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The Karatsuba algorithm is the first fast computational algorithm, Merge-sort from isn't!!! The note below is written by a person who is not a specialist in this field.
In his book D. Knuth, The art of computer programming. Donald Knuth wrote when he described the A. Karatsuba method , that "this idea was not known before" and wrote that even people who multiplied numbers very fast in mind and all other sources didn't know this idea. But here you try to prescribe the idea to von Neumann!! It's an absolutely wrong statement, von Neumann didn't know the Karatsuba method and the Karatsuba idea! By the way, it's impossible to write that "Karatsuba algorithm is a notable example of "Divide and Conquer" or "Binary Splitting", since just Karatsuba invented this idea, the names "Divide and Conquer", "Binary Splitting" were called much later.
Somebody called the Karatsuba idea "Divide and Conquer", so Karatsuba didn't use somebody else' idea, somebody else used the Karatsuba idea. See you the difference? To write "Karatsuba algorithm is a notable example of "Divide and Conquer" means to write that Karatsuba used the method "Divide and Conquer" to create a fast multiplication, but Karatsuba just invented this method in computational mathematics.
The Karatsuba multiplication algorithm, The Karatsuba algorithm can be considered to be the earliest example of a binary splitting algorithm, or more generally, a divide and conquer algorithm.
I think this is wrong. I think, this is an absurd to compare von Neumann Merge Sort and the Karatsuba fast multiplication algorithm. Only after the Karatsuba algorithm the history of fast algorithms began. Von Neumann or anybody else results can not help here.
That is why the Karatsuba algorithm can be considered as the frist FAST algorithm in the history of computations. I read the Divide and Conquer topic now what is a dreadful content! I wrote the remarks which I will add also below: What relation "Merge Sort" has to fast computational algorithms? Can you increase the efficiency of calculation of, say, sinus of x, applying the von Neumann idea? So it is a great difference between the Karatsuba idea and the von Neumann idea.
Using the Karatsuba you obtain a tool for calculation of a great number of functions, intagrals and series much more effectively. Using von Neumann you obtain almoust nothing. How it is possible not only to compare these two approaches, but even to "put them in one box"? Karatsuba didn't use the "divide and Conquer" paradigma to invent his method, he just invented such a general method that permit to increase the efficiency of the computer work.
After the Karatsuba invention the name "Divide and Conquer" was introduced, not before. To equal such different by importance results as the Karatsuba method and von Neumann Sorting means to diminish the advantage of the Karatsuba result. Knuth in his "Art of Computer Programming" is writing that the Karatsuba idea was not known till years of calculations and multiplications.
Claiming that a fast sort algorithm is not important is uninformed at best. In fact, most applications depend on sorting rather than on multiplication of long numbers. None of the references really talked about any "rule of thumb" I moved those references. I doubt this rule of thumb is any close to truth I think there was a mistake, I think in this case n refers to the operands themselves and not to the number of digits Cause if Karatsuba's algorithm only got faster after such an insane amount of digits it wouldn't have any practical use Perhaps this claim should be removed or at least fixed to remove this ambiguity not to mention including an actual reference Well, either way the amguity about "n" remains, I will change n to "the operands" —Preceding unsigned comment added by Consider merging material from The Karatsuba multiplication , which has some descriptions of variants and some more sources.
Now, I do not know what's wrong, the formula or just the result. Someone might want to look into this issue. Actually, i was confused as well. Should we move excerpt from talks to article itself? Actually multiplication by 10 9 corresponds to shifting the array of "digits" each stored in one bit word by one full word. The last two choices allow decimal output and input by table look-up, without divisions or multiplications.
All the best, -- Jorge Stolfi talk Fast algorithms, and this first fast algorithm for multiplication, just been created to be useful not only for computers of today but to any computer in future, without essential changing the algorithm.
Today "multiplication by 10 N is not realizable by bit-shifts" and tomorrow "multiplication by 10 N is not realizable by bit-shifts", but fast algorithms can not loss or can not obtain any advantage from it: Florin Matei, Romania However, it doesn't explains the algorithm for compute that.
Is there the time for applying long multiplication, or just apply Karatsuba algorithm again to that expression? The lead of our article claims that this algorithm was "invented by Karatsuba". Given that the original publication was joint with Ofman, is there reliable sourcing independent of Karatsuba himself that this algorithm was invented solely by Karatsuba?
If so, what was Ofman's contribution? The history section is problematic, even if we ignore the plausible claims for Karatsuba, because that section also says "Kolmogorov was very agitated about the discovery; he communicated it at the next meeting of the seminar, which was then terminated.
This sentence should either be sourced or removed. The opening paragraphs states that Karatsuba discovered this algorithm. Although the principles or properties underlying the algorithm may indeed have been discovered, the algorithm itself would have been invented , unless it was somehow preexistent. I posted this instead of modifying the article directly in case somebody has a justification for the current wording.
From Wikipedia, the free encyclopedia. Did you intend to describe something like Ancient Egyptian multiplication? The idea to reduce multiplication to two squarings is old. How do you propose to achieve the squarings? Do you resolve the mixed scale multiplication again to a difference of two squares?
What is the complexity relation, that is, how is T 2n expressed in terms of n and T n? Did you check that? Multidigit multiplication for mathematicians. If the formalism is right, the language doesn't matter that much. But even then this discussion page is not the right place to talk about this fundamentally different "idea".
Please read and understand the paper by Bernstein, as I told You before. Now You are proposing the widely explored application of numerical FFT to polynomial and integer multiplication. What you want is to leave the formulation of a problem to the reader. This is crank style. Come back when you have an implementation with demonstrable runtimes. Retrieved from " https: Mathematics articles related to applied mathematics C-Class mathematics articles Low-Priority mathematics articles.