SparseM.solve

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Linear Equation Solving for Sparse Matrices

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.

Keywords
algebra
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.

Note

Because the sparse Cholesky algorithm re-orders the positive definite sparse matrix A, the value of x <- backsolve(C, b) does not equal the solution to the triangular system \(Cx = b\), but is instead the solution to the system \(CPx = Pb\) for some permutation matrix \(P\) (and analogously for x <- forwardsolve(C, b)). However, a little algebra easily shows that backsolve(C, forwardsolve(C, b), twice = FALSE) is the solution to the equation \(Ax=b\). Finally, if C <- chol(A) for some sparse covariance matrix A, and z is a conformable standard normal vector, then the product y <- as.matrix.csr(C) %*% z is normal with covariance matrix A irrespective of the permutation of the Cholesky factor.

References

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

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

See Also

slm for sparse version of lm

Aliases
  • SparseM.solve
  • chol,ANY-method
  • chol,matrix.csr-method
  • chol,matrix.csc-method
  • chol,matrix-method
  • chol
  • backsolve-methods
  • backsolve,ANY-method
  • backsolve
  • forwardsolve
  • solve
  • backsolve,matrix.csr.chol-method
  • forwardsolve,matrix.csr.chol-method
  • solve,ANY-method
  • solve,matrix.csr-method
Examples
# NOT RUN {
data(lsq)
class(lsq) # -> [1] "matrix.csc.hb"
model.matrix(lsq)->design.o
class(design.o) # -> "matrix.csr"
dim(design.o) # -> [1] 1850  712
y <- model.response(lsq) # extract the rhs
length(y) # [1] 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))
# }
Documentation reproduced from package SparseM, version 1.78, License: GPL (>= 2)

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