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Suggested extra credit questions for a future edition of Golub & Van Loan

Jim Demmel (UC Berkeley)

A generation of researchers and students has benefitted immensely from the textbook "Matrix Computations'' by Gene Golub and Charlie Van Loan. In this talk we imagine what a set of "extra credit'' questions for a future edition might look like. Here are two examples for section 2.4:

1. You have an array of n double precision (64-bit) floating point numbers, and are allowed to do n-1 double extended (80-bit) floating point additions to compute their sum (and no other arithmetic operations or "bit-fiddling''). How small can you make the error bound?
2. We know that you can multiply matrices in O(n^ω) operations if and only if you can invert matrices in O(n^ω) operations.
   True or false: You can multiply matrices in O(n^{ω + ε}) operations for any ε > 0 if and only if you can invert matrices "stably'' in O(n^{ω + ε}) operations for any ε > 0.