Linear Program Polynomial Interpolation Examples
.Introduction to Numerical Analysis:Polynomial Interpolation Introduction to Polynomial InterpolationPolynomial interpolation is the procedure of fitting a polynomial of degree to a set of data points. For example, if we have two data points, then we can fit a polynomial of degree 1 (i.e., a linear function) between the two points. If we have three data points, then we can fit a polynomial of the second degree (a parabola) that passes through the three points. Shows fitting polynomials of degrees 1, 2, 3, and 4, to sets of 2, 3, 4, and 5 data points. The polynomial equation for each curve is shown in the title for each graph.Polynomial interpolation relies on the fact that for every data points, there is a unique polynomial of degree that passes through these data points. This fact relies on the that states that every degree polynomial has exactly roots. So, if two degree polynomials and agree on points, then, their difference is a polynomial of degree but has roots which implies that.To find the unique polynomial of degree that passes through data points, we need to solve a linear set of equations with the coefficients of the polynomial being the unknowns.
Let be the data point where. Then, we have:Which can be written as follows in matrix form:(1)The above system can be solved to find the values of the coefficients which would ensure that the polynomial curve passes through the data points.
Linear Interpolation Example Problem
The simplest method of interpolation is to draw straight lines between the known data points and consider the function as the combination of those straight lines. This method, called linear interpolation, usually introduces considerable error. A more precise approach uses a polynomial function to connect the points. A is a mathematical expression comprising a sum of terms, each term including a variable or variables raised to a power and multiplied by a. The simplest polynomials have one variable. Polynomials can exist in factored form or written out in full. For example:( x - 4) ( x + 2) ( x + 10)x 2 + 2 x + 13 y 3 - 8 y 2 + 4 y - 2The value of the largest exponent is called the degree of the polynomial.If a set of data contains n known points, then there exists exactly one polynomial of degree n-1 or smaller that passes through all of those points.
Polynomial Interpolation Calculator
The polynomial's graph can be thought of as 'filling in the curve' to account for data between the known points. This methodology, known as polynomial interpolation, often (but not always) provides more accurate results than linear interpolation.The main problem with polynomial interpolation arises from the fact that even when a certain polynomial function passes through all known data points, the resulting graph might not reflect the actual state of affairs.
Polynomial Interpolation Pdf
It is possible that a polynomial function, although accurate at specific points, will differ wildly from the true values at some regions between those points. This problem most often arises when 'spikes' or 'dips' occur in a graph, reflecting unusual or unexpected events in a real-world situation. Such anomalies are not reflected in the simple polynomial function which, even though it might make perfect mathematical sense, cannot take into account the chaotic nature of events in the physical universe.