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In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation.Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Horner himself, and can be traced back many hundreds of years to Chinese and Persian mathematicians. [1]
In numerical analysis, the Weierstrass method or Durand–Kerner method, discovered by Karl Weierstrass in 1891 and rediscovered independently by Durand in 1960 and Kerner in 1966, is a root-finding algorithm for solving polynomial equations. [1] In other words, the method can be used to solve numerically the equation f(x) = 0,
Recently, the polynomial method has led to the development of remarkably simple solutions to several long-standing open problems. [1] The polynomial method encompasses a wide range of specific techniques for using polynomials and ideas from areas such as algebraic geometry to solve combinatorics problems.
Hermite interpolation problems are those where not only the values of the polynomial p at the nodes are given, but also all derivatives up to a given order. This turns out to be equivalent to a system of simultaneous polynomial congruences, and may be solved by means of the Chinese remainder theorem for polynomials.
The graph of a polynomial function of degree 3. The x occurring in a polynomial is commonly called a variable or an indeterminate. When the polynomial is considered as an expression, x is a fixed symbol which does not have any value (its value is "indeterminate").
This image shows, for four points ((−9, 5), (−4, 2), (−1, −2), (7, 9)), the (cubic) interpolation polynomial L(x) (dashed, black), which is the sum of the scaled basis polynomials y 0 ℓ 0 (x), y 1 ℓ 1 (x), y 2 ℓ 2 (x) and y 3 ℓ 3 (x). The interpolation polynomial passes through all four control points, and each scaled basis ...
Given a quadratic polynomial of the form + + it is possible to factor out the coefficient a, and then complete the square for the resulting monic polynomial. Example: + + = [+ +] = [(+) +] = (+) + = (+) + This process of factoring out the coefficient a can further be simplified by only factorising it out of the first 2 terms.
The polynomial x 2 + 1 = 0 has roots ± i. Any real square matrix of odd degree has at least one real eigenvalue. For example, if the matrix is orthogonal, then 1 or −1 is an eigenvalue. The polynomial + has roots , +,, and thus can be factored as
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