jacobi: Solve linear systems using the Jacobi method

In fastmatrix: Fast Computation of some Matrices Useful in Statistics

Solve linear systems using the Jacobi method

Description

Jacobi method is an iterative algorithm for solving a system of linear equations.

Usage

jacobi(a, b, start, maxiter = 200, tol = 1e-7)

Arguments

a	a square numeric matrix containing the coefficients of the linear system.
b	a vector of right-hand sides of the linear system.
start	a vector for initial starting point.
maxiter	the maximum number of iterations. Defaults to 200
tol	tolerance level for stopping iterations.

Details

Let \bold{D}, \bold{L}, and \bold{U} denote the diagonal, lower triangular and upper triangular parts of a matrix \bold{A}. Jacobi's method solve the equation \bold{Ax} = \bold{b}, iteratively by rewriting \bold{Dx} + (\bold{L} + \bold{U})\bold{x} = \bold{b}. Assuming that \bold{D} is nonsingular leads to the iteration formula

\bold{x}^{(k+1)} = -\bold{D}^{-1}(\bold{L} + \bold{U})\bold{x}^{(k)} + \bold{D}^{-1}\bold{b}

Value

a vector with the approximate solution, the iterations performed are returned as the attribute 'iterations'.

References

Golub, G.H., Van Loan, C.F. (1996). Matrix Computations, 3rd Edition. John Hopkins University Press.

See Also

seidel

Examples

a <- matrix(c(5,-3,2,-2,9,-1,3,1,-7), ncol = 3)
b <- c(-1,2,3)
start <- c(1,1,1)
z <- jacobi(a, b, start)
z # converged in 15 iterations

fastmatrix documentation built on Oct. 12, 2023, 5:14 p.m.