Ads
related to: fast exponentiation algorithm calculator
Search results
Results From The WOW.Com Content Network
In mathematics and computer programming, exponentiating by squaring is a general method for fast computation of large positive integer powers of a number, or more generally of an element of a semigroup, like a polynomial or a square matrix. Some variants are commonly referred to as square-and-multiply algorithms or binary exponentiation.
Inputs An integer b (base), integer e (exponent), and a positive integer m (modulus) Outputs The modular exponent c where c = b e mod m. Initialise c = 1 and loop variable e′ = 0; While e′ < e do Increment e′ by 1; Calculate c = (b ⋅ c) mod m; Output c; Note that at the end of every iteration through the loop, the equation c ≡ b e ...
To calculate 16 n−k mod (8k + 1) quickly and efficiently, the modular exponentiation algorithm is done at the same loop level, not nested. When its running 16x product becomes greater than one, the modulus is taken, just as for the running total in each sum. Now to complete the calculation, this must be applied to each of the four sums in turn.
In mathematics and computer science, optimal addition-chain exponentiation is a method of exponentiation by a positive integer power that requires a minimal number of multiplications. Using the form of the shortest addition chain , with multiplication instead of addition, computes the desired exponent (instead of multiple) of the base .
The algorithm begins with a multiprecision integer T and reduces it one word at a time. First an appropriate multiple of N is added to make T divisible by B. Then a multiple of N is added to make T divisible by B 2, and so on. Eventually T is divisible by R, and after division by R the algorithm is in the same place as REDC was after the ...
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 ...
The Schönhage–Strassen algorithm is an asymptotically fast multiplication algorithm for large integers, published by Arnold Schönhage and Volker Strassen in 1971. [1] It works by recursively applying fast Fourier transform (FFT) over the integers modulo 2 n + 1 {\displaystyle 2^{n}+1} .
Let us first calculate exp(J). We have = () The exponential of a 1×1 matrix is just the exponential of the one entry of the matrix, so exp(J 1 (4)) = [e 4]. The exponential of J 2 (16) can be calculated by the formula e (λI + N) = e λ e N mentioned above; this yields [23]