二分法与Regula Falsi法的区别

二分法用于寻找形式为 f(x) = 0 的非线性方程的方程的根，它基于中间值属性的重复应用。设 f(x) 为闭区间 [x1,x2] 内的连续函数，若 f(x1), f(x2) 符号相反，则区间 (x1,x2) 内至少有一个根 α，使得 f(α) = 0。

公式

X2= (X0 + X1) / 2

例子

问题：求方程 f(x)=x3-x-1 的根

解决方案：

给定方程 f(x)=x3-x-1

让 x = 0, 1, 2

在第一次迭代中：

f(1)=-1<0 and f(2)=5>0

Root lies between these two points 1 and 2

x0=1+2/2 = 1.5

f(x0)=f(1.5)=0.875>0

在第二次迭代中：

f(1)=-1<0 and f(1.5)=0.875>0

Root lies between these two points 1 and 1.5

x1=1+1.5/2 =1.25

f(x1)=f(1.25)=-0.29688<0

在第三次迭代中：

f(1.25)=-0.29688<0 and f(1.5)=0.875>0

Root lies between these two points 1.25 and 1.5

x2=1.25+1.5/2 = 1.375

f(x2)=f(1.375)=0.22461>0

在第四次迭代中：

f(1.25)=-0.29688<0 and f(1.375)=0.22461>0

Root lies between these two points 1.25 and 1.375

x3=1.25+1.375/2=1.3125

f(x3)=f(1.3125)=-0.05151<0

在第 5 次迭代中：

f(1.3125)=-0.05151<0 and f(1.375)=0.22461>0

Root lies between these two points 1.3125 and 1.375

x4=1.3125+1.375/2=1.34375

f(x4)=f(1.34375)=0.08261>0

在第 6 次迭代中：

f(1.3125)=-0.05151<0 and f(1.34375)=0.08261>0

Root lies between these two points 1.3125 and 1.34375

x5=1.3125+1.34375/2=1.32812

f(x5)=f(1.32812)=0.01458>0

在第 7 次迭代中：

f(1.3125)=-0.05151<0 and f(1.32812)=0.01458>0

Root lies between these two points 1.3125 and 1.32812

x6=1.3125+1.32812/2 =1.32031

f(x6)=f(1.32031)=-0.01871<0

在第 8 次迭代中：

f(1.32031)=-0.01871<0 and f(1.32812)=0.01458>0

Root lies between these two points 1.32031 and 1.32812

x7=1.32031+1.32812/2=1.32422

f(x7)=f(1.32422)=-0.00213<0

在第 9 次迭代中：

 f(1.32422)=-0.00213<0 and f(1.32812)=0.01458>0

Root lies between these two points 1.32422 and 1.32812

x8=1.32422+1.32812/2=1.32617

f(x8)=f(1.32617)=0.00621>0

在第 10 次迭代中：

f(1.32422)=-0.00213<0 and f(1.32617)=0.00621>0

Root lies between these two points 1.32422 and 1.32617

x9=1.32422+1.32617/2=1.3252

f(x9)=f(1.3252)=0.00204>0

在第 11 次迭代中：

f(1.32422)=-0.00213<0 and f(1.3252)=0.00204>0

Root lies between these two points 1.32422 and 1.3252

x10=1.32422+1.3252/2=1.32471

f(x10)=f(1.32471)=-0.00005<0

使用二分法的方程 x3-x-1=0 的近似根是 1.32471

Regula Falsi 方法：

Regula Falsi 是找到方程 f(x) = 0 的实根的最古老的方法之一，与二分法非常相似。它需要较少的计算量，因为我们每次迭代只需要评估一个函数。

公式

X3 = X1(fX2) - X2(fX1)/ f(X2) -f(X1)

例子

问题：求方程 f(x)=x3-x-1 的根

解决方案：

给定方程， x3-x-1=0

让 x = 0, 1, 2

在第一次迭代中：

 f(1)=-1<0 and f(2)=5>0

Root lies between these two points x0=1 and x1=2

x2=x0-f(x0)

= x1-x0

f(x1)-f(x0)

 x2=1-(-1)⋅

= 2-1

= 5-(-1)

 x2=1.16667

f(x2)=f(1.16667)=-0.5787<0

在第二次迭代中：

 f(1.16667)=-0.5787<0 and f(2)=5>0

Root lies between these two points x0=1.16667 and x1=2

x3=x0-f(x0)

x1-x0

f(x1)-f(x0)

x3=1.16667-(-0.5787)

2-1.16667

5-(-0.5787)

x3=1.25311

f(x3)=f(1.25311)=-0.28536<0

在第三次迭代中：

 f(1.25311)=-0.28536<0 and f(2)=5>0

Root lies between these two points x0=1.25311 and x1=2

x4=x0-f(x0)⋅

x1-x0

f(x1)-f(x0)

x4=1.25311-(-0.28536)⋅

2-1.25311

5-(-0.28536)

x4=1.29344

f(x4)=f(1.29344)=-0.12954<0

在第四次迭代中：

f(1.29344)=-0.12954<0 and f(2)=5>0

Root lies between these two points x0=1.29344 and x1=2

x5=x0-f(x0)⋅

x1-x0

f(x1)-f(x0)

x5=1.29344-(-0.12954)⋅

2-1.29344

5-(-0.12954)

x5=1.31128

f(x5)=f(1.31128)=-0.05659<0

在第 5 次迭代中：

 f(1.31128)=-0.05659<0 and f(2)=5>0

Root lies between these two points x0=1.31128 and x1=2

x6=x0-f(x0)⋅

x1-x0

f(x1)-f(x0)

x6=1.31128-(-0.05659)⋅

2-1.31128

5-(-0.05659)

x6=1.31899

f(x6)=f(1.31899)=-0.0243<0

在第 6 次迭代中：

 f(1.31899)=-0.0243<0 and f(2)=5>0

Root lies between these two points x0=1.31899 and x1=2

x7=x0-f(x0)⋅

x1-x0

f(x1)-f(x0)

x7=1.31899-(-0.0243)⋅

2-1.31899

5-(-0.0243)

x7=1.32228

f(x7)=f(1.32228)=-0.01036<0

在第 7 次迭代中：

 f(1.32228)=-0.01036<0 and f(2)=5>0

Root lies between these two points x0=1.32228 and x1=2

x8=x0-f(x0)⋅

x1-x0

f(x1)-f(x0)

x8=1.32228-(-0.01036)⋅

2-1.32228

5-(-0.01036)

x8=1.32368

使用 Regula Falsi 方法的方程 x3-x-1=0 的近似根为 1.32368

二分法与正则假法的区别

Basis	Bisection Method	Regula Falsi Method
Definition	In mathematics, the bisection method is a root-finding method that applies to continuous function for which knows two values with opposite signs.	In mathematics, the false position method is a very old method for solving equations with one unknown this method is modified form is still in use.
Simplicity	it is simple to use and easy to implement.	Simple to use as compared to Bisection Method
Computational Efforts	Less as compared to Regula Falsi Method	More as compared to Bisection Method
Iteration required	In the bisection method, if one of the initial guesses is closer to the root, it will take a large number of iterations to reach the root.	Less as compared to Bisection Method. This method can be less precise than bisection – no strict precision is guaranteed.
Convergence	The order of convergence of the bisection method is slow and linear.	This method faster order of convergence than the bisection method.
General Iterative Formula	Formula is : X3 =( X1 + X2)/2	Formula is : X3 = X1(fx2) – x2(fx1)/ f(x2) -f(x1)
Other Names	It is also known as the Bolzano method, Binary chopping method, half Interval method.	It is also known as the False Position method.