seidel: Solve linear systems using the Gauss-Seidel method In fastmatrix: Fast Computation of some Matrices Useful in Statistics

Description

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

Usage

 1 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.