Lecture 6 Summary

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Reviewed stability definition, and covered the special (stronger) condition of backwards stability, which is true of many algorithms and often not too hard to prove. Showed that floating-point summation of n numbers is backwards stable.

When quantifying errors, a central concept is a norm. Defined norms (as in lecture 3 of Trefethen), gave examples of Lp norms (usually p = 1, 2, or ∞).

More norms: weighted norms, Frobenius norm, and induced matrix norms. Bounded induced square-matrix norm in terms of matrix eigenvalues (we will give a more precise bound later in terms of SVDs). Showed that unitary matrices preserve L2 norms and induced norms, and also the L2 Frobenius matrix norm.

Equivalence of norms. Sketched proof that any two norms are equivalent up to a constant bound, and that this means that stability in one norm implies stability in all norms. The proof involves: (i) showing that all norms are continuous; (ii) showing that we can reduce the problem of showing any norm is equivalent to L2 on the unit circle; and (iii) a result from real analysis: a continuous function on a closed/bounded set achieves its maximum and minimum.