Search results
Results from the WOW.Com Content Network
The Karatsuba algorithm is a fast multiplication algorithm. It was discovered by Anatoly Karatsuba in 1960 and published in 1962. [ 1 ] [ 2 ] [ 3 ] It is a divide-and-conquer algorithm that reduces the multiplication of two n -digit numbers to three multiplications of n /2-digit numbers and, by repeating this reduction, to at most n log 2 3 ...
For fast Fourier transforms (FFTs) (or any linear transformation) the complex multiplies are by constant coefficients c + di (called twiddle factors in FFTs), in which case two of the additions (d−c and c+d) can be precomputed. Hence, only three multiplies and three adds are required. [28]
As making the partial products is () and the final addition is (), the total multiplication is (), not much slower than addition. From a complexity theoretic perspective, the Wallace tree algorithm puts multiplication in the class NC 1. The downside of the Wallace tree, compared to naive addition of partial products, is its much higher ...
Magma contains asymptotically fast algorithms for all fundamental integer and polynomial operations, such as the Schönhage–Strassen algorithm for fast multiplication of integers and polynomials. Integer factorization algorithms include the Elliptic Curve Method, the Quadratic sieve and the Number field sieve. Algebraic number theory
In general, Toom-k runs in Θ(c(k) n e), where e = log(2k − 1) / log(k), n e is the time spent on sub-multiplications, and c is the time spent on additions and multiplication by small constants. [1] The Karatsuba algorithm is equivalent to Toom-2, where the number is split into two smaller ones.
It is faster than the standard matrix multiplication algorithm for large matrices, with a better asymptotic complexity, although the naive algorithm is often better for smaller matrices. The Strassen algorithm is slower than the fastest known algorithms for extremely large matrices, but such galactic algorithms are not useful in practice, as ...
The Schönhage–Strassen algorithm was the asymptotically fastest multiplication method known from 1971 until 2007. It is asymptotically faster than older methods such as Karatsuba and Toom–Cook multiplication , and starts to outperform them in practice for numbers beyond about 10,000 to 100,000 decimal digits. [ 2 ]
The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop: