Algorithms for Programmers - Ideas, Source Code by J. Arndt

By J. Arndt

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G. the revbin-permuted) index k. h]. The generated codes can be of great use when one wants to spot parts of the original code that need further optimization. Especially repeated trigonometric values and unused symmetries tend to be apparent in the unrolled code. g. multiplications, additions, load/stores) of the code it emits. It is even better if those numbers are compared to the corresponding values found in the compiled assembler code. It is possible to have GCC produce the assembler code with the original source interlaced (which is a great tool with code optimization.

Memory access is without skips) 6. Multiply each matrix element (index r, c) by exp(−σ 2 π i r c/n). 7. Apply a (length R) FFT on each column (of the transposed matrix). (memory access with C-skips) Note that steps 3, 4 and 5 constitute a length-C convolution on each row. cc] The algorithm can be used for mass storage convolutions if care is taken to minimize the number of disk seeks. Minimization of the number of disk seeks The number of disk seeks has to be kept minimal because these are slow operations which, if occur too often, degrade performance unacceptably.

That is, a R × C ‘matrix’ of R rows (of length C) and C columns (of length R). 10: Multidimensional FTs 29 For a m-dimensional array ax (where x = (x1 , x2 , x3 , . . , xm ) and xi ∈ 0, 1, 2, . . , Si ) the m-dimensional Fourier transform ck (where k = (k1 , k2 , k3 , . . , km ) and ki ∈ 0, 1, 2, . . k ... x1 =0 x2 =0 where z = eσ 2 π i/n , n = S 1 S 2 . . k where S = (S1 − 1, S2 − 1, . . 56b) x=0 The inverse transform is again the complex conjugate transform. 57) y=0 which shows that the 2-dimensional FT can be obtained by first applying 1-dimensional transforms on the rows and then applying 1-dimensional transforms on the columns.

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