Last Updated on September 24, 2026 by Maged kamel
Definition of Interpolation and linear Interpolation?
As defined by Prof. Gilat Amos, reference book, Numerical Methods for Engineers and Scientists, 3rd Edition: An Introduction with Applications Using MATLAB, Interpolation is a procedure in which a mathematical formula is used to represent a given set of data points such that the formula gives the exact value at all the data points and an estimated value between the points.
This section shows how to do this using a single polynomial, regardless of the number of points. As mentioned in the previous section, for any number of points n, a polynomial of order n—1 passes through all points.
For two points, the polynomial is first order (a straight line connecting the points).
For three points, the polynomial is of second order (a parabola that connects the points), and so on. The following slide shows the same function represented by different polynomials based on the given points.
For instance, if only two points, A and B, are given, the first graph will be represented by linear Interpolation.
The second graph gives three points: the polynomial is quadratic, etc.
Fig. 6-11 illustrates how first-, second-, third-, and fourth-order polynomials connect two, three, four, and five points, respectively.
What are polynomials?
According to the Math is Fun site, the term “polynomial” is composed of two words: poly (meaning “many”) and nominal (in this case, meaning “term”), so it means “many terms.”
A polynomial can have constants, variables, and exponents, but never be divided by a variable.
In the next slide, the first-degree polynomial comprises two terms: one constant and one variable. Second-degree and third-degree polynomials are expressed.
What is the leading coefficient for a polynomial?
The leading coefficient of a polynomial is the coefficient of the term with the highest variable degree. Here is the polynomial calculator for the DEGREE AND LEADING COEFFICIENT CALCULATOR.
A solved example of the leading coefficient and leading term.
For a given example of f(x)= 5×4+2×3-x+4, the leading coefficient is the coefficient with the highest power. The highest power is 4, and its coefficient is 5. The highest term is (5*x4).
What is linear Interpolation?
Linear Interpolation estimates a missing functional value between two data points by constructing a line between them. You can then use the line’s linear function to find the desired value for the data point of interest.
Using the initial point A, with coordinates (x0, y0), and another point B, with coordinates (x1, y1), the line joining the two points can be estimated by calculating the slope as the difference in y divided by the difference in x.
The final form of the polynomial used for the linear Interpolation can be written as P(x)=y=y0+((y1-y0)*(x-x0)/(x1-x0).
Then, we will substitute the given values of points A and B. Thus, we get the final equations for x and y.
Solved example#1.
To obtain a linear interpolation of f(x) = sqrt(x), we require two points: point 1, with coordinates (1,1), and point 2, with coordinates (9,3).
We substitute into the equation P(x)=y=y0+((y1-y0)*(x-x0)/(x1-x0)) with x0=1, y0=1, and (x1=9, y1=3), giving the final linear interpolation equation.
For point c, where x = 5, we can set P(5) to 2.00. For F(x) as the square root of 5, it will equal =2.236. We can estimate the error between the original function and its linear interpolation. A graph shows the difference between the two functions.
You can view or download the PDF containing this post’s content from the following link.
The next post is post 2a- an Easy introduction to quadratic Interpolation.
This is an excellent resource for the background of Interpolation.
Linear interpolation calculator tool.