对超松弛迭代法求解线性方程组

功能： 对超松弛迭代法求解线性方程组。

function [x, k] = LinearEquations_SSOR(A, b, x0, MaxIters, err, w)
%{
函数功能：对超松弛迭代法求解线性方程组；
输入：
  A：系数矩阵
  b：常数矩阵；
  x0：初始解；
  MaxIters：最大迭代次数；
  err：精度阈值；
  w：松弛因子；
输出：
  x：近似解；
  k：迭代次数；
示例：
clear; clc;
A = [1, 1, 1; 1, 2, 3; 2, 1, 3];
b = [3; 6; 6];
x0 = [0; 0; 0];
MaxIters = 1000;
err = 1e-6;
w = 0.5;
[x, k] = LinearEquations_SSOR(A, b, x0, MaxIters, err, w);
%} 
% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 
n = length(x0);
x1 = x0;
x2 = zeros(n, 1); 
x3 = zeros(n, 1);
r = max(abs(b - A*x1));
k = 0;
while r > err
    for i = 1 : n
        sum = 0;
        for j = 1 : n
            if j > i
                sum = sum + A(i, j) * x1(j);
            elseif j < i
                sum = sum + A(i, j) * x2(j);
            end
        end
        x2(i) = (1 - w)*x1(i) + w*(b(i) - sum) / (A(i, i) + eps);
    end
    for i = n : -1 : 1
        sum = 0;
        for j = 1 : n
            if j > i
                sum = sum + A(i, j) * x3(j);
            elseif j < i
                sum = sum + A(i, j) * x2(j);
            end
        end
        x3(i) = (1 - w) * x2(i) + w * (b(i) - sum) / A(i, i);
    end
    r = max(abs(x3 - x1));
    x1 = x3;
    k = k + 1;
    if k > MaxIters
        x = [];
        return;
    end
end
x = x1;