DIRECT METHOD INTERACTIVE E-BOOK

INTERPOLATION

MAJOR GENERAL

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

Table of Contents

Background

What is Interpolation

Direct Method of Interpolation

PowerPoint presentation

Examples

Example 1: First order polynomial

Example 2: Second order polynomial

Example 3: Third order polynomial

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

Problem Sets

Multiple choice question examination

A set of homework assignment problems

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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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 an 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.

Polynomial interpolation involves finding a polynomial of order ‘n’ that passes through the ‘n+1’ points. One of the methods is called the direct method of interpolation. Other methods include Newton’s divided difference polynomial method and Lagrangian interpolation method. We discuss the direct method in this section.

Figure 1: Interpolation of discrete data

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DIRECT METHOD

The direct method of interpolation is based on the following premise. Given 'n+1' data points, fit a polynomial of order 'n' as given below

(1)

through the data, where a₀, a₁, . . ., a_n are n+1 real constants. Since n+1 values of y are given at n+1 values of x, one can write n+1 equations. Then the 'n+1' constants, a₀, a₁, . . ., a_n, can be found by solving the n+1 simultaneous linear equations. To find the value of y at a given value of x, simply substitute the value of x in equation (1).

But, it is not necessary to use all the data points. How does one then choose the order of the polynomial and what data points to use? This concept and the direct method of interpolation are best illustrated using examples.

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Simulation ([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 the direct method and 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 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. Then

gives

Writing the equations in matrix form

and solving the above two equations gives,

Hence

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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 velocity at t = 16 seconds using direct method and a second order polynomial.

Solution:

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

Figure 4: Quadratic interpolation

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

gives

Writing the three equations in matrix form

and the solution of the above three equations gives

Hence

At t = 16,

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 direct method and a third order polynomial interpolation using direct method. Find the absolute relative approximate error for the third order polynomial approximation.

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

c) Using the third order polynomial interpolant for velocity from part (a), 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 that also bracket to evaluate it.

The four points are t₀=10, t₁=15, t₂=20 and t₃=22.5.

such that

Writing the four equations in matrix form, we have

Solving the above four equations gives

Hence

The absolute percentage relative approximate error, for the value obtained for v(16) between second and third order polynomial is

b) The distance covered by the rocket between t=11s and t=16s can be calculated from the interpolating polynomial

Note that the polynomial is valid between t=10 and t=20 and hence includes the limits of integration of t=11 and t=16.

So

=

c) The acceleration at t=16 is given by

Given that ,

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Problem Set

1. You are given data for the upward velocity of a rocket as a function of time in the table below.

t	v(t)
[s]