Every algorithm in this unit — Gram-Schmidt, elimination, the SVD's construction — was analysed assuming exact real-number arithmetic. This note opens Stage 4 by describing what a computer actually does instead: represent numbers in a finite, discrete set (floating point), with every arithmetic operation rounding its result to the nearest representable value. This single fact — not any flaw in the algorithms themselves — is the root cause of every numerical subtlety flagged earlier in the unit, from Gram-Schmidt's cancellation problem onward.
Learning objectives. By the end of this note you should be able to:
- describe the floating-point number system and machine epsilon;
- state the basic rounding-error model for a single arithmetic operation;
- distinguish absolute from relative error and explain why relative error is the natural measure;
- identify catastrophic cancellation directly from the rounding-error model.
In this lesson
- The floating-point number system
- The rounding-error model
- Catastrophic cancellation
Before you start
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