spam (version 0.20-3)

spam solve: Linear Equation Solving for Sparse Matrices

Description

backsolve and forwardsolve solve a system of linear equations where the coefficient matrix is upper or lower triangular. solve solves a linear system or computes the inverse of a matrix if the right-hand-side is missing.

Usage

solve(a, b, ...)
backsolve(r, x, ...)forwardsolve(l, x, ...)

Arguments

a

symmetric positive definite matrix of class spam or a Cholesky factor as the result of a chol call.

l,r

object of class spam or spam.chol.method returned by the function chol.

x,b

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

...

further arguments passed to or from other methods, see Details below.

Details

We can solve A %*% x = b by first computing the Cholesky decomposition A = t(R)%*%R), then solving t(R)%*%y = b for y, and finally solving R%*%x = y for x. solve combines chol, a Cholesky decomposition of a symmetric positive definite sparse matrix, with forwardsolve and then backsolve. In case a is from a chol, then solve is an efficient way to calculate backsolve(a,forwardsolve( t(a),b)).

forwardsolve and backsolve solve a system of linear equations where the coefficient matrix is lower (forwardsolve) or upper (backsolve) triangular. Usually, the triangular matrix is result from a chol call and it is not required to transpose it for forwardsolve. Note that arguments of the default methods k, upper.tri and transpose do not have any effects here.

If the right-hand-side in solve is missing it will compute the inverse of a matrix. For details about the specific Cholsesky decomposition, see chol.

Recall that the Cholesky factors are from ordered matrices.

References

See references in chol.

See Also

det, chol and ordering.

Examples

# Generate multivariate form a covariance inverse:
# (usefull for GRMF)
set.seed(13)
n <- 25    # dimension
N <- 1000  # sample size
Sigmainv <- .25^abs(outer(1:n,1:n,"-"))
Sigmainv <- as.spam( Sigmainv, eps=1e-4)


Sigma <- solve( Sigmainv)  # for verification 
iidsample <- array(rnorm(N*n),c(n,N))

mvsample <- backsolve( chol(Sigmainv), iidsample)
norm( var(t(mvsample)) - Sigma, type="HS")

# compare with:
mvsample <- backsolve( chol(as.matrix( Sigmainv)), iidsample)
norm( var(t(mvsample)) - Sigma, type="HS")



# 'solve' step by step:
b <- rnorm( n)
R <- chol(Sigmainv)
norm( backsolve( R, forwardsolve( R, b))-
      solve( Sigmainv, b),type="HS") 
norm( backsolve( R, forwardsolve( R, diag(n)))- Sigma,type="HS")