Electronic Materials for Numerical Methods 

 

This page accompanies a presentation of the same title delivered as part of the session THE BEST APPROXIMATION OF A GOOD NUMERICAL METHODS COURSE, organized by Kyle Riley (South Dakota School of Mines & Technology) at the 2006 MathFest.  The presenter and creator of this page (and of the materials) is Cindy Wyels, California State University Channel Islands.

 

Textbook references are to Burden and Faires’ Numerical Analysis, 8th Ed.  

 

Background:  Math 448, Scientific Computing, is an upper division interdisciplinary course at CSU Channel Islands.  It is required of Computer Science majors and is an elective for Mathematics majors.  These materials were created by a mathematician with no special expertise in the area, teaching the course for the first time. 

 

Feel free to adopt and/or adapt these materials for your own use.  I request that you note credit appropriately.  (E.g. a common footnote for many materials I’ve used is of the sort “© K. Fogel, Spring ’04;  adapted by C. Wyels, F’06.”)

 

 

Classification of Materials by Topic

Computer arithmetic

  • Rounding
  • Chopping
  • Approximating irrationals
  • Rounding while performing Gaussian elimination

Calculus

  • First-year calculus topics
  • Taylor polynomials, infinite series
  • Root finding and fixed-point iteration

Polynomial Interpolation

·        Cubic splines

Linear Algebra

  • Intro to basic concepts
  • Gaussian elimination
  • Partial pivoting and scaled partial pivoting

Differential Equations

  • Intro to basic concepts
  • Euler’s method
  • Higher-order Taylor’s methods

Numerical Integration

  • Lagrange interpolating polynomials
  • Newton-Cotes methods
  • Composite numerical integration
  • Adaptive quadrature

Classification of Materials by Type

 

Lab:  students are expected to work in pairs.  Each is to work through the material provided while engaging her partner in discussion, using the partner as a resource, etc.  Students are expected (and often directed) to refer to the textbook and supplemental materials as necessary.

 

Lecture aid:  the material is constructed as the backbone of an interactive lecture.  It typically consists of demonstrations, often interspersed with questions.

 

Demo/ student resource (Demo/ SR):  the material is at most demonstrated in class;  its main use is for students to use as a resource out-of-class (e.g. while working on homework).  (All other materials are posted for students as well.)

 

Mixed:  usually alternates elements of lab (students work individually or in pairs) and interactive lecture.

 

Finding Materials

Use the main section headers (left) to jump to materials pertaining to specific topics.

 

 

 

Topic(s)

Activity Type

Resource Type

Comments

Computer arithmetic

Rounding

Lab

Worksheet

Problem #3 (to motivate future discussion);  problem #4

Demo/ SR

Maple, html

Notes and discusses Maple’s default use of bankers’ rounding.

Chopping

Demo/ SR

Maple, html

Implements the chopping algorithm provided by the authors;  illustrates its use in summing 1/n! for n from 1 to 10 first forwards then backwards.

Approximating irrationals

Lab

Maple, html Worksheet

#1 illustrates and compares three methods of approximating Pi.

Rounding while performing Gaussian elimination

Demo/ SR

Maple, html

Provides an algorithm to incorporate rounding arithmetic (to user-specified SF) while performing row-by-row Gaussian elimination.

Calculus

First-year calculus topics

 

Taylor polynomials, infinite series

Lab

Worksheet

Problems #1 and #2 ask students to review/ explore basic calculus concepts.

Lecture

Maple, html

Reviews calculus concepts (uses student work from worksheet above), includes optimization, limits, integrals, FToC, Taylor polynomials, infinite series.  Resource for students re Maple commands.

Root finding and fixed-point iteration

Lab

Worksheet

Problem  #5 asks students to work with and compare root-finding via Bisection and Newton’s methods (preprogrammed).

Lab

Maple, html

Worksheet

Explores the Bisection Method and Fixed-Point methods;  introduces relationship between fixed points and roots of associated functions.

Lecture

Maple, html

Example giving graphical “meaning” of Newton’s method and the secant method

Polynomial Interpolation

Cubic splines

Lecture

Maple, html

Shows how the process of constructing cubic splines can be inferred from the pieces of the definition.

Mixed lecture/ lab

Maple, html

·        Pictures and commands to explain what splines are, advantages over Lagrange interpolation.

·        Section w/ commands that might be needed (cheat sheet).  Instructions to walk through construction of cubic spline – challenging!

Linear Algebra

Intro to basic concepts

Lecture

Worksheet

Uses heat transfer example to motivate need to solve systems of linear equations; walks through process and explores computational complexity.

Gaussian elimination

 

Lecture

Maple, html

Builds on paper/pencil example above to develop an algorithm for Gaussian elimination;  provides situation that provokes questions about accuracy of GE and other numerical methods.

Demo/ SR

Maple, html

Provides an algorithm to incorporate rounding arithmetic (to user-specified SF) while performing row-by-row Gaussian elimination.

Partial pivoting and scaled partial pivoting

Lecture

Maple, html

Worksheet

Compares accuracy of four methods for solving a linear system (by example);  teaches partial pivoting;  teaches scaled partial pivoting; introduces condition number.

Differential Equations

Intro to basic concepts

Lecture

Maple, html

Provides commands to solve 1st-order DEs and plot solutions together with direction fields.  Last example is of unstable DE.

Euler’s method

Lecture

Maple, html

Demonstrates Euler’s method and develops idea behind higher-order Taylor’s methods.

Higher-order Taylor’s methods

 

 

 

Numerical Integration

Via Lagrange interpolating polynomials

 

Newton-Cotes methods

Lab

Maple, html

Worksheet

Works through approximating def. integrals via integrating the first and second Lagrange interpolating poly.   Explores degrees of polynomials vis-ŕ-vis numerical methods providing exact results.  Introduces more general Newton-Cotes open and closed approaches.

Composite numerical integration

Lecture

Maple, html

 

Introduces composite numerical methods;  works out algorithm for composite midpoint rule.

Adaptive quadrature

Lecture

Maple, html

 

Example using composite Simpson’ rule to achieve specified accuracy;  step-by-step demonstration of adaptive quadrature