02 二分法迭代求零點

感謝Python與機器學習算法頻道。
接着來學二分法迭代求零點。

01

對於區間$[a,b]$上單調連續函數$y=f(x）$，若$f(a)×f(b)<0$,通過不斷地把函數 $f(x)$的零點所在區間一分爲二，使區間的兩個端點逐步逼近零點（一個解），進而得到零點近似值的方法叫二分法。

02

這種方法的侷限性：如上，二分求解，給定的初始區間$[a,b]$，必須滿足$f(a)×f(b)<0$，並且這種方法只能找到一個單根。

二分求解最重要是的找出縮小區間的條件。

二分區間$[a,b]$,得到區間的中間點$mid = a + (b-a)/2.0$;

• 如果$f（mid）· f（b）< 0$，則可以縮小區間，$a = mid$;
• 如果 $f（a）· f（mid）< 0$，則可以縮小區間，$b = mid$
• 如果$f(mid) < threshold$，則認爲 mid 就是方程的根；

3 Python實現

def biSection(a,b,threshold,f):
    iter=0
    while a<b:
        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))
s = biSection(5, 50,1e-10, lambda x: x*x-11*x+10 )
print("solve= "+str(s))

$f(x) = x*x-11*x+10,a=5, b=50, threshold=1e-10$，這個方程的解： 1，10，因此在（5，50）的解爲10；
結果爲：

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
solve= 10.000000000009095