Lecture 11 Summary

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Considered Householder algorithm in more detail, including the detail that one has a choice of Householder reflectors...we choose the sign to avoid taking differences of nearly-equal vectors. Gave flop count, showed that we don't need to explicitly compute Q if we store the Householder reflector vectors.

Returned to Gaussian elimination. Introduced partial pivoting, and pointed out (omitting bookkeeping details) that this can be expressed as a PA=LU factorization where P is a permutation. Discussed backwards stability of LU, and gave example where U matrix grows exponentially fast with m to point out that the backwards stability result is practically useless here, and that the (indisputable) practicality of Gaussian elimination is more a result of the types of matrices that arise in practice.

Brief discussion of Cholesky factorization, and more generally of the fact that one can often take advantage of special structure if it is present in your matrix.