Math 210 Introduction to Mathematical Computing

Instructors: R. Israel, israel@math.ubc.ca, and D. Sjerve, sjer@math.ubc.ca.

Grading: Final exam 45%, midterms 35% , homework 20%.

This course is intended for second year students majoring in mathematics. It gives an introduction to algebraic, numerical and graphical computational tools which are useful in many areas of mathematics, using the Maple computer algebra system. Some elementary programming in Maple will be included, but the main focus is on the use of Maple’s built-in commands.

Text (Recommended): M.L. Abell and J.P. Braselton, Maple V by Example, Academic Press 1994, ISBN 0-12-041545-3. Other references (on 1 day reserve in Math library):

Bauldry, W. and Fiedler, J., Calculus laboratories with Maple: a tool, not an oracle, QA 303.5 D37 B38 1991

Blachman, N. and Mossinghoff, M., Maple V quick reference, QA 76.95 B52 1994 Devitt, J., Calculus with Maple V, QA 303.5 D37 D48 1993 Heck, A., Introduction to Maple, QA 155.7 E4 H43 1993 Israel, R.B., Calculus the Maple Way, (to appear shortly).

Outline of Topics

1. Solution of nonlinear equations and systems:

(a) Equations in one variable. Introduction to “solve”. Floating point computations. Plotting. Newton’s Method and iteration. Cobweb diagrams. Stability and instability of fixed points and cycles.

(b) Systems of polynomials in several variables. Resultants and elimination. Graph-

ical methods. Newton’s Method.

2. Integration:

(a) Symbolic and numerical integration. Methods of numerical quadrature: midpoint, trapezoid, Simpson’s and Newton-Cotes rules. Richardson extrapolation.

(b) Fourier series. Oscillatory integrals.

(c) Improper integrals. Removal of singularities.

3. Series:

(a) Convergence of series. Evaluation of sums and errors.

(b) Power series. Computation of Taylor expansions for explicit and implicit functions and integrals.

(c) Asymptotic series. Euler-Maclaurin summation formula.

Robert B. Israel Jan. 15, 1996