AMATH 301: Beginning Scientific Computing

Course Description

Textbook and Notes

Syllabus

1. Review of Applied Linear Algebra: Basis, range, rank vector norms, matrix norms. Special matrices: symmetric, orthogonal, lower and upper triangular, tridiagonal.
2. Introduction to Matlab: How to create and manipulate vectors and matrices, create loops and logic statements, and output data in plots and data files.
3. Direct Methods for Solving Dense Systems of Linear Equations: Gaussian elimination with partial pivoting. Solution of triangular systems, multiple right hand sides. Tridiagonal systems.
4. Solution of a Single Nonlinear Equation: Bisection method. Newton's method. Convergence to a root.
5. Interpolation: Polynomial interpolation by Lagrange polynomials. Cubic splines.
6. Numerical Quadrature: Trapezoid and Simpson quadrature. Richardson extrapolation. Infinite limits of integration.
7. Ordinary Differential Equations - Initial Value Problems: Euler's method. Accuracy and stability. Trapezoid method. Runge-Kutta method.
8. Partial Differential Equations: Basic time and space stepping techniques, both implicit and explicit.

Schedule

• Week 1
  • Lecture 1 (3/31): Introduction to matrices and vectors
  • Lecture 2 (4/1): Vectors and Matrices in Matlab
  • Lecture 3 (4/2): Logic, Loops and Iterations bisection.m
  • Lecture 4 (4/4): Plotting, importing and exporting data plot_script.m
• Week 2 - Linear Systems
  • Lecture 5 (4/7): Direct solution methods
  • Lecture 6 (4/8): More matlab practice, scoreit how to: golden_ratio.m matrix_construction.m
  • Lecture 7 (4/9): Iterative solution methods: jacobi_example.m gs_example.m
  • Lecture 8 (4/11): Eigenvalues, eigenvectors and solvability
• Week 3 - Curve Fitting
  • Lecture 9 (4/14): Least-square fitting methods iteration_example.m
  • Lecture 10 (4/15): Matlab programming practice jacobi2.m
  • Lecture 11 (4/16): Polynomial fits and splines
  • Lecture 12 (4/18): Data fitting with MATLAB fitting_demo.m gafit.m linefit.dat gaussfit.dat
• Week 4 - Numerical Differentiation and Integration
  • Lecture 13 (4/21): Numerical Differentiation
  • Lecture 14 (4/22): Homework 1 review and round-off error example round_off.m
  • Lecture 15 (4/23): Numerical Integration int_examples.m
  • Lecture 16 (4/25): COE Open House - NO CLASS
• Week 5 - Differential Equations
  • Lecture 17 (4/28): Initial value problems: Euler, Runge-Kutta and Adams methods
  • Lecture 18 (4/29): Numerical Differntiation examples diff_demo.m spring.dat
  • Lecture 19 (4/30): Error analysis for time-stepping routines
  • Lecture 20 (5/2): Boundary value problems: The Shooting Method bvp_shooting.m bvp_shooting_rhs.m bvp_shooting_error.m
  • Supplementary Lecture: Matlab ODE Solver Examples Example Code
• Week 6 - Midterm and Differential Equations
  • Lecture 21 (5/5): Midterm Review Solutions
  • Midterm in-class (5/6)
  • Lecture 22 (5/7): Implementation of the shooting method shoot_demo.m shoot2.m
  • Lecture 23 (5/9): Boundary value problems: direct solve and relaxation
• Week 7 - Fourier Transforms
  • Lecture 24 (5/12): Basics of the Fourier Transform
  • Lecture 25 (5/13): BVP - Implementing the direct solve method
  • Lecture 26 (5/14): Spectral Analysis
  • Lecture 27 (5/16): Applications of the FFT

Homework and Exams

Homework Submission

Grading

You may view your homework and exam grades on-line.

Matlab Resources

In this course we will make extensive use of Matlab, a technical computing environment for numerical computions and visualization produced by The MathWorks, Inc. A Matlab manual is available in the ICL Lab. If you are working in the Windows environment, be sure to check out the Matlab notebook feature that integrates Matlab with Microsoft Word.

Here is a list of possible useful internet resources as well:

• Matlab Hypertext Reference, Portland State University