R: Cholesky Decomposition of a Sparse Matrix

Cholesky {Matrix} | R Documentation

## Cholesky Decomposition of a Sparse Matrix

### Description

Computes the Cholesky (aka “Choleski”) decomposition of a sparse, symmetric, positive-definite matrix. However, typically `chol()` should rather be used unless you are interested in the different kinds of sparse Cholesky decompositions.

### Usage

    Cholesky(A, perm = TRUE, LDL = !super, super = FALSE, Imult = 0, ...)

### Arguments

`A` | sparse symmetric matrix. No missing values or IEEE special values are allowed.
`perm` | logical scalar indicating if a fill-reducing permutation should be computed and applied to the rows and columns of `A`. Default is `TRUE`.

`LDL` | logical scalar indicating if the decomposition should be computed as LDL' where `L` is a unit lower triangular matrix. The alternative is LL' where `L` is lower triangular with arbitrary diagonal elements. Default is `TRUE`. Setting it to `NA` leaves the choice to a CHOLMOD-internal heuristic.
`super` | logical scalar indicating if a supernodal decomposition should be created. The alternative is a simplicial decomposition. Default is `FALSE`. Setting it to `NA` leaves the choice to a CHOLMOD-internal heuristic.
`Imult` | numeric scalar which defaults to zero. The matrix that is decomposed is A+m*I where m is the value of `Imult` and `I` is the identity matrix of order `ncol(A)`.
`...` | further arguments passed to or from other methods.

### Details

This is a generic function with special methods for different types of matrices. Use `showMethods("Cholesky")` to list all the methods for the `Cholesky` generic.

The method for class `dsCMatrix` of sparse matrices — the only one available currently — is based on functions from the CHOLMOD library.

Again: If you just want the Cholesky decomposition of a matrix in a straightforward way, you should probably rather use `chol(.)`.

Note that if `perm=TRUE` (default), the decomposition is

A = P' L~ D L~' P = P' L L' P,

where L can be extracted by `as(*, "Matrix")`, P by `as(*, "pMatrix")` and both by `expand(*)`, see the class `CHMfactor` documentation.

Note that consequently, you cannot easily get the “traditional” cholesky factor R, from this decomposition, as

R'R = A = P'LL'P = P' R~' R~ P = (R~ P)' (R~ P),

but R~ P is not triangular even though R~ is.

### Value

an object inheriting from either `"CHMsuper"`, or `"CHMsimpl"`, depending on the `super` argument; both classes extend `"CHMfactor"` which extends `"MatrixFactorization"`.

In other words, the result of `Cholesky()` is not a matrix, and if you want one, you should probably rather use `chol()`, see Details.

### References

Yanqing Chen, Timothy A. Davis, William W. Hager, and Sivasankaran Rajamanickam (2008) Algorithm 887: CHOLMOD, Supernodal Sparse Cholesky Factorization and Update/Downdate. ACM Trans. Math. Softw. 35, 3, Article 22, 14 pages. doi: 10.1145/1391989.1391995

Timothy A. Davis (2006) Direct Methods for Sparse Linear Systems, SIAM Series “Fundamentals of Algorithms”.

### See Also

Class definitions `CHMfactor` and `dsCMatrix` and function `expand`. Note the extra `solve(*, system = . )` options in `CHMfactor`.

Note that `chol()` returns matrices (inheriting from `"Matrix"`) whereas `Cholesky()` returns a `"CHMfactor"` object, and hence a typical user will rather use `chol(A)`.

### Examples

    data(KNex)
    mtm <- with(KNex, crossprod(mm))
    str(mtm@factors) # empty list()
    (C1 <- Cholesky(mtm))             # uses show(<MatrixFactorization>)
    str(mtm@factors) # 'sPDCholesky' (simpl)
    (Cm <- Cholesky(mtm, super = TRUE))
    c(C1 = isLDL(C1), Cm = isLDL(Cm))
    str(mtm@factors) # 'sPDCholesky'  *and* 'SPdCholesky'
    str(cm1  <- as(C1, "sparseMatrix"))
    str(cmat <- as(Cm, "sparseMatrix"))# hmm: super is *less* sparse here
    cm1[1:20, 1:20]

    b <- matrix(c(rep(0, 711), 1), nc = 1)
    ## solve(Cm, b) by default solves  Ax = b, where A = Cm'Cm (= mtm)!
    ## hence, the identical() check *should* work, but fails on some GOTOblas:
    x <- solve(Cm, b)
    stopifnot(identical(x, solve(Cm, b, system = "A")),
              all.equal(x, solve(mtm, b)))

    Cn <- Cholesky(mtm, perm = FALSE)# no permutation -- much worse:
    sizes <- c(simple = object.size(C1),
               super  = object.size(Cm),
               noPerm = object.size(Cn))
    ## simple is 100, super= 137, noPerm= 812 :
    noquote(cbind(format(100 * sizes / sizes[1], digits=4)))

    ## Visualize the sparseness:
    dq <- function(ch) paste('"',ch,'"', sep="") ## dQuote(<UTF-8>) gives bad plots
    image(mtm, main=paste("crossprod(mm) : Sparse", dq(class(mtm))))
    image(cm1, main= paste("as(Cholesky(crossprod(mm)),\"sparseMatrix\"):",
                            dq(class(cm1))))

    ## Smaller example, with same matrix as in  help(chol) :
    (mm <- Matrix(toeplitz(c(10, 0, 1, 0, 3)), sparse = TRUE)) # 5 x 5
    (opts <- expand.grid(perm = c(TRUE,FALSE), LDL = c(TRUE,FALSE), super = c(FALSE,TRUE)))
    rr <- lapply(seq_len(nrow(opts)), function(i)
                 do.call(Cholesky, c(list(A = mm), opts[i,])))
    nn <- do.call(expand.grid, c(attr(opts, "out.attr")$dimnames,
                  stringsAsFactors=FALSE,KEEP.OUT.ATTRS=FALSE))
    names(rr) <- apply(nn, 1, function(r)
                       paste(sub("(=.).*","\\1", r), collapse=","))
    str(rr, max=1)

    str(re <- lapply(rr, expand), max=2) ## each has a 'P' and a 'L' matrix

    R0 <- chol(mm, pivot=FALSE)
    R1 <- chol(mm, pivot=TRUE )
    stopifnot(all.equal(t(R1), re[[1]]$L),
              all.equal(t(R0), re[[2]]$L),
              identical(as(1:5, "pMatrix"), re[[2]]$P), # no pivoting
    TRUE)

    # Version of the underlying SuiteSparse library by Tim Davis :
    .SuiteSparse_version()

[Package Matrix version 1.3-4 Index]