SparseM (version 1.81)

# SparseM.solve: Linear Equation Solving for Sparse Matrices

## Description

`chol` performs a Cholesky decomposition of a symmetric positive definite sparse matrix `x` of class `matrix.csr`. `backsolve` performs a triangular back-fitting to compute the solutions of a system of linear equations in one step. `backsolve` and `forwardsolve` can also split the functionality of `backsolve` into two steps. `solve` combines `chol` and `backsolve` and will compute the inverse of a matrix if the right-hand-side is missing.

## Usage

```chol(x, …)
# S4 method for matrix.csr.chol
backsolve(r, x, k = NULL, upper.tri = NULL,
transpose = NULL, twice = TRUE, …)
forwardsolve(l, x, k = ncol(l), upper.tri = FALSE, transpose = FALSE)
solve(a, b, …)```

## Arguments

a

symmetric positive definite matrix of class `matrix.csr`.

r

object of class `matrix.csr.chol` returned by the function `chol`.

l

object of class `matrix.csr.chol` returned by the function `chol`.

x,b

vector(regular matrix) of right-hand-side(s) of a system of linear equations.

k

inherited from the generic; not used here.

upper.tri

inherited from the generic; not used here.

transpose

inherited from the generic; not used here.

twice

Logical flag: If true backsolve solves twice, see below.

further arguments passed to or from other methods.

## Details

`chol` performs a Cholesky decomposition of a symmetric positive definite sparse matrix `a` of class `matrix.csr` using the block sparse Cholesky algorithm of Ng and Peyton (1993). The structure of the resulting `matrix.csr.chol` object is relatively complicated. If necessary it can be coerced back to a `matrix.csr` object as usual with `as.matrix.csr`. `backsolve` does triangular back-fitting to compute the solutions of a system of linear equations. For systems of linear equations that only vary on the right-hand-side, the result from `chol` can be reused. Contrary to the behavior of `backsolve` in base R, the default behavior of `backsolve(C,b)` when C is a `matrix.csr.chol` object is to produce a solution to the system \(Ax = b\) where `C <- chol(A)`, see the example section. When the flag `twice` is `FALSE` then backsolve solves the system \(Cx = b\), up to a permutation -- see the comments below. The command `solve` combines `chol` and `backsolve`, and will compute the inverse of a matrix if the right-hand-side is missing. The determinant of the Cholesky factor is returned providing a means to efficiently compute the determinant of sparse positive definite symmetric matrices.

There are several integer storage parameters that are set by default in the call to the Cholesky factorization, these can be overridden in any of the above functions and will be passed by the usual "dots" mechanism. The necessity to do this is usually apparent from error messages like: Error in local(X...) increase tmpmax. For example, one can use, `solve(A,b, tmpmax = 100*nrow(A))`. The current default for tmpmax is `50*nrow(A)`. Some experimentation may be needed to select appropriate values, since they are highly problem dependent. See the code of chol() for further details on the current defaults.

## References

Koenker, R and Ng, P. (2002). SparseM: A Sparse Matrix Package for R, http://www.econ.uiuc.edu/~roger/research/home.html

Ng, E. G. and B. W. Peyton (1993), "Block sparse Cholesky algorithms on advanced uniprocessor computers", SIAM J. Sci. Comput., 14, pp. 1034-1056.

`slm` for sparse version of `lm`

## Examples

Run this code
``````# NOT RUN {
data(lsq)
class(lsq) # ->  "matrix.csc.hb"
model.matrix(lsq)->design.o
class(design.o) # -> "matrix.csr"
dim(design.o) # ->  1850  712
y <- model.response(lsq) # extract the rhs
length(y) #  1850

t(design.o) %*% design.o -> XpX
t(design.o) %*% y -> Xpy
chol(XpX) -> chol.o

b1 <- backsolve(chol.o,Xpy) # least squares solutions in two steps
b2 <- solve(XpX,Xpy)        # least squares estimates in one step
b3 <- backsolve(chol.o, forwardsolve(chol.o, Xpy),
twice = FALSE) # in three steps
## checking that these three are indeed equal :
stopifnot(all.equal(b1, b2), all.equal(b2, b3))
# }
``````

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