Generic Toom-3 multiplication for gr_poly #2071
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Interesting! |
doc/source/gr_poly.rst
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| int gr_poly_mul_toom33(gr_poly_t res, const gr_poly_t poly1, const gr_poly_t poly2, gr_ctx_t ctx); | ||
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| Balanced Toom-3 multiplication with interpolation in five points, | ||
| using the Bodrato evaluation scheme. Assumes commutativity and division by 3. |
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What is meant by "division by 3"?
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We require that the ring supports exact division by 3, i.e. (3x)/3 = x; in other words the characteristic should be 0 or not divisible by 3.
Actually, division by 2 is required too; I should update this comment.
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Are there any requirements on the length, i.e. minimal length, for any of the functions? |
No, but this algorithm is most efficient when the operands are approximately balanced. For unbalanced products one should use an unbalanced Toom rule instead. |
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For fun, this is the theoretical speedup of Karatsuba vs classical and Toom-3 vs Karatsuba for an n x n multiply, counting arithmetic operations (in Toom, multiplication or division by 2 or 3 counts as an arithmetic operation here):
This is when using the brute forced optimal cutoffs classical -> Karatsuba -> Toom-3 for the recursive multiplications (in particular, Karatsuba and Toom-3 are not used at all when suboptimal). Of course, counting all arithmetic operations as having unit cost isn't realistic; this just gives a rough idea. The curves aren't quite smooth because Karatsuba performs a bit better when n is even, and Toom performs a bit better when n is divisible by 3. Raw number of arithmetic operations: |
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Should be a bit better asymptotically than Karatsuba, for rings where we don't have FFT.
This can be tidied up a bit (avoiding zero-padding, etc). Not optimized for squaring, nor for unbalanced products.
Would be interesting to see how Bodrato's sequence (used here) compares to a completely generic Toom-(n,m).