Fundamentals of Numerical Mathematics for Physicists and Engineers. Alvaro Meseguer
(1.2) starting from the same initial guess
Table 1.1 Iterates resulting from using bisection
and Newton–Raphson when solving (1.2), with added shading of converged digits.
|
Bisection |
Newton–Raphson |
|
|
0 | 1.5 | 1.5 |
|
|
1 | 1.75 | 1.869 565 215 355 94 |
|
|
2 | 1.875 | 1.799 452 405 786 30 |
|
|
3 | 1.812 5 | 1.796 327 970 874 37 |
|
|
4 | 1.781 25 | 1.796 321 903 282 30 |
|
|
5 | 1.796 875 | 1.796 321 903 259 44 |
|
|
6 | 1.789 062 5 | 1.796 321 903 259 44 |
|
|
7 | 1.792 968 75 |
|
||
8 | 1.794 921 875 |
|
||
|
|
|
||
46 | 1.796 321 903 259 45 |
|
||
47 | 1.796 321 903 259 44 | |||
48 | 1.796 321 903 259 44 |
1.4 Order of a Root‐Finding Method
From Table 1.1, it is clear that the Newton–Raphson algorithm converges much faster than the bisection method. In numerical mathematics it is crucial