Talk:Toom–Cook multiplication

message: I'd like to propose renaming this article to "Toom-Cook multiplication" and generalize it a bit.  I've never renamed an article, and don't know what problems that might cause from external sites, so I'm not going to do it myself.
My main objection is that the Toom-Cook algorithm doesn't just specify a three-way split of the numbers; it's more general than that.  The article makes it sound like three-way is the only way.
In addition, I believe that it should be noted that any computer implementation of this algorithm (and who's doing Toom-Cook by hand?) will use a power-of-two base, instead of the base 10 used in the article.--Eliasen 23:32, 3 Oct 2004 (UTC)
thread_title:  Proposal to rename / generalize article 

message: I'm trying to understand the example.
*"To compute c(x) = a(x) * b(x), we evaluate the two polynomials a and b in five points, and multiply the results, for our example we perform the following:"
**How were the points in this example chosen, and why?
**What does c(1/0) mean?  It can't possibly mean c(x) where x is 1 divided by 0?
*"Then solve the interpolation problem:"
**What interpolation problem?  
**How is this solved by performing a number of operations?
**What, exactly, are these operations anyway?  Well, what they are isn't that hard to figure out, but why replace w3 with a value calculated from w3 and w1?  Why not create a new set of variables instead, to avoid confusion over which w3 is used?
*"The result can be recomposed with:"
**Right...  Again, how were the order et c here chosen?  Obviously it depends on the choices made in the first and second steps, but how?
/85.228.39.223 15:58, 7 September 2007 (UTC)
replies: text: These are all great questions - the existing example was sparse and omitted a lot of detail about how the algorithm works. I hope the new explanation makes more sense - please take a look and let me know if there's anything you still don't understand. Dcoetzee 01:29, 14 October 2007 (UTC)
thread_title:  The example 

message: Why were the Vandermonde-like evaluation and interpolation matrices chosen to have ascending degree along the rows in the worked example? This is the opposite of the convention used by the Bodrato paper that is often referred to in the very same example. Was it more didactic this way? It makes it a bit confusing when reading the two.  — Preceding unsigned comment added by 178.208.131.82 (talk) 20:59, 20 December 2011 (UTC)
thread_title:  Evaluation and Interpolation Vandermonde Matrices 

message: ;ideas that might help abt that coetient that slows O() -- possible emprovements abt solving the system of equations:
main creative idea is to use O(k) equations that can b written directly in the coetients that r we looking for
practicaly use special pondered sum of products from the term spitting a to term spitting b where a n b are at least one 0 or k-1 (for k*k problem ), the terms coetients let them be 2^i or 2^(-i)
we can observe that there is possibility of forming entirely a system of equation that works directly in desired coetients
maybe this might b helfull to solve faster the equation system but idk how, at least not yet, sorry
93.118.212.93 (talk) 15:26, 2 February 2013 (UTC)
thread_title:  Original research by 93.118.212.93 (Florin Matei), collapsed 

message: If you use the derivative p'(0) instead of p(-2) you get a simpler matrix inversion, for the price of reduction to 6 multiplications instead of 5. This is, because the derivative of a product needs the computation of two products: r'(0) = p'(0) \cdot q(0) + p(0) \cdot q'(0). Nonetheless this seems to be a reasonable approach to me. Does someone know whether and where this was explored? HenningThielemann (talk) 10:33, 11 August 2013 (UTC)
replies: text: I haven't worked out what you're up to but I don't see the point. The whole point of Toom-Cook is to cut down the number of multiplications. The constants divisions are much faster. Anyway Karatsuba algorithm already reduces four multiplications to three. Dmcq (talk) 12:59, 11 August 2013 (UTC)
thread_title:  Toom-Cook using derivatives 

message: IMO,  Toom-1.5 is some kind of long multiplication, except that it doesn't immediately subdivide matters into into MxN elementary multiplications.  Instead, it cuts the problem in half.  Still, close enough.
Toom-2 is basically a systematic way to derive Karatsuba's algorithm, instead of matching coefficients by brute force, which would get impractical beyond 3 even using computers.  No problem there, either.
Toom-3 is of course the logical extension of Toom-2, useful for bigger numbers of approximately the same length, and Toom-2.5 is a version optimized for numbers where one is significantly longer than the other but less than twice as long.
But "Toom-1" looks like a purely theoretical extension of Toom's idea to me: it doesn't cut the problem down, but merely replaces it with itself.   Unlike the others, it doesn't form a useful algorithm, but would result in infinite recursion if implemented.  Yet, section 2.1 claims that it "degenerates to long multiplication" — which a viable algorithm surely would, but only because such algorithm wouldn't resort to actual Toom-1 for any input.
Which means that "Toom-1" is like 1 in comparison to primes; unlike the rest, it doesn't reduce a problem any further: running Toom-1 is a non-operation just like dividing an integer by 1.  In both cases, any input is a fixed point.
84.148.246.92 (talk) 21:53, 7 October 2020 (UTC)
thread_title:  What would "Toom-1" actually be?