1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
|
# Root Finding Methods
Root Finding Methods or non-linear equation solvers.
In this section we will cover four different methods to find a root of a function. Let us take a step back and think algorithmically. To find a root numerically, we can implement a search algorithm.
Let's consider a phone book - a database of names and numbers sorted alphabetically by last name. Our goal is to find the phone number of James Watt (our root). We could make an algorithm that starts at the the first page, checks to see if the current name equals our target (Watt), if it doesn't, check the next entry. This is method is called *incremental search*. As you may see, this can take very long if the database is very large.
> [!NOTE] Example Phonebook Database Entries
> **A** – Archimedes
> **B** – Bernoulli
> **C** – Curie
> **D** – Darwin
> **E** – Einstein
> **F** – Feynman
> **G** – Galileo
> **H** – Hubble
> **I** – Ibn Sina
> **J** – Joule
> **K** – Kelvin
> **L** – Leonardo
> **M** – Maxwell
> **N** – Newton
> **O** – Oppenheimer
> **P** – Pascal
> **Q** – Quetelet
> **R** – Röntgen
> **S** – Schrödinger
> **U** – Urey
> **V** – Volta
> **W** – Watt
> **X** – Xenophon
> **Y** – Yukawa
> **Z** – Zuse
Another approach may be to index the book and jump straight to the 'W' names however this requires two steps. Let us introduce *Bisection*.
## Incremental Search
Incremental search
## Bisection
This method start by sectioning the database into two halves. We can do this with a phone book by simply opening it in the middle of the book. In our data set, we have find the middle entry to be 'M - Maxwell'. Since all names are sorted alphabetically we can conclude that we will find our target in the second half, essentially eliminating the first half. Doing this a second time gives us our new midpoint 'S – Schrödinger' and a third time get us to our target 'W - Watt'. In only 3 iterations we have reached out target goal. This is a huge improvement from the Incremental search (22 iterations) versus our new 3 iterations.
This exact same concept can be applied to finding roots of functions. Instead of comparing strings we are comparing floats. For computers this is a lot easier to do. Take a moment to think about how you would program this.
```python
import numpy as np
def my_bisection(f, a, b, tol):
# approximates a root, R, of f bounded
# by a and b to within tolerance
# | f(m) | < tol with m the midpoint
# between a and b Recursive implementation
# check if a and b bound a root
if np.sign(f(a)) == np.sign(f(b)):
raise Exception(
"The scalars a and b do not bound a root")
# get midpoint
m = (a + b)/2
if np.abs(f(m)) < tol:
# stopping condition, report m as root
return m
elif np.sign(f(a)) == np.sign(f(m)):
# case where m is an improvement on a.
# Make recursive call with a = m
return my_bisection(f, m, b, tol)
elif np.sign(f(b)) == np.sign(f(m)):
# case where m is an improvement on b.
# Make recursive call with b = m
return my_bisection(f, a, m, tol)
```
### Example
```python
f = lambda x: x**2 - 2
r1 = my_bisection(f, 0, 2, 0.1)
print("r1 =", r1)
r01 = my_bisection(f, 0, 2, 0.01)
print("r01 =", r01)
print("f(r1) =", f(r1))
print("f(r01) =", f(r01))
```
## Modified Secant
We can do better.
## Newton-Raphson
|
