Polynomial Interpolation
Introduction
Polynomial interpolation is a method of estimating values between two known values. It is a form of interpolation which involves the use of polynomials. Polynomial interpolation is a fundamental subject in numerical analysis and has wide applications in computer graphics, physics, and engineering.
Definition
Polynomial interpolation is the approximation of a polynomial of degree n that passes through n+1 distinct points in the Cartesian plane. For any given set of points and numbers, there exists a polynomial of degree n or less that passes through all the given points and has the given numbers as its coefficients.
Mathematical Formulation
Given a set of n+1 data points
- (x_0, y_0), (x_1, y_1), ..., (x_n, y_n)
where no two x_i are the same, one is looking for a polynomial P of degree at most n
- P(x) = a_0 + a_1 x + a_2 x^2 + ... + a_n x^n
that passes through these points, i.e.
- P(x_i) = y_i
for all i = 0, ..., n.
Methods of Polynomial Interpolation
There are several methods to perform polynomial interpolation. These include:
1. Lagrange Interpolation 2. Newton's Interpolation 3. Hermite Interpolation 4. Spline Interpolation
Each of these methods has its own advantages and disadvantages, and the choice of method often depends on the specific problem at hand.
Applications of Polynomial Interpolation
Polynomial interpolation is used in a wide range of applications, including:
1. Computer Graphics: In computer graphics, polynomial interpolation is used to create curves and surfaces from control points. 2. Physics: In physics, polynomial interpolation is used to approximate the values of a function at certain points when only a discrete set of values is known. 3. Engineering: In engineering, polynomial interpolation is used in signal processing and control systems.
Limitations and Problems
While polynomial interpolation is a powerful tool, it is not without its limitations and problems. These include:
1. Runge's Phenomenon: This is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced interpolation points. 2. Overfitting: This is a problem where the interpolating polynomial fits the given points too well, to the point that it does not represent the underlying function well.