已知函数表达式，估计海塞矩阵(Hessian Matrix)

最新推荐文章于 2024-08-05 08:30:41 发布

Marc Pony

最新推荐文章于 2024-08-05 08:30:41 发布

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分类专栏：数值计算文章标签：有限差分估算海塞矩阵海塞矩阵

本文链接：https://blog.youkuaiyun.com/maple_2014/article/details/128809192

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该文介绍了使用MATLAB进行Hessian矩阵的数值计算方法，通过有限中心差分法估算函数的二阶偏导数。提供了一个MATLAB函数`est_hess`，用于计算给定函数在特定点的Hessian矩阵，并通过一个符号计算的例子和实际数值计算的例子进行验证。

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一、参考资料出处

1.Computing the elements of a Hessian matrix with finite difference
2.Forward finite difference approximation for second order cross derivatives
3.4-point-like central finite difference for second partial derivatives

二、matlab实例验证

%有限中心差分估算海塞矩阵
function hess = est_hess(fun, p, nvars, delta)
hess = zeros(nvars, nvars);
for i = 1 : nvars
    for j = 1 : nvars
        if i <= j
            p1 = p;
            p1(i) = p1(i) - delta(i);
            p1(j) = p1(j) - delta(j);
            f1 = fun(p1);
            
            p2 = p;
            p2(i) = p2(i) - delta(i);
            p2(j) = p2(j) + delta(j);
            f2 = fun(p2);
            
            p3 = p;
            p3(i) = p3(i) + delta(i);
            p3(j) = p3(j) - delta(j);
            f3 = fun(p3);
            
            p4 = p;
            p4(i) = p4(i) + delta(i);
            p4(j) = p4(j) + delta(j);
            f4 = fun(p4);
            
            hess(i,  j) = (f4 - f3 - f2 + f1) / (4.0 * delta(i) * delta(j));
        else
            hess(i,  j) = hess(j,  i);
        end
    end
end

end

clc
clear

syms x y z real
f = x*y*sin(z) + x*z*sin(y) + y*z*sin(x);
hess_syms = hessian(f,[x,y,z]);
x = 1;
y = 2;
z = 3;
hess = eval(hess_syms)

fun = @(p) p(1)*p(2)*sin(p(3)) + p(1)*p(3)*sin(p(2)) + p(2)*p(3)*sin(p(1));  % x  -> p(1),  y  -> p(2),  z  -> p(3)
delta = [0.01, 0.01, 0.01];
p = [x, y, z];
nvars = 3;
est_hess = est_hess(fun, p, nvars, delta)

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