二次方程优化

Posted 2020-10-12 Mr.ZeroW

在本文中，我们初步讨论一下二次方程的最优化，先贴出结论，感兴趣的同学可以继续往下看推论的过程，不想看推论的可以直接跳到最后看例子如何应用

对于最小化方程

解是

最小值是

发现matlab中没有LDU分解的函数，只有LU分解，先贴上LDU分解的代码，需要的同学可以复制下来。然后help 一下，有说明和例子。关于原理，这里就不赘述了，都是初等行变换

之后会稍微用到这个函数，不过喜欢手算的同学可以跳过这个

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% --------------------------
%INPUT
%A: matrix;
%--------------------------
%OUTPUT
%L: lower triangular matrix
%D: diagonal matrix
%U: upper triangular matrix
%--------------------------
%Example 1
%A = [4 -1 -3/2; -1 3 1];
%ldu(A)
%
%Example 2
%syms a b c d e f
%ldu([a b c; b d e; c e f])
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function ldu(A)
nn = size(A);
n = nn(1);
A = [A, eye(n)];
r = 0;
for j = 1:(n - 1)
    r = r + 1;
    for i = (j + 1) : n
        A(i, :) = A(i, :) - A(i, j)/A(r, j)*A(r, :);      
    end
end
L = inv(A(1 :n, (n+1) : 2*n))
D = diag(diag(A))
U = D\\A(1: n , 1: n)
end

 

下面进入正题

贴上代码和运行结果

syms a b c 
ldu([a b; b c]);

 

 所以方程变成