A practical introduction to numerical methods at an elementary level. It exposes students to a range of possibilities to scientific computing. Although oriented towards Mathematica, the book can be used with other programming languages. It contains lessons on Mathematica but also assumes reasonable access to the Mathematica manual. Avoiding partial derivatives which many students study but fail to master, it covers systems of ordinary differential equations to give the student an accurate picture of scientific computing. The main purpose of the book is to teach the principles of numerical analysis. In addition, it sets out to teach some simple and useful numerical methods; to indicate the sort of techniques that are used in actual numerical software and thereby suggest what kind of performance migh be expected; to give practical advice on the assessment and enhancement of accuracy; and to show how problems requiring numerical computation arise in application. Also available is an instructor's manual (0-07-057821-4).
 

Applications
computational errors
algorithms
error
error propagation
floating-point computation
positional number systems
floating-point numbers
rounding; basic operations
numerical instability
other difficulties
rootfinding
roots
bisection method
Newton's method
functional interaction
secant method
systems of linear equations
matrices
norms and sensitivity
Gaussian elimination
accuracy of partial pivoting
linear equation solvers
large sparse systems
LU factorization
interpolation
Taylor's series
big-oh notation
existence and uniqueness
Lagrange form
inverse interpolation
least squares approximation
the least squares problem
orthogonality
the normal equations
numerical differentiation and integration
numerical differentiation
simple quadrature rates
Gaussian quadrature rules
numerical error estimates
global adaptive quadrature
ordinary differential equators
a single ODE
Euler's method
systems of ODE's
Taylor series methods
Runge-Kutta methods
overview. 
Appendices: Getting started with Mathematica
exact and approximate numerical calculations
algebraic calculations
lists
defining functions
procedural programming
expressions
defining rules
how evaluation works
delayed evaluation.

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