LAGRANGIAN METHOD

INTERACTIVE E-BOOK

INTERPOLATION

MAJOR

GENERAL

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Title	An interactive e-book for illustrating the Lagrangian Method for interpolation.
Creator	Autar K Kaw
Subject and Keywords	Lagrangian Method, Interpolation, Mathcad, Maple, Mathematica, Matlab, Simulations.
Description	An interactive E-book for illustrating Lagrangian Method for interpolation. It includes links to simulations in Mathcad, Maple, Mathematica, and Matlab for the algorithm, multiple choice quizzes, and a PowerPoint presentation.
Publisher	Holistic Numerical Methods Institute, College of Engineering, University of South Florida, Tampa, FL 33620-5350.
Contributors	Autar Kaw, Michael Keteltas
Format	Text/HTML
Last Revised	June 14, 2004
Identifier	http://numericalmethods.eng.usf.edu/ebooks/largrange_05inp_ebook.pdf
Language	English
Rights	http://numericalmethods.eng.usf.edu/rights.htm

Table of Contents

Background

What is Interpolation

Lagrangian Method of Interpolation

Quadratic interpolation

PowerPoint presentation

Examples

Example 1: First order polynomial interpolation

Example 2: Second order polynomial interpolation

Example 3: Third order polynomial interpolation

Simulation ([MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB])

Problem Sets

Multiple choice question examination

Background

This is an example of how an instructor of a Numerical Methods course can develop an E-book on a single topic by using resources that are only available at the Holistic Numerical Methods Institute (HNMI) Website.

Although one of the primary advantages of a web-based resource is that one can use documents outside of its own domain, we are particularly using this example as a way to illustrate the holistic nature of the resources available at HNMI. You are welcome to modify and edit this e-book to suit your purpose.

Using the textbook document written in MS Word 2000 as a foundation, it was made interactive within MS Word 2000 by putting bookmarks and hyperlinks to background information, PowerPoint presentations, Mathcad simulations, historical anecdotes, multiple choice tests, problem sets, etc. Then the MS Word 2000 file was saved as a webpage and that is what you are about to read here.

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LAGRANGIAN INTERPOLATION

What is interpolation?

Many a times, a function is given only at discrete points such as . How does one find the value of ‘y’ at any other value of ‘x’? Well, a continuous function may be used to represent the ‘n+1’ data values with passing through the ‘n+1’ points. Then one can find the value of y at any other value of x. This is called interpolation. Of course, if 'x' falls outside the range of 'x' for which the data is given, it is no longer interpolation but instead is called extrapolation.

So what kind of function should one choose? A polynomial is a common choice for interpolating function because polynomials are easy to

a) evaluate

b) differentiate, and

c) integrate

as opposed to other choices such as a sine or exponential series.

Figure 1: Interpolation of discrete data

Polynomial interpolation involves finding a polynomial of order ‘n’ that passes through the ‘n+1’ points. One of the methods to find this polynomial is called Lagrangian Interpolation. Other methods include the direct method and the Newton’s Divided Difference Polynomial method.

Lagrangian interpolating polynomial is given by

where ‘’ in stands for the order polynomial that approximates the function given at data points as , and

is a weighting function that includes a product of terms with terms of omitted. The application would be clear using an example.
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Simulations [MAPLE] [MATHCAD] [MATHEMATICA] [MATLAB]

Example 1

The upward velocity of a rocket is given as a function of time in Table 1.

Table 1: Velocity as a function of time

t	v(t)
s	m/s
0	0
10	227.04
15	362.78
20	517.35
22.5	602.97
30	901.67

Determine the value of the velocity at t=16 seconds using a first order polynomial.

Figure 2: Velocity vs. time data for the rocket example

Solution

For the first order polynomial (also called linear interpolation), we choose the velocity as given by

Figure 3: Linear interpolation

Since we want to find the velocity at t=16, we choose two data points that are closest to t=16 and that also bracket t=16. Those two points are t_o=15 and t₁=20.

m/s.

You can see that and are like weightages given to the velocities at t=15 and t=20 to calculate the velocity at t=16.

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Quadratic Interpolation

For the second order polynomial interpolation (also called quadratic interpolation), we choose the velocity given by

Figure 4: Quadratic interpolation

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Example 2

The upward velocity of a rocket is given as a function of time in Table 2.

Table 2: Velocity as a function of time

t	v(t)
s	m/s
0	0
10	227.04
15	362.78
20	517.35
22.5	602.97
30	901.67

Determine the value of the velocity at t=16 seconds using second order polynomial interpolation using Lagrangian polynomial interpolation. Find the absolute relative approximate error for approximation from second order polynomial.

Solution:

Since we want to find the velocity at t=16, we need to choose data points that are closest to t=16 as well as bracket t=16. These three points are t₀=10, t₁=15, t₂=20.

gives

m/s.

The absolute relative approximate error obtained between the results from the first and second order polynomial is

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Example 3

The upward velocity of a rocket is given as a function of time in Table 3.

Table 3: Velocity as a function of time

t	v(t)
s	m/s
0	0
10	227.04
15	362.78
20	517.35
22.5	602.97
30	901.67

a) Determine the value of the velocity at t=16 seconds using third order polynomial interpolation using Lagrangian polynomial interpolation. Find the absolute relative approximate error for the third order polynomial approximation.

b) Using the third order polynomial interpolant for velocity, find the distance covered by the rocket from t=11s to t=16s.

c) Using the third order polynomial interpolant for velocity, find the acceleration of the rocket at t=16s.

Solution:

a) For the third order polynomial (also called cubic interpolation), we choose the velocity given by

Since we want to find the velocity at t=16, and we are using a third order polynomial, we need to choose the four points closest to and the bracket to evaluate it.