comphy (version 1.0.5)

PJacobi: The Jacobi method

Description

Implementation of the Jacobi iterative method to solve a system \(Ax=b\) of \(n\) linear equations in \(n\) unknowns.

Usage

PJacobi(A, b, x0 = NULL, tol = 1e-06, nmax = 1e+05, ddominant = TRUE)

Value

A numeric vector of length \(n\) with values approximating the system's solution.

Arguments

A

The \(n \times n\) matrix of coefficients of the unknowns in the linear system.

b

A vector of \(n\) constants representing the right-hand side of the linear system. This function does not work out solutions of homogeneous systems, where the b is a vector of zeros (null vector). Therefore input with b equal to a null vector is rejected.

x0

A vector of \(n\) starting numeric values for the iterations. If no values are entered for x0, a column of zero will be adopted by default.

tol

A real number indicating the threshold under which the relative increment from one solution approximation to the next is small enough to stop iteration. The default value is tol=1e-6.

nmax

An integer. The maximum number of iterations allowed, if convergence according to the criterion is not reached.

ddominant

A logical variable. If FALSE, the method is applied also if the matrix of coefficients is not diagonally dominant (default is TRUE).

Details

The Jacobi method guarantees a finite solution for linear systems characterised by a diagonally dominant matrix \(A\) of coefficients. This means that each element on its diagonal must be, in absolute value, larger than the sum of the absolute value of all the elements in the corresponding row.

Examples

# Simple system with solution 1,2,3
A <- matrix(c(3,1,2,-1,-4,2,1,1,7),ncol=3)
b <- c(4,-4,27)

# Solution
x <- PJacobi(A,b)
print(x)

# Start from a different point
x0 <- c(-1,2,8)
x <- PJacobi(A,b,x0)
print(x)