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The m-th term of any constant-recursive sequence (such as Fibonacci numbers or Perrin numbers) where each term is a linear function of k previous terms can be computed efficiently modulo n by computing A m mod n, where A is the corresponding k×k companion matrix. The above methods adapt easily to this application.
For algorithms describing how to calculate the remainder, see Division algorithm.) The remainder, as defined above, is called the least positive remainder or simply the remainder . [ 2 ] The integer a is either a multiple of d , or lies in the interval between consecutive multiples of d , namely, q ⋅ d and ( q + 1) d (for positive q ).
The functions deg() and rem() denote the degree of a polynomial and the remainder of the Euclidean division. In the algorithm, this remainder is always in Z [ X ] . Finally the divisions denoted / are always exact and have their result either in Z [ X ] or in Z .
The computational expense per step is associated chiefly with finding q k, since the remainder r k can be calculated quickly from r k−2, r k−1, and q k. r k = r k−2 − q k r k−1. The computational expense of dividing h-bit numbers scales as O(h(ℓ + 1)), where ℓ is the length of the quotient. [93]
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, called the modulus of the operation. Given two positive numbers a and n, a modulo n (often abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor. [1]
Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.
Polynomial long division can be used to find the equation of the line that is tangent to the graph of the function defined by the polynomial P(x) at a particular point x = r. [3] If R ( x ) is the remainder of the division of P ( x ) by ( x – r ) 2 , then the equation of the tangent line at x = r to the graph of the function y = P ( x ) is y ...
In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size. This is used for defining the exponential of a matrix , which is involved in the closed-form solution of systems of linear differential equations .