Quadratic Interpolation Formula: Function & Examples
Interpolation is considered as the estimation of an unknown quantity or data point between two known quantities. Interpolation is simple if the seasonality, trends and long term cycles are available. The concept can be shown in regression analysis and series analysis in statistics. Though the interpolation algorithm is available as an in-built function in computers/ calculators, its formula has significance if the tabular form is available. Also represents an introduction for a bigger application of finite differences.
Key Takeaways: Quadratic Interpolation, Function, Extrapolation, Lagrange Interpolation
Quadratic Interpolation
Interpolation approaches are a common methods to the more general area of line search for
Optimization. In quadratic interpolation, the critical value of a function is bracketed, and a quadratic interpolant is fitted to the arc contained in the interval. Following that, the interpolant is minimized, and a new interval is determined on the basis of relation of the minimizer to the actual endpoints of the interval.
Interpolation is used to calculate the value of f(x) or a function of x from the known values of a function. If the values are known as x? <….< xn and y? = f(x?)…..yn = f(xn) and x? < x < xn, then the estimated values of f(x) are considered as interpolation. Similarly, if x > xn or x < x? then the f(x) approximate value is known to be an extrapolation.
Formula for Quadratic Interpolation
There are lagrange interpolation formulas and linear interpolation formulas that are used in order to identify the unknown values of a specific set of data points. The linear interpolation formula is known as follows:
In addition, Lagrange interpolation formula is as follows:
Examples of Quadratic Interpolation
Interpolation helps to calculate the value of an unknown quantity between two known quantities. Similarly, polynomial interpolation is used to calculate the values between known data points. An example for interpolation is given as follows.
Consider a person who planted a tree and started measuring its height from day 1. He records the measure each day and on the fourth day decides to determine the height. The tabular data of the plant’s height is recorded below:
| Days | Height (mm) |
|---|---|
| 1 | 0 |
| 3 | 4 |
| 5 | 8 |
| 7 | 12 |
| 9 | 16 |
| 11 | 20 |
Using the given data, one can determine the height of the plant on any random day. The height on day four is estimated as 6 mm. Also, with the data we can say that the plant grows in a linear pattern. The pattern shows a straight line and the height can be determined by plotting the data on a graph. In case the plant stops growing in a linear direction, then the pattern takes the form of a curve. Linear interpolation formula is an effective way to solve this mathematical problem.
Things to Remember
- Interpolation is considered as the estimation of an unknown data point between two known quantities.
- Interpolation concept can be expressed in regression analysis and series analysis in statistics.
- The formula is used to determine the unknown values for any data related to geography such as elevation, rainfall, noise level, etc.
- There are various types of interpolation methods such as biharmonic interpolation method, thin-plate spline method, cubic spline method, linear interpolation, etc.
Sample Questions
Ques: Estimate the value of y at x=8 and given a set of values (2,6), (5,9) with the help of an interpolation formula? (2 marks)
Ans: Consider the data x?=8, x?=2; x?=5; y?=6, and y?=9
With the help of interpolation formula, we can calculate the value of ay at x=6
Use the interpolation formula as given
y = 6 + ((8 – 2)/(5 – 2) * (9 – 6))
y = 6 + 6
Hence, y = 12
Ques: Calculate the values of y at x=4,with the given sets (2,4), (6,7) using an interpolation formula? (2 marks)
Ans: Consider the given data, x?=4, x?=2; x?=6; y?=4, and y?=7
With interpolation formula, we can determine the value of ay at x=6
Use the formula
y = 4 + (4 – 2)/ (6 – 2) * (7 – 4)
y = 4 + 3/ 2
Hence, y = 11/ 2
Ques: Explain Lagrange interpolation formula? (2 marks)
Ans: Lagrange interpolation formula can be used to determine a polynomial known as Lagrange polynomial, which takes on specific values at arbitrary points. It is an nth degree polynomial approximation to f(x).
Ques: What is an interpolation? (2 marks)
Ans: Interpolation helps in calculating an unknown value that comes between known values. Suppose, a straight passes through two known points. Using the known values, we can estimate the point of the unknown value.
Ques: What are some advantages and disadvantages of Lagrange interpolation? (2 marks)
Ans: It is simple and easy to remember. The application of the formula is not swift. The disadvantage is that the chances of getting errors is high. Also, one cannot determine if the functional values used for the calculation are correct or not.
Ques: Define linear interpolation? (2 marks)
Ans: Linear interpolation is used to calculate values at a point in between a given point. The points are conjoined by a simple line segment. Each segment can be interpolated individually. The values on the interrelated line can be estimated via the parameter mu. Linear interpolation is used to calculate the values of internet rate for a point or security in which no data is provided.
Ques: How to identify the interpolation between two numbers? (2 marks)
Ans: Consider the formula y = y1 + ((x – x1) / (x2 – x1)) * (y2 – y1). The known value is x and y is the unknown value. The coordinates below the known value x are x1 and y1. The coordinates above the x value are x2 and y2.
Ques: What produces smooth interpolation? (2 marks)
Ans: The smoother interpolants are being produced by polynomial interpolation and spline interpolation.
METHOD OF QUADRATIC INTERPOLATION
Quadratic Polynomial Interpolation
Recall from the Linear Polynomial Interpolation page that given two points, (x0,y0) and (x1,y1) where x0≠x1 we can construct a line P1 that passes through point points to approximate a function that passes through these two points. The line P1 has the equation:
As we saw on the Linear Polynomial Interpolation page, the accuracy of approximations of certain values using a straight line dependents on how straight/curved the function is originally, and on how close we are to the points (x0,y0) and (x1,y1). We will now look at quadratic interpolation which in general is more accurate.
This time we will need three points of interest. Let (x0,y0), (x1,y1), and (x2,y2) be distinct points. Define the functions L0(x), L1(x), and L2(x) as follows:
We then define the quadratic polynomial interpolation through the points (x0,y0), (x1,y1), and (x2,y2) as the following function:
Note that P2(x) does in fact pass through all the points specified above since P2(x0)=y0, P2(x1)=y1, and P2(x2)=y2. Also note that in general, for i=0,1,2 and j=0,1,2 we have that:
Let’s now look at some examples of constructing a quadratic interpolation.
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