Matlab:查找由微分方程给出的y值的根 [英] Matlab: Find a root of y-values given by a differential equation

问题描述

我已经解决了一个初始值微分方程，并绘制了ODE45给定的Y值.从图中可以隐约地知道根应该在哪里，但是在给定的任务中，我需要非常精确地找到它.

I've solved an initial value differential equation and plotted the Y values given by ODE45. From the plot I can vaguely tell where the root should be, but in the given task I need to find it with great accuracy.

我的第一个猜测是将多项式调整为X和Y值，然后求解多项式方程.但是我使用了polyfit并有69个已知值，这给了我68级的多项式，我无法解决.因此，有人知道我如何在不知道实际方程的情况下找到一组给定的Y值的根"吗?在任务中写道，应该使用插值！

My first guess was to adjust a polynom to my X and Y values, and then solve the polynom equation. But I used polyfit and had 69 know values which gave me a polynom of the 68grade which I couldn't solve. So, does anyone know how I could find a "root" to a set of given Y values without knowing the actual equation? It's written in the task that interpolation should be used!

提前谢谢！

推荐答案

给定一个Y值的向量(按照相应的X值稳定增长的顺序排列)，您可能会很容易发现根位于哪个X值附近.根要么是Y值为零，要么是两个连续Y值之间的正负号，它们会更改符号.这个代码片段说明了这个想法:

Given a vector of Y values (ordered in the sense that the corresponding X values are steadily increasing), you may find easily near which X values the roots are located. The roots are either where a Y value is zero or between two consecutive Y values that change sign. The idea is illustrated in this code snippet:

X = -1:0.1:1;
Y = X.*X - 0.4;

root_exact_pos  = find(Y==0);
root_approx_pos = find(diff(sign(Y))~=0);

根在X值中，或者在X(root_exact_pos(k))中，或者在X(root_approx_pos(k))和X(root_approx_pos(k)+1)之间，k从1到各个根位置数组的元素数.

The roots are in the X values, either in X(root_exact_pos(k)), or between X(root_approx_pos(k)) and X(root_approx_pos(k)+1), k going from 1 to the number of elements of the respective root position array.

从现在开始，您可以应用任何想要的插值来找到更好的根近似值(我想在两点之间使用线性).

From here on you may apply whatever interpolation you'd like to find a better approximation of the root (I would go with linear, between the 2 points).

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