【数值分析】choleskey分解，matlab实现

平方根分解（Choleskey分解）

$$A=G{G}^{\mathrm{T}},A对称正定$$
$$\begin{array}{r}A=LDM=LD{L}^{\mathrm{T}}=(L{D}^{1/2})(L{D}^{1/2}{)}^{\mathrm{T}}=G{G}^{\mathrm{T}}\end{array}$$
$$\{\begin{array}{l}Gy=b\\ \\ G^{\mathrm{T}}x=y\end{array}$$
手算的话根据转置的性质直接把 $G$ 矩阵写出来就行，非常好写，编程直接参考同济《现代数值计算》算法2.2.3。
matlab编程实现：

%% Choleskey分解解线性方程组
function [x,G] = cholesSolve(A,b)
    n = size(A);
    for i = 1:n
        t = 0;
        for s = 1:i-1
            t = t+ G(i,s)^2;
        end
        G(i,i) = sqrt(A(i,i)-t);
        for k = i+1:n
            t = 0;
            for s = 1:i-1
                t = t+G(i,s)*G(k,s);
            end
            G(k,i) = (A(k,i)-t)/G(i,i);
        end
    end
    % 回代
    for i = 1:n
        t = 0;
        for j = 1:i-1
            t = t+G(i,j)*y(j);
        end
        y(i)=(b(i)-t)/G(i,i);
    end
    for i = n:-1:1
        t = 0;
        for j = i+1:n
            t = t+G(j,i)*x(j);
        end
        x(i) = (y(i)-t)/G(i,i);
    end
    x = x';
end