二分法求零点的步骤(什么是二分法求零点)(二分法求零点是什么意思)

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python代码

"""

bisection to find solve to f(x) when x in (a,b) and f(a)*f(b)<0

"""

def biSection(a,b,threshold,f):

iter=0

while a

mid = a + abs(b-a)/2.0

if abs(f(mid))< threshold:

return mid

if f(mid)*f(b)< 0:

a = mid

if f(a)*f(mid)< 0:

b=mid

iter+=1

print(str(iter)+ " a= "+str(a)+ ", b= "+str(b))

调用：a=5, b=50, threshold=1e-10, f(x) = x*x-11*x+10, 这个方程的解： 1，10，因此在（5，50）的解为10；

下面用二分法求解：

s = biSection(5, 50,1e-10, lambda x: x*x-11*x+10 )

print("solve= "+str(s))

下面是迭代求解的过程，经过38次迭代，最终得到解。

1 a= 5, b= 27.5

2 a= 5, b= 16.25

3 a= 5, b= 10.625

4 a= 7.8125, b= 10.625

5 a= 9.21875, b= 10.625

6 a= 9.921875, b= 10.625

7 a= 9.921875, b= 10.2734375

8 a= 9.921875, b= 10.09765625

9 a= 9.921875, b= 10.009765625

10 a= 9.9658203125, b= 10.009765625

11 a= 9.98779296875, b= 10.009765625

12 a= 9.998779296875, b= 10.009765625

13 a= 9.998779296875, b= 10.0042724609375

14 a= 9.998779296875, b= 10.00152587890625

15 a= 9.998779296875, b= 10.000152587890625

16 a= 9.999465942382812, b= 10.000152587890625

17 a= 9.999809265136719, b= 10.000152587890625

18 a= 9.999980926513672, b= 10.000152587890625

19 a= 9.999980926513672, b= 10.000066757202148

20 a= 9.999980926513672, b= 10.00002384185791

21 a= 9.999980926513672, b= 10.000002384185791

22 a= 9.999991655349731, b= 10.000002384185791

23 a= 9.999997019767761, b= 10.000002384185791

24 a= 9.999999701976776, b= 10.000002384185791

25 a= 9.999999701976776, b= 10.000001043081284

26 a= 9.999999701976776, b= 10.00000037252903

27 a= 9.999999701976776, b= 10.000000037252903

28 a= 9.99999986961484, b= 10.000000037252903

29 a= 9.999999953433871, b= 10.000000037252903

30 a= 9.999999995343387, b= 10.000000037252903

31 a= 9.999999995343387, b= 10.000000016298145

32 a= 9.999999995343387, b= 10.000000005820766

33 a= 9.999999995343387, b= 10.000000000582077

34 a= 9.999999997962732, b= 10.000000000582077

35 a= 9.999999999272404, b= 10.000000000582077

36 a= 9.99999999992724, b= 10.000000000582077

37 a= 9.99999999992724, b= 10.000000000254659

38 a= 9.99999999992724, b= 10.00000000009095

得到方程在区间(5,50)的解： 10.000000000009095