Polynomial evaluation in the context of Polynomial ring

⭐ Core Definition: Polynomial evaluation

In mathematics and computer science, polynomial evaluation refers to computation of the value of a polynomial when its indeterminates are substituted for some values. In other words, evaluating the polynomial $P(x_{1},x_{2})=2x_{1}x_{2}+x_{1}^{3}+4$ at $x_{1}=2,x_{2}=3$ consists of computing $P(2,3)=2\cdot 2\cdot 3+2^{3}+4=24.$ See also Polynomial ring § Polynomial evaluation

For evaluating the univariate polynomial $a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0},$ the most naive method would use $n$ multiplications to compute $a_{n}x^{n}$ , use $n-1$ multiplications to compute $a_{n-1}x^{n-1}$ and so on for a total of ${\tfrac {n(n+1)}{2}}$ multiplications and $n$ additions.Using better methods, such as Horner's rule, this can be reduced to $n$ multiplications and $n$ additions. If some preprocessing is allowed, even more savings are possible.

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⭐ Core Definition: Polynomial evaluation
Polynomial evaluation in the context of Chinese mathematics
Polynomial evaluation in the context of Horner's method

Polynomial evaluation in the context of Chinese mathematics

Mathematics emerged independently in China by the 11th century BCE. The Chinese independently developed a real number system that includes significantly large and negative numbers, more than one numeral system (binary and decimal), algebra, geometry, number theory and trigonometry.

Since the Han dynasty, as diophantine approximation being a prominent numerical method, the Chinese made substantial progress on polynomial evaluation. Algorithms like regula falsi and expressions like simple continued fractions are widely used and have been well-documented ever since. They deliberately find the principal nth root of positive numbers and the roots of equations. The major texts from the period, The Nine Chapters on the Mathematical Art and the Book on Numbers and Computation gave detailed processes for solving various mathematical problems in daily life. All procedures were computed using a counting board in both texts, and they included inverse elements as well as Euclidean divisions. The texts provide procedures similar to that of Gaussian elimination and Horner's method for linear algebra. The achievement of Chinese algebra reached a zenith in the 13th century during the Yuan dynasty with the development of tian yuan shu.

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Polynomial evaluation in the context of Horner's method

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. After the introduction of computers, this algorithm became fundamental for computing efficiently with polynomials.

The algorithm is based on Horner's rule, in which a polynomial is written in nested form: ${\begin{aligned}&a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots +a_{n}x^{n}\\={}&a_{0}+x{\bigg (}a_{1}+x{\Big (}a_{2}+x{\big (}a_{3}+\cdots +x(a_{n-1}+x\,a_{n})\cdots {\big )}{\Big )}{\bigg )}.\end{aligned}}$

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Polynomial evaluation in the context of Polynomial ring

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⭐ Core Definition: Polynomial evaluation

Polynomial evaluation in the context of Chinese mathematics

Polynomial evaluation in the context of Horner's method