Polynomial

In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. Here, simple means they are constructed using only multiplication and addition. Smooth means they are infinitely differentiable, i.e., they have derivatives of all finite orders.

Numerical analysis

Polynomials and calculus

One important aspect of calculus is the project of analyzing complicated functions by means of approximating them with polynomials. The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial, and the Stone-Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial. Polynomials are also frequently used to interpolate functions.

Related Topics:
Taylor's theorem - Differentiable - Stone-Weierstrass theorem - Continuous - Compact - Interval - Interpolate

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Quotients of polynomials are called rational functions. Piecewise rationals are the only functions that can be evaluated directly on a computer, since typically only the operations of addition, multiplication, division and comparison are implemented in hardware. All the other functions that computers need to evaluate, such as trigonometric functions, logarithms and exponential functions, must then be approximated in software by suitable piecewise rational functions.

Related Topics:
Quotient - Rational function - Piecewise - Computer - Hardware - Trigonometric function - Logarithm - Exponential function - Software

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Evaluation of polynomials

The fast and numerically stable evaluation of a polynomial for a given x is a very important topic in numerical analysis. Several different algorithms have been developed for this problem. Which algorithm is used for a given polynomial depends on the form of the polynomial and the chosen x.

Related Topics:
Numerically stable - Numerical analysis - Algorithm

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To evaluate a polynomial in monomial form one can use the Horner scheme. For a polynomial in Chebyshev form the Clenshaw algorithm can be used. If several equidistant xn have to be calculated one would use Newton's difference method.

Related Topics:
Monomial form - Horner scheme - Chebyshev form - Clenshaw algorithm - Newton's difference method

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Finding roots

As there is no general closed formula to calculate the roots of a polynomial of degree 5 and higher, root-finding algorithms are used in numerical analysis to approximate the roots. Approximations for the real roots of a given polynomial can be found using Newton's method, or more efficiently using Laguerre's method which employs complex arithmetic and can locate all complex roots.

Related Topics:
Root-finding algorithm - Newton's method - Laguerre's method

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