Generally speaking, the evaluation of an unary polynomial of degree n requires (n+ 1)*n/2 multiplications and n additions, while Qin's algorithm only requires n multiplications and n additions. In manual calculation, the operation process is greatly simplified at one time.

Put a polynomial of degree n:

Rewrite it in the following form:

To calculate the value of a polynomial, first calculate the value of the polynomial in the innermost bracket, that is

Then the value of the polynomial is calculated layer by layer from the inside out, that is

In this way, finding the value of n-degree polynomial f(x) is transformed into finding the value of n-degree polynomial.

Conclusion: For polynomial of degree n, multiply and add degree n at most. ..

Extended data:

Qin algorithm is a polynomial simplification algorithm proposed by Qin, a mathematician in the Southern Song Dynasty. It is called Horner algorithm in the west. Qin (about 1202- 126 1 year) was born in Lu Jun (now a native of Qufu, Shandong) at the end of the Southern Song Dynasty.

In his early years, he studied mathematics with a recluse gentleman. Later, because his father went to Sichuan to be an official, he moved with his father. He was also a native of Anyue, Zhou Pu (now Anyue County, Sichuan Province).

Qin algorithm is an algorithm that transforms the evaluation problem of n-degree unary polynomials into n linear expressions. It greatly simplifies the calculation process. Even in modern times, Qin algorithm is still the best algorithm when solving polynomial evaluation problems with computers.

Known as Horner algorithm in the west, it is named after the British mathematician Horner.

Qin, Yang Hui and Zhu Shijie are also called the four great mathematicians in Song and Yuan Dynasties. (The Qin Memorial Hall in Anyue County officially started construction in September 2000, 1998, and was completed in February 2000. )

Qin was clever and diligent, and Song Shaoding was a scholar in the fourth year of Qin Dynasty (AD 123 1). He has served as county commandant, judge, counselor and state defender. He was an official in Hubei, Anhui, Jiangsu, Zhejiang and other places. Jong-won Lee was the magistrate of Meizhou in the Southern Song Dynasty (A.D. 1260) and died in Meizhou the following year.

According to historical records, he "knows everything about his spirit, astrology, temperament, arithmetic and even creation", and he also tried to learn poetry from Li. In his spare time, he devoted himself to studying mathematics, which has a wide range of applications: astronomical calendar, water conservancy and hydrology, architecture, surveying and mapping, farming, military, commercial finance and so on.

Qin is one of the outstanding representatives of ancient algebra in China. His "Shu Shu Jiu Zhang" summarized the main achievements of China's traditional mathematics in Song and Yuan Dynasties, especially systematically summarized and developed the numerical solution of higher-order equations and the solution of a congruence problem, and put forward quite complete "extraction of positive and negative squares" and "seeking a skill by seeking a big derivative". It has a wide influence on the development of mathematics.

Qin is a scientist who pays equal attention to theory and practice and is good at inheritance and innovation. He is called "one of the greatest mathematicians in his country, that era and all times" by foreign historians of science.

References:

Baidu encyclopedia-Qin algorithm