在什么情况下需要多项式的泰勒级数? [英] In what situation would a taylor series for a polynomial be necessary?

问题描述

我很难理解为什么对函数使用泰勒级数以获得函数的近似值会有用，而不是在编程时仅使用函数本身.如果我可以告诉我的计算机计算e ^(.1)并给我一个确切的值，为什么我要采用近似值呢?

I'm having a hard time understanding why it would be useful to use the Taylor series for a function in order to gain an approximation of a function, instead of just using the function itself when programming. If I can tell my computer to compute e^(.1) and it will give me an exact value, why would I take an approximation instead?

推荐答案

泰勒级数通常不用于近似函数.通常，使用某种形式的极大极小多项式.

Taylor series are generally not used to approximate functions. Usually, some form of minimax polynomial is used.

Taylor级数收敛缓慢(需要很多项才能获得所需的精度)并且效率低下(它们在中心位置附近更准确，而在其中心处更不准确).泰勒级数的最大用途可能是在数学课和论文中，它们对于检查函数的性质和学习微积分很有用.

Taylor series converge slowly (it takes many terms to get the accuracy desired) and are inefficient (they are more accurate near the point around which they are centered and less accurate away from it). The largest use of Taylor series is likely in mathematics classes and papers, where they are useful for examining the properties of functions and for learning about calculus.

要逼近函数，通常使用极小极大多项式.最小最大多项式在特定情况下具有最小可能的最大误差(函数的近似间隔，可用于该多项式的次数).通常没有找到极小极大多项式的解析解.使用 Remez算法以数字方式找到它们.可以对Minimax多项式进行调整以适合特定需求，例如使相对误差或绝对误差最小化，在特定时间间隔内逼近函数等等. Minimax多项式需要比Taylor级数更少的项才能获得可接受的结果，并且它们在整个区间内分散"误差，而不是在中心变好而在末尾变差.

To approximate functions, minimax polynomials are often used. A minimax polynomial has the minimum possible maximum error for a particular situation (interval over which a function is to be approximated, degree available for the polynomial). There is usually no analytical solution to finding a minimax polynomial. They are found numerically, using the Remez algorithm. Minimax polynomials can be tailored to suit particular needs, such as minimizing relative error or absolute error, approximating a function over a particular interval, and so on. Minimax polynomials need fewer terms than Taylor series to get acceptable results, and they "spread" the error over the interval instead of being better in the center and worse at the ends.

当调用exp函数计算e ^x 时，您可能使用的是minimax多项式，因为有人为您完成了工作，并构造了评估该多项式的库例程.在大多数情况下，唯一的算术计算机处理器只能执行加，减，乘和除运算.因此，必须从这些操作中构造其他功能.前三个为您提供多项式，多项式足以逼近许多函数，例如正弦，余弦，对数和乘幂(还有一些将事物移入和移出浮点值的指数字段的附加操作).除法添加了有理函数，这对于诸如反正切的函数很有用.

When you call the exp function to compute e^x, you are likely using a minimax polynomial, because somebody has done the work for you and constructed a library routine that evaluates the polynomial. For the most part, the only arithmetic computer processors can do is addition, subtraction, multiplication, and division. So other functions have to be constructed from those operations. The first three give you polynomials, and polynomials are sufficient to approximate many functions, such as sine, cosine, logarithm, and exponentiation (with some additional operations of moving things into and out of the exponent field of floating-point values). Division adds rational functions, which is useful for functions like arctangent.

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