matlab雅克比迭代法程序收敛性代码实现
时间: 2023-10-31 11:29:14 浏览: 260
以下是 MATLAB 实现雅可比迭代法并检查其收敛性的代码:
% 设置问题参数
n = 100; % 问题大小
A = gallery('poisson', n); % 创建离散 Poisson 问题的系数矩阵
b = ones(n^2, 1); % 创建 RHS 向量
% 预备迭代法参数
x0 = zeros(n^2, 1); % 初始猜测向量
maxit = 1000; % 迭代最大次数
tol = 1e-8; % 迭代容错
% Jacobi 迭代法主循环
x = x0;
for k = 1:maxit
xprev = x;
for i = 1:n^2
aii = A(i,i); % 得到系数矩阵对角线元素
xi = x(i); % 得到 X 中未知数的当前值
xi_new = (b(i) - A(i,:) * x) / aii + xi; % 雅可比迭代
x(i) = xi_new; % 更新 X 中的未知数
end
% 检查是否达到收敛容错
err = norm(x - xprev, inf);
if err < tol
break;
end
end
% 输出 Jacobi 迭代的结果
fprintf('通过 %d 个迭代,Jacobi 法求解的渐近误差为 %e\n', k, err);
如果收敛,算法应该在指定的步数内停止(这里是 1000 步)并输出最终结果。如果它不收敛,则可能需要考虑使用其他迭代法,例如 Gauss-Seidel 或 SOR。
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