closed-form solution(闭合解)，analytical solution(解析解)，numerical solution(数值解)

1. 闭合解也称解析解

"An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally accepted set. For example, an infinite sum would generally not be considered closed-form. However, the choice of what to call closed-form and what not is rather arbitrary since a new "closed-form" function could simply be defined in terms of the infinite sum." 

--Wolfram Alpha

"In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a finite number of certain "well-known" functions. Typically, these well-known functions are defined to be elementary functions—constants, one variable x, elementary operations of arithmetic (+ − × ÷), nth roots, exponent and logarithm (which thus also include trigonometric functions and inverse trigonometric functions). Often problems are said to be tractable if they can be solved in terms of a closed-form expression." 

-- Wikipedia

简单来说，解析解就是给出解的具体函数形式，y = f (x) , 给一个 x , 可以计算出对应的 y .

2. 数值解

简单来说，数值解是以数值的方式求解，给出一系列对应的自变量和解。数值的方式有数值逼近，数值插值等 .

3. 例子一

问题 ： x ^ 2 = 5

解析解：x = sqrt (5)

数值解：x = 2.236

4. 例子二

问题：求函数的根 f(x)=x−5

解析解： 令 f(x)=x−5=0得到x=5

数值解 :

我们先猜 x=1发现 f(1)=1−5=−4为一个负数

我们再猜 x = 6  发现 f(6) = 6 - 5 = 1 为一个正数

所以解在 1 到 6 中间

然后猜测 x = (1+6)/2 继续进行解的逼近 ...

这也叫做二分求解

数值方法看起来挺麻烦的，之所以用之，是因为有时候当我们面对一个复杂的方程时候，该方程的解析解是不存在，或者，获得该方程的解析精确解需要 15个小时的计算时间，而获得该方程的数值近似解只需要 15 秒 ，此时我们选择后者或许更好。