【数值分析】LU分解解Ax=b，matlab自己编程实现

LU分解（直接三角分解，Doolittle分解）

$$Ax=b,A=LU$$
$$\{\begin{array}{l}Ly=b\\ Ux=y\end{array}$$
矩阵 $L$ 的对角元素为 $1$ ，矩阵 $U$ 的第一行和 $A$ 相同。
步骤：
$$\begin{array}{rl}1.& 矩阵L的对角元素为1，矩阵U的第一行和A相同。\\ & \\ 2.& 迭代,j=1,2,\cdots n-1\\ & \\ & 算L的第j列,L_{i,j}=\frac{A_{i,j}-{\sum }_{r=1}^{j-1}{L}_{i,r}{U}_{r,j}}{U_{j,j}},i=j+1,j+2,\cdots ,n\\ & \\ & 算U的第j+1行,U_{j+1,k}=\frac{A_{j+1,k}-{\sum }_{r=1}^j{L}_{j+1,r}{U}_{r,k}}{L_{j+1,j+1}},k=j+1,j+2,\cdots ,n\\ & \\ 3.& 回代,\\ & \\ & y_i=b_i-\sum _{j=1}^{i-1}{L}_{i,j}{y}_j,i=1,2,\cdots ,n\\ & \\ & x_i=\frac{y_i-{\sum }_{j=i+1}^n{x}_j\cdot U_{i,j}}{U_{i,i}},i=n,n-1,\cdots ,1\end{array}$$
matlab实现

%% Ax=b例子
A = [16 -12 2 4;
    12 -8 6 10;
    3 -13 9 23;
    -6 14 1 -28];
b = [17 36 -49 -54]';
[x,L,U] = LUsolve(A,b)

%% LU分解解Ax=b
% 输入方阵A,向量b
% 输出解x,L、U矩阵
function [x,L,U] = LUsolve(A,b)
    n = size(A);
    L = eye(n);
    U(1,[1:n]) = A(1,[1:end]);
    for j = 1:n-1 % 对U是行号，对L是列号
        for i = j+1:n % 算L第i行j列
            L(i,j) = A(i,j);
            for r = 1:j-1
                L(i,j) = L(i,j)- L(i,r)*U(r,j);
            end
            L(i,j) = L(i,j)/U(j,j);
        end
        for k = j+1:n % 算U第j+1行k列
            U(j+1,k) = A(j+1,k);
            for r = 1:j
                U(j+1,k) = U(j+1,k)-L(j+1,r)*U(r,k);
            end
            U(j+1,k) = U(j+1,k)/L(j+1,j+1);
        end
    end
    % 回代
    for i = 1:n
        y(i) = b(i);
        for j = 1:i-1
            y(i) = y(i)-L(i,j)*y(j);
        end
    end
    for i=n:-1:1 
        x(i) = y(i);
        for j=n:-1:i+1
            x(i) = x(i)-U(i,j)*x(j);
        end
        x(i) = x(i)/U(i,i);
    end
    x = x';
end