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α is a root of a polynomial p if and only if α −1 is a root of p ∗. [4] If p(x) ≠ x then p is irreducible if and only if p ∗ is irreducible. [5] p is primitive if and only if p ∗ is primitive. [4] Other properties of reciprocal polynomials may be obtained, for instance:
We calculate each respective numerator by (1) taking the root of the denominator (i.e. the value of x that makes the denominator zero) and (2) then substituting this root into the original expression but ignoring the corresponding factor in the denominator. Each root for the variable is the value which would give an undefined value to the ...
Finding roots in a specific region of the complex plane, typically the real roots or the real roots in a given interval (for example, when roots represents a physical quantity, only the real positive ones are interesting). For finding one root, Newton's method and other general iterative methods work generally well.
This consists in using the last computed approximate values of the root for approximating the function by a polynomial of low degree, which takes the same values at these approximate roots. Then the root of the polynomial is computed and used as a new approximate value of the root of the function, and the process is iterated.
A primitive polynomial of degree m has m different roots in GF(p m), which all have order p m − 1, meaning that any of them generates the multiplicative group of the field. Over GF(p) there are exactly φ(p m − 1) primitive elements and φ(p m − 1) / m primitive polynomials, each of degree m, where φ is Euler's totient function. [1]
% The fixed point iteration function is assumed to be input as an % inline function. % This function will calculate and return the fixed point, p, % that makes the expression f(x) = p true to within the desired % tolerance, tol. format compact % This shortens the output. format long % This prints more decimal places. for i = 1: 1000 % get ready ...
Root-finding algorithms are used to solve nonlinear equations (they are so named since a root of a function is an argument for which the function yields zero). If the function is differentiable and the derivative is known, then Newton's method is a popular choice. [16] [17] Linearization is another technique for solving nonlinear equations.
For example, the function f(x) = x 20 − 1 has a root at 1. Since f ′(1) ≠ 0 and f is smooth, it is known that any Newton iteration convergent to 1 will converge quadratically. However, if initialized at 0.5, the first few iterates of Newton's method are approximately 26214, 24904, 23658, 22476, decreasing slowly, with only the 200th ...