Every algorithm built across Stages 1–3 — Gram-Schmidt, elimination, the SVD — was analysed as exact mathematics: given exact real-number inputs, the algorithms produce exact outputs. This topic opens Stage 4 by confronting the gap between that mathematical ideal and what a computer actually does: arithmetic performed in finite-precision floating point, where every single operation introduces a small rounding error. The topic builds the vocabulary — condition number, backward stability — needed to reason rigorously about when those small errors stay small and when they explode, closing the running caveat that has trailed every algorithm since Topic 2's Gram-Schmidt failure note.
By the end of this topic you should be able to:
- explain the basic model of floating-point arithmetic and machine epsilon;
- define and compute the condition number of a matrix via its singular values;
- state the concept of backward stability and explain why it is the right question to ask;
- compare the stability of Gaussian elimination with partial pivoting against Householder .
In this lesson
- Contents
- The big picture
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