Algorithms for Programmers by Arndt J

By Arndt J

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Variant of the MFA is called ‘four step FFT’ in [34]. CHAPTER 1. THE FOURIER TRANSFORM 33 modern CPUs. The matrix Fourier algorithm has a much better memory locality (cf. [34]), because the work is done in the short FFTs over the rows and columns. For the reason given above the computation of the column FFTs should not be done in place. One can insert additional transpositions in the algorithm to have the columns lie in contiguous memory when they are worked upon. The easy way is to use an additional scratch space for the column FFTs, then only the copying from and to the scratch space will be slow.

Note that once one has routines for both cyclic and negacyclic convolution the parts h(0) and h(1) can be computed as sum and difference, respectively. Thereby all expressions of the form α h(0) + β h(1) can be trivially computed. 4 Half cyclic convolution for half the price ? 7 (without computing h(1) ) is called half cyclic convolution. Apparently, one asks for less information than one gets from the acyclic convolution. One might hope to find an algorithm that computes h(0) and uses only half the memory compared to the linear convolution or that needs half the work, possibly both.

CONVOLUTIONS +-| 0: 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 44 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0- 2 3 4 5 6 7 8 9 10 11 12 13 14 15 01- 3 4 5 6 7 8 9 10 11 12 13 14 15 012- 4 5 6 7 8 9 10 11 12 13 14 15 0123- 5 6 7 8 9 10 11 12 13 14 15 01234- 6 7 8 9 10 11 12 13 14 15 012345- 7 8 9 10 11 12 13 14 15 0123456- 8 9 10 11 12 13 14 15 01234567- 9 10 11 12 13 14 15 012345678- 10 11 12 13 14 15 0123456789- 11 12 13 14 15 012345678910- 12 13 14 15 01234567891011- 13 14 15 0123456789101112- 14 15 012345678910111213- 15 01234567891011121314- Here the products that enter with negative sign are indicated with a postfix minus at the corresponding entry.

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