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.

1 |

`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 |

`LDL` |
logical scalar indicating if the decomposition should be
computed as LDL' where |

`super` |
logical scalar indicating if a supernodal decomposition
should be created. The alternative is a simplicial decomposition.
Default is |

`Imult` |
numeric scalar which defaults to zero. The matrix that is
decomposed is |

`...` |
further arguments passed to or from other methods. |

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.

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.

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.
http://doi.acm.org/10.1145/1391989.1391995

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

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)`

.

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 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 | ```
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()
``` |

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