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MANE 3351

Lecture 10, September 22

Classroom Management

Agenda

  • Update on Raspberry Pi VM
  • Lecture
  • Lab today!

Resources

Handouts

Calendar

Lecture Date Content
1 August 23 Welcome to Class, Syllabus
2 August 25 Error Analysis
3 August 30 Introduction to Jupyter Notebook
4 September 1 Markdown
5 September 6 Labor Day Holiday
6 September 8 Taylor Polynomials, Homework 1
7 September 13 Roots of Equations, Bisection Method
8 September 15 Bisection Error Analysis
9 September 20 False Position Algorithm
10 September 22 Newton Raphson Algorithm

Assignments

  • Homework 2 (assigned 9/15/2021, due 9/23/2021)
  • Homework 3 (assigned 9/22/2021, due 9/30/2021)

Lecture 10, September 22

Both the bisection and False Position methods were bracketing approaches to find roots that required two starting points that "bracketed" the root. Today's topic, Newton's method or Newton-Raphson method, require only one starting point. Newton's method also requires the knowledge of the derivative.

Geometric Inspiration

Brin (2020)1 demonstrate the geometric inspiration for Newton's method

Newton's Method

Newton's Method

  • The formula is simply
\[ x_{i+1}=x_i - \frac{f(x_i)}{f^\prime(x_i)} \]

New Example Problem

  • Consider a new function, \(f(x)=e^x+2^{-x}+2\cos(x)-6=0\)

Example Function for Newton's method

First Derivative

  • Newton's Method requires the first derivative:
\[ \frac{d}{dx}\left[e^x+2^{-x}+2\cos(x)\right]=e^x-2^{-x}\ln(x)-2\sin(x) \]

Review of Derivatives

Pseudo-code

Brin (2020)1provides the following pseudo-code

Newton's Method pseudocode

Jupyter Notebook Demonstration

Convergence

  • Cheney and Kincaid (2004)2 study the performance of Newton's methods
  • Assumptions
  • \(f\) contains two continuous derivatives, \(f^\prime\) and \(f^{\prime\prime}\)
  • \(r\) is a simple root, \(f^\prime(r)\neq 0\)
  • If \(r\) is started sufficiently close to \(r\), converges quadratically to \(r\)

  • \(|r-x_{n+1}|\leq c|r-x_n|^2\)

  • In other words, \(x_{n+1}\) has approximately twice as many correct digits as \(x_n!\)

Other Comments

Kiusalaas (2013)3 provides the following introduction to the Newton-Raphson Method

The Newton-Raphson algorithm is the best known method of finding roots for a good reason: It is simple and fast. The only drawback of the methods is that it uses the derivative \(f^\prime(x)\) of the function as well as the function \(f(x)\) itself. Therefore, the Newton-Raphson method is usable only in problems where \(f^\prime(x)\) can be readily computed.

Importance of Good Starting Point

Cheney and Kincaid (2004)2, provide several illustrations of bad starting points and the problems that can occur.

Bad Starting Points

  1. Brin, L, (2020), Tea Time Numerical Analysis: Experiences in Mathematics, 3rd edition ↩↩

  2. Cheny, W., and Kincaid, D., (2004), Numerical Mathematics and Computer, 5th edition ↩↩

  3. Kiusalaas, J., (2013), Numerical Methods in Engineering with Python 3 ↩