Algorithms by Jeff Erickson

By Jeff Erickson

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3 But we can speed this up by implicitly hard-coding the sample positions into the algorithm, To convert from samples to coefficients, we can simply multiply the sample vector by the inverse of V , again in O(n2 ) time. 4 3 In fact, Lagrange’s formula is just a special case of Cramer’s rule for solving linear systems. Actually, it is possible to invert an n × n matrix in o(n3 ) time, using fast matrix multiplication algorithms that closely resemble Karatsuba’s sub-quadratic divide-and-conquer algorithm for integer/polynomial multiplication.

Edu/~jeffe/teaching/algorithms/ for the most recent revision. 18 Lecture 2: Fast Fourier Transforms [Fa’10] Algorithms Calvin: Here’s another math problem I can’t figure out. What’s 9+4? Hobbes: Ooh, that’s a tricky one. You have to use calculus and imaginary numbers for this. Calvin: IMAGINARY NUMBERS?! Hobbes: You know, eleventeen, thirty-twelve, and all those. It’s a little confusing at first. Calvin: How did YOU learn all this? You’ve never even gone to school! Hobbes: Instinct. Tigers are born with it.

B) Exactly how many blits does the algorithm perform when n is a power of two? (c) Describe how to modify the algorithm so that it works for arbitrary n, not just powers of two. How many blits does your modified algorithm perform? 13 Lecture 1: Recursion [Fa’10] Algorithms C A D B A B C D A B C D B D A C Two algorithms for rotating a pixel map. Black arrows indicate blitting the blocks into place; white arrows indicate recursively rotating the blocks. The first rotation algorithm (blit then recurse) in action.

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