seidel {fastmatrix}

Solve linear systems using the Gauss-Seidel method

Description

Gauss-Seidel method is an iterative algorithm for solving a system of linear equations.

Usage

seidel(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}. Gauss-Seidel method solve the equation \bold{Ax} = \bold{b}, iteratively by rewriting (\bold{L} + \bold{D})\bold{x} + \bold{Ux} = \bold{b}. Assuming that \bold{L} + \bold{D} is nonsingular leads to the iteration formula

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

jacobi

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 <- seidel(a, b, start)
z # converged in 10 iterations