Search results
Results from the WOW.Com Content Network
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]
Analysis technique using bitfilters [1] allows one to very efficiently determine the properties of a given generator polynomial. The results are the following: All burst errors (but one) with length no longer than the generator polynomial can be detected by any generator polynomial + +. This includes 1-bit errors (burst of length 1).
As an example of implementing polynomial division in hardware, suppose that we are trying to compute an 8-bit CRC of an 8-bit message made of the ASCII character "W", which is binary 01010111 2, decimal 87 10, or hexadecimal 57 16. For illustration, we will use the CRC-8-ATM polynomial + + +.
The polynomial is written in binary as the coefficients; a 3rd-degree polynomial has 4 coefficients (1x 3 + 0x 2 + 1x + 1). In this case, the coefficients are 1, 0, 1 and 1. The result of the calculation is 3 bits long, which is why it is called a 3-bit CRC. However, you need 4 bits to explicitly state the polynomial.
In statistics, polynomial regression is a form of regression analysis in which the relationship between the independent variable x and the dependent variable y is modeled as an nth degree polynomial in x. Polynomial regression fits a nonlinear relationship between the value of x and the corresponding conditional mean of y, denoted E(y |x).
In numerical analysis, polynomial interpolation is the interpolation of a given bivariate data set by the polynomial of lowest possible degree that passes through the points of the dataset. [ 1 ]
The original construction of Reed & Solomon (1960) interprets the message x as the coefficients of the polynomial p, whereas subsequent constructions interpret the message as the values of the polynomial at the first k points , …, and obtain the polynomial p by interpolating these values with a polynomial of degree less than k.
The examples in the article use a generator polynomial where the first consecutive root is α : (X-α) (X-α 2) (X-α 3) ... (X-α t) If the first consecutive root of a generator polynomial isn't α, then the method used to calculate Ω(x) in the Euclidean example would need to be modified.