`jacobi`

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

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

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

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.

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

jacobi