix" object, say A, (with the same structure of nonzeros as the matrix that was decomposed to produce object) or a general "dgCMatrix", say M, where A := M M' (== tcrossprod(parent)) is used for A. Further it provides an optional argument mult, a numeric scalar. This method updates the numeric values in object to the decomposition of A+mI where A is the matrix above (either the parent or M M') and m is the scalar mult. Because only the numeric values are updated this method should be faster than creating and decomposing A+mI. It is not uncommon to want, say, the determinant of A+mI for many different values of m. This method would be the preferred approach in such cases.

See Also

Cholesky, also for examples; class dgCMatrix.

Examples

## An example for the expand() method
n <- 1000; m <- 200; nnz <- 2000
set.seed(1)
M1 <- spMatrix(n, m,
               i = sample(n, nnz, replace = TRUE),
               j = sample(m, nnz, replace = TRUE),
               x = round(rnorm(nnz),1))
XX <- crossprod(M1) ## = M1'M1  = M M'  where M <- t(M1)
CX <- Cholesky(XX)
isLDL(CX)
str(CX) ## a "dCHMsimpl" object
r <- expand(CX)
L.P <- with(r, crossprod(L,P))  ## == L'P
PLLP <- crossprod(L.P)          ## == (L'P)' L'P == P'LL'P  = XX = M M'
b <- sample(m)
stopifnot(all.equal(PLLP, XX), 
          all(as.vector(solve(CX, b, system="P" )) == r$P %*% b),
          all(as.vector(solve(CX, b, system="Pt")) == t(r$P) %*% b) )

u1 <- update(CX, XX,    mult=pi)
u2 <- update(CX, t(M1), mult=pi) # with the original M, where XX = M M'
stopifnot(all.equal(u1,u2, tol=1e-14))

   ## [ See  help(Cholesky)  for more examples ]
   ##        -------------