1 Mathematical Preliminaries and Error Analysis

Numerical analysis designs methods that produce numbers, and then asks how wrong those numbers can be. Every method in this course replaces an exact problem (a root, an integral, a solution of a differential equation) by a finite computation, and every computation is carried out with numbers that have only finitely many digits. So two kinds of error are always present:

• truncation error, made by the method itself, for example when an infinite series is cut off or a derivative is replaced by a difference quotient;
• round-off error, made by the arithmetic, because the computer stores and combines numbers to a fixed number of digits.

This section collects the calculus we shall use to bound truncation error, describes how computers represent numbers, and measures the effect of round-off.

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