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Time Narration
00:01 Dear Friends, Welcome to the Spoken Tutorial on Numerical Interpolation.
00:06 At the end of this tutorial, you will learn how to:
00:10 Develop Scilab code for different Numerical Interpolation algorithms
00:16 Calculate new value of function from given data points.
00:21 To record this tutorial, I am using
00:24 Ubuntu 12.04 as the operating system
00:27 and Scilab 5.3.3 version.
00:31 To practice this tutorial, a learner should have
00:34 basic knowledge of Scilab
00:36 and should know Numerical Interpolation.
00:40 To learn Scilab, please refer to the relevant tutorials available on the Spoken Tutorial website.
00:47 Numerical interpolation is a method of
00:51 constructing new data points
00:53 within the range of a discrete set of known data points.
00:59 We can solve interpolation problems using numerical methods.
01:05 In Lagrange interpolation,
01:07 We pass a polynomial of degree N – 1 through N points.
01:12 Then, we find the unique N order polynomial y of x which interpolates the data samples.
01:22 We are given the natural logarithm values for nine, nine point five and eleven.
01:29 We have to find the value of natural logarithm of nine point two.
01:35 Let us solve this problem using Lagrange interpolation method.
01:41 Let us look at the code for Lagrange interpolation.
01:46 We define the function Lagrange with arguments x zero, x, f and n.
01:53 X zero is the unknown interpolation point.
01:57 x is the vector containing the data points.
02:01 f is the vector containing the values of the function at corresponding data points.
02:08 And n is the order of the interpolating polynomial.
02:14 We use n to initialize m and vector N.
02:19 The order of the interpolating polynomial determines the number of nodes created.
02:25 Then, we apply Lagrange interpolation formula
02:29 to find the value of the numerator and denominator.
02:35 Then we divide the numerator and denominator to get the value of L.
02:41 We use L to find the value of the function y at the given data point.
02:48 Finally we display the value of L and f of x f(x).
02:53 Let us Save and execute the file.
02:57 Switch to Scilab console to solve the example problem.
03:02 Let us define the data points vector.
03:05 On the console, type:
03:07 x equal to open square bracket nine point zero comma nine point five comma eleven point zero close square bracket.
03:18 Press Enter.
03:21 Then type: f equal to open square bracket two point one nine seven two comma two point two five one three comma two point three nine seven nine close square bracket
03:39 Press Enter .
03:41 Then type x zero equal to nine point two
03:46 Press Enter.
03:48 Let us use a quadratic polynomial interpolating polynomial.
03:53 Type n equal to two.
03:58 Press Enter.
04:00 To call the function, type:
04:02 y equal to Lagrange open parenthesis x zero comma x comma f comma n close parenthesis
04:14 Press Enter.
04:16 The value of the function y at x equal to nine point two is displayed.
04:22 Let us look at Newton's Divided Difference Method.
04:26 In this method, Divided Differences recursive method is used.
04:32 It uses lesser number of computation than Lagrange method.
04:38 In spite of this, the same interpolating polynomial, as in Lagrange method, is generated.
04:47 Let us solve this example using Divided Difference method.
04:52 We are given the data points and
04:54 the corresponding values of the function at those data points.
05:00 We have to find the value of the function at x equal to three.
05:05 Let us look at the code for Newton Divided Difference method.
05:11 Open the file Newton underscore Divided dot sci on Scilab Editor.
05:18 We define the function Newton underscore Divided with arguments x, f and x zero.
05:29 x is a vector containing the data points,
05:33 f is the corresponding function value and
05:36 x zero is the unknown interpolation point.
05:41 We find the length of vector and then equate it to n.
05:46 The first value of vector is equated to a of one a(1).
05:51 Then we apply divided difference algorithm and compute the divided difference table.
05:57 Then we find the coefficient list of the Newton polynomial.
06:03 We sum the coefficient list to find the value of the function at given data point.
06:10 Save and execute the file Newton underscore Divided dot sci.
06:16 Switch to Scilab console.
06:19 Clear the screen by typing c l c.
06:22 Press Enter.
06:24 Let us enter the data points vector.
06:27 Type: x equal to open square bracket two comma two point five comma three point two five comma four close square bracket
06:39 Press Enter.
06:41 Then type values of the function
06:44 f equal to open square bracket zero point five comma zero point four comma zero point three zero seven seven comma zero point two five close square bracket
07:01 Press Enter.
07:03 Type x zero equal to three
07:06 Press Enter.
07:08 Then call the function by typing
07:11 I P equal to Newton underscore Divided open parenthesis x comma f comma x zero close parenthesis
07:23 Press Enter.
07:25 The value of y at x equal to three is shown.
07:30 Let us summarize this tutorial.
07:33 In this tutorial, we have learnt to develop Scilab code for interpolation methods.
07:40 We have also learnt to find the value of a function at new data point.
07:46 Solve this problem on your own using Lagrange method and Newton's Divided Difference method.
07:54 Watch the video available at the link shown below.
07:57 It summarizes the Spoken Tutorial project.
08:00 If you do not have good bandwidth, you can download and watch it.
08:05 The spoken tutorial project Team:
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08:33 More information on this mission is available at
08:38 This is Ashwini Patil, signing off.
08:41 Thank you for joining.

Contributors and Content Editors

Gaurav, PoojaMoolya, Sandhya.np14