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MANE 3351

Lecture 17, October 18

Classroom Management

Agenda

  • Lecture
  • Homework 4 Assignment

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
11 September 27 Secant Method
12 September 29 Numerical Integration
13 October 4 Simpson's Rule
14 October 6 Romberg Integration
  • I will update soon

Assignments

Looking Ahead

Lecture Date Content
17 October 18 Gaussian Quadrature (end of material for Test 1)
18 October 20 Vectors
19 October 25 Vector Operations
  • Tentative, schedule! I will update soon

Lecture 17 Content

Today's topic is Gaussian Quadrature

  • \(\int_{-1}^1f(x)dx\approx\sum_{i=1}^nw_if(x_i)+R_n\)
  • $x_i $ is the abscissa is the \(i\)-th zero of \(P_n(x)\) (Legendre polynomial)
  • \(w_i\) is the weight associate with \(x_i\)
  • \(R_n=\frac{2^{2n+1}(n!)^4}{(2n+1)[(2n)!]^3}f^{(2n)}(\xi)\)
  • Abscissas and weights are commonly tabled6

Jupyter Notebook, part 1

Gauss' Formula, Arbitrary Interval

  • \(\int_{a}^bf(y)dy\approx\frac{b-a}{2}\sum_{i=1}^nw_if(y_i)+R_n\)
  • \(y_i=\left(\frac{b-a}{2}\right)x_i+\left(\frac{b+a}{2}\right)\)
  • \(R_n=\frac{(b-a)^{2n+1}(n!)^4}{(2n+1)[(2n)!]^3}f^{(2n)}(\xi)\)
  • Note that \(x_i\) and \(w_i\) are the points and weights of Gaussian quadrature

Jupyter Notebook, part 2

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

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

  3. Chapra, S., and Canale, R., (2015), Numerical Methods for Engineers, 7th edition ↩

  4. https://www.codesansar.com/numerical-methods/integration-simpson-1-3-method-algorithm.htm ↩

  5. https://www.codesansar.com/numerical-methods/integration-simpson-3-8-method-pseudocode.htm ↩

  6. Abramowitz, M., and Stegun, I. (1972), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables ↩