Mathematics · University
Numerical approximation and conditioning · revision notes
Truncation error comes from approximation; round-off comes from finite-precision arithmetic.
Key ideas and conditions
Error sources
Truncation error comes from approximation; round-off comes from finite-precision arithmetic.
Conditioning
A problem can amplify input perturbations even when the algorithm is carefully implemented.
Newton iteration
Use x_new=x−f(x)/f′(x) only when the derivative is nonzero and the iteration is meaningful.
Convergence claims
An asymptotic convergence order requires hypotheses and sufficiently close iterates, not just one successful step.
Explain without looking
- Why can Newton's method fail when starting far from a root?
- A method has error about C h². By what factor should the error decrease when h is halved in its asymptotic regime?
Further reading and review references
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