基于MATLAB的Cholesky分解法

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Method

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Let L=⎡⎣⎢⎢⎢⎢⎢l11l21⋮ln10l22⋮ln2⋯ ⋯ ⋱⋯ 00⋮lnn⎤⎦⎥⎥⎥⎥⎥ $L=\left[\begin{array}{cccc}l11& 0& \cdots & 0\\ l21& l22& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ ln1& ln2& \cdots & lnn\end{array}\right]$
Use Squre Root Method, when A=LLT $A=L{L}^T$, we have:

uii=aii−∑k=1i−1u2ki−−−−−−−−−−⎷,i=1,…,n $$u_{ii}=\sqrt{a_{ii}-\underset{k=1}{\overset{i-1}{\sum }}{u}_{ki}^2},i=1,\dots ,n$$

uij=aij−∑i−1k=1uikujkuii,j=i+1,…,n $$u_{ij}=\frac{a_{ij}-\underset{k=1}{\overset{i-1}{\sum }}{u}_{ik}{u}_{jk}}{u_{ii}},j=i+1,\dots ,n$$

Using these two function to realize Cholesky factorization.

function [L]=Cholesky(A) %平方根法
    [N,N]=size(A);
    X=zeros(N,1);
    Y=zeros(N,1);
    for i=1:N
        A(i,i)=sqrt(A(i,i)-A(i,1:i-1)*A(i,1:i-1)'); if A(i,i)==0 'A is singular, no unique solution';
            break;
        end
        for j=i+1:N
            A(j,i)=(A(j,i)-A(j,1:i-1)*A(i,1:i-1)')/A(i,i); end end for x=1:N for y=1:N B(x,y)=A(x,y); %右上角元素归0 if x<y B(x,y)=0; break; end end end L=B