Curve Fitting with Polynomials

Curve Fitting

• Curve fitting involves finding a functional relationship between one or more independent variables and a dependent variable for a given set of data points.
• The most commonly used functions for curve fitting are polynomials.

Types of Curve Fitting

• Exact polynomial fitting: The number of data points is exactly equal to (n+1), where n is the degree of the polynomial.
• Inexact polynomial fitting: The number of data points is greater than (n+1), and a single polynomial of degree n cannot pass through all the data points exactly.

Exact Polynomial Fitting

• Given (n+1) points, there is a unique polynomial of degree n that passes through these points.
• The coefficients of the polynomial can be determined by solving a linear system of equations.

Linear System of Equations

• The linear system of equations can be written in matrix form as [A]{B} = {Y}.
• The elements of [A] are given by aij = (xi-1)j-1, where i = 1, 2, ..., n+1, and j = 1, 2, ..., n+1.

Example of Exact Polynomial Fitting

• Example 2.1: Determine the 3rd degree polynomial that passes through the points (0, 1), (1, 0), (2, 1), and (3, 0).
• Solution: The coefficients of the polynomial are b0 = 1.0, b1 = -3.3333, b2 = 3.0, and b3 = -0.66666.

Mathematical Background

Statistics

• Descriptive statistics provide a summary of the data and convey information about specific characteristics of the data set.
• Common descriptive statistics include:
  • Arithmetic mean: The sum of individual data points divided by the number of points.
  • Standard deviation: A measure of the spread of the data.

Arithmetic Mean

• The arithmetic mean is calculated as Σ yi / n, where yi is the individual data point and n is the number of points.

Standard Deviation

• The standard deviation is calculated as S y = √[Σ (yi - y)^2 / (n - 1)], where yi is the individual data point, y is the arithmetic mean, and n is the number of points.