嵌套数值积分 [英] Nested numerical integration

问题描述

链接中的问题: 可以进行分析集成，答案是4，但是我对使用Matlab进行数值集成感兴趣，因为它的形式类似于我无法解析地集成的问题.由于两个内部积分中的函数是x，y和z和z的函数，因此出现了数值积分的困难.

The problem in the link: can be integrated analytically and the answer is 4, however I'm interested in integrating it numerically using Matlab, because it's similar in form to a problem that I can't integrate analytically. The difficulty in the numerical integration arises because the function in the two inner integrals is a function of x,y and z and z can't be factored out.

推荐答案

好吧，这很奇怪，因为在发布者之前类似的问题上，我声称无法做到这一点，而现在，在查看了Guddu的回答后，我意识到了没有那么复杂.我之前写的数字积分结果是数字而不是函数，这是正确的–但是，要点是:一个人可以定义一个函数来评估每个给定参数的积分，这样，一个就可以有效地做到这一点. 具有数字积分的功能.

Well, this is strange, because on the poster's similar previous question I claimed this can't be done, and now after having looked at Guddu's answer I realize its not that complicated. What I wrote before, that a numerical integration results in a number but not a function, is true – but beside the point: One can just define a function that evaluates the integral for every given parameter, and this way effectively one does have a function as a result of a numerical integration.

无论如何，都在这里

function q = outer

    f = @(z) (z .* exp(inner(z)));
    q = quad(f, eps, 2);

end

function qs = inner(zs)
% compute \int_0^1 1 / (y + z) dy for given z

    qs = nan(size(zs));
    for i = 1 : numel(zs)
        z = zs(i);
        f = @(y) (1 ./ (y + z));
        qs(i) = quad(f, 0 , 1);
    end

end

我在注释中应用了我自己建议的简化，消除了x.函数inner计算y上的内部积分的值，作为z的函数.然后，函数external计算z上的外部积分.通过让积分从eps而不是0运行，我避免了z = 0时的极点.结果是

I applied the simplification suggested by myself in a comment, eliminating x. The function inner calculates the value of the inner integral over y as a function of z. Then the function outer computes the outer integral over z. I avoid the pole at z = 0 by letting the integration run from eps instead of 0. The result is

4.00000013663955

inner必须使用for循环来实现，因为赋予quad的函数需要能够同时为多个参数值返回其值.

inner has to be implemented using a for loop because a function given to quad needs to be able to return its value simultaneously for several argument values.

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