Lesson 253
Numerical Methods & Stability
Round-off · Root-finding · Stability
1:00Why computers cannot do exact real-number arithmetic, and the core methods — bisection, Newton, integration rules, and Gaussian pivoting — that keep floating-point errors under control.
By the end, you can
- Explain why 0.1 + 0.2 is not exactly 0.3 in IEEE-754 floating point.
- Distinguish round-off error from truncation error and identify which source each arises from.
- Recognize catastrophic cancellation and describe how to avoid it.
- Define condition number and stability, and explain why they are separate concepts.
- Trace two steps of bisection on a given bracket and state its convergence rate.
- Apply one Newton iteration given the update rule and state its convergence rate.
- Compare the error orders of the trapezoidal rule and Simpson's rule and compute the improvement from refinement.
- Explain why partial pivoting is the standard stable default for Gaussian elimination.
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