Math 428: Course Outline for Exam #1 Preparation

The following material has been covered in the first part of the course. Your notes should follow the chronological development of these topics. I tend to use a ``just in time philosophy'' so that theoretical ideas are introduced right before we need them. Hopefully, everyone is starting to see the Big Picture, and so I am listing the topics in conceptual order rather than chronological order.

• Foundations & Essentials.

  • Taylor's Theorem.

  • Triangle inequality.

  • Prolongations and restrictions.

  • Big O notation.

  • Lipschitz continuity.

  • Existence and uniqueness for ODEs.

• Quadratures.

  • Methods of integration.

    • Midpoint rule.

    • Trapezoid rule.

    • Simpson's rule.

    • Simpson's $\frac{3}{8}$ rule.

  • Analysis of quadratures.

    • Taylor series.

    • Polynomial test function.

    • Combining two methods of the same order to make a more accurate method.

    • Designing new quadratures by finding interpolating polynomials.

    • Gaussian quadratures.

• Numerical solution to ODEs.

  • Algorithms.

    • Euler's method.

    • Backward Euler's method.

    • Trapezoid method.

    • New methods using other quadratures.

    • Taylor series methods.

  • Analysis of methods.

    • Accuracy via local truncation error and manipulating norms.

    • Stability via the scalar test equation and solution of recurrence relations.