Wolfram

Wolfram-Cloud / Mathematica，有效地处理递归函数(Wolfram-Cloud/Mathematica, effective working with recursive functions)

我正在使用Чебышёв-polynomials，递归定义的多项式。 对于极有可能的情况，你以前从未见过它们：

f[0,x_] := 1; f[1,x_] := x; f[n_,x_] := 2 * x * f[n-1, x] - f[n-2, x]; Plot[{f[9, x],f[3, x]},{x, -1, 1}]

我发现自己在问，因为我经常使用python，如果有一种方法可以在wolfram-cloud中构建一系列函数，那么就可以简化这个过程。

因此，我必须只计算一次f[n]一次，这样我可以相当多地改善运行时间，并允许我扩展n的范围。

I am working with Чебышёв-polynomials at the moment, recursive defined polynomials. For the very likely case you never saw them before:

f[0,x_] := 1; f[1,x_] := x; f[n_,x_] := 2 * x * f[n-1, x] - f[n-2, x]; Plot[{f[9, x],f[3, x]},{x, -1, 1}]

And I found myself asking, since I usually work with python, if there is a way to build an array of functions in wolfram-cloud, to ease the process.

Thus I have to calculate every f[n] only once, allowing me to improve the run-time quite a bit and also allowing me to extend the range of n.

最满意答案

使用memoization 。

在这种情况下，memoization比平常更棘手，因为我们使用函数，而不是函数值。

Clear[cheb] cheb[0] = 1 &; cheb[1] = # &; cheb[n_] := cheb[n] = Evaluate@Expand[2 # cheb[n - 1][#] - cheb[n - 2][#]] &

Evaluate确保在提供和参数之前对Function的内部进行评估。

Use memoization.

In this case memoization is trickier than usual because we work with functions, not function values.

Clear[cheb] cheb[0] = 1 &; cheb[1] = # &; cheb[n_] := cheb[n] = Evaluate@Expand[2 # cheb[n - 1][#] - cheb[n - 2][#]] &

The Evaluate makes sure that the insides of the Function get evaluated even before supplying and argument.