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| | ! <center>Visual Cue</center> | | ! <center>Visual Cue</center> |
| | ! <center>Narration</center> | | ! <center>Narration</center> |
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| | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| At the end of this tutorial, you will learn how to: | | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| At the end of this tutorial, you will learn how to: |
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| − | * Develop Scilab code for different Numerical Interpolation algorithms | + | * Develop '''Scilab code''' for different '''Numerical Interpolation algorithms''' |
| − | * Calculate new value of function from given data points | + | * Calculate new value of '''function''' from given data points |
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| | * '''Ubuntu 12.04''' as the operating system | | * '''Ubuntu 12.04''' as the operating system |
| | * and '''Scilab 5.3.3''' version | | * and '''Scilab 5.3.3''' version |
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| | * a '''discrete set''' of known '''data points'''. | | * a '''discrete set''' of known '''data points'''. |
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| − | We can solve '''interpolation''' problems using numerical methods. | + | We can solve '''interpolation''' problems using '''numerical methods'''. |
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| | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 6- Lagrange Interpolation | | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 6- Lagrange Interpolation |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| In '''Lagrange interpolation''', we pass a '''polynomial '''of''' degree N – 1''' through '''N''' points. | + | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| In '''Lagrange interpolation''', |
| − | | + | *We pass a '''polynomial '''of''' degree N – 1''' through '''N''' points. |
| − | | + | *Then we find the unique '''N''' '''order polynomial y of x''' which '''interpolates''' the '''data''' samples. |
| − | Then we find the unique '''N''' '''order polynomial y of x''' which '''interpolates''' the '''data''' samples. | + | |
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| − | We have to find the value ofIs this correct? Pls check.''Reply to nancy (10/06/2013, 16:20): "..."'' | + | We have to find the value of natural logarithm of nine point two. |
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| − | Change made. natural logarithm of''' nine point two'''.
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| | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Show Lagrange.sci code on Scilab editor | | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Show Lagrange.sci code on Scilab editor |
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| | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us look at the code for '''Lagrange interpolation.''' | | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us look at the code for '''Lagrange interpolation.''' |
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| | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight | | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight |
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| − | m <nowiki>=</nowiki> n + 1<nowiki>;</nowiki> | + | m = n + 1<nowiki>;</nowiki> |
| | | | |
| − | N <nowiki>=</nowiki> ones(1,m)<nowiki>;</nowiki> | + | N = ones(1,m)<nowiki>;</nowiki> |
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| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight | + | | style="border-top:n |
| − | | + | |
| − | Newton_Divided(x,f,x0)
| + | |
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| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| We define the '''function Newton underscore Divided''' with '''arguments x, f '''and''' x zero'''.
| + | |
| − | | + | |
| − | | + | |
| − | '''X''' is a '''vector''' containing the '''data points''', '''f''' is the corresponding '''function''' '''value''' and''' x zero''' is the unknown '''interpolation point'''.
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| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight
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| − | | + | |
| − | n <nowiki>=</nowiki> length(x)<nowiki>;</nowiki> | + | |
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| − | | + | |
| − | | + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| We find the length of '''vector''' and then equate it to''' n.'''
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| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight
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| − | a(1) <nowiki>=</nowiki> f(1)<nowiki>;</nowiki>
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| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The first value of '''vector''' is equated to '''a of one.'''
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| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight
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| − | | + | |
| − | for k <nowiki>=</nowiki> 1 : n - 1
| + | |
| − | | + | |
| − | D(k, 1) <nowiki>=</nowiki> (f(k+1) - f(k))/(x(k+1) - x(k))<nowiki>;</nowiki>
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| − | | + | |
| − | end
| + | |
| − | | + | |
| − | for j <nowiki>=</nowiki> 2:n-1
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| − | for k <nowiki>=</nowiki> 1:n-j
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| − | | + | |
| − | D(k, j) <nowiki>=</nowiki> (D(k+1, j-1) - D(k, j-1))/(x(k+j) - x(k))
| + | |
| − | | + | |
| − | end
| + | |
| − | | + | |
| − | end
| + | |
| − | | + | |
| − | disp(D, 'The Divided Difference Table')
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| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Then we apply '''divided difference algorithm''' and compute the '''divided difference table'''
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| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight
| + | |
| − | | + | |
| − | Df(1) <nowiki>=</nowiki> 1<nowiki>;</nowiki>
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| − | | + | |
| − | c(1) <nowiki>=</nowiki> a(1)<nowiki>;</nowiki>
| + | |
| − | | + | |
| − | | + | |
| − | for j <nowiki>=</nowiki> 2 : n
| + | |
| − | | + | |
| − | Df(j)<nowiki>=</nowiki>(x0 - x(j-1)).*Df(j-1)<nowiki>;</nowiki>
| + | |
| − | | + | |
| − | c(j) <nowiki>=</nowiki> a(j).*Df(j)<nowiki>;</nowiki>
| + | |
| − | | + | |
| − | end
| + | |
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| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Then we find the '''coefficient list''' of the '''Newton polynomial'''
| + | |
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| − | |-
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| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight
| + | |
| − | | + | |
| − | IP <nowiki>=</nowiki> sum(c)<nowiki>;</nowiki>
| + | |
| − | | + | |
| − | | + | |
| − | | + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| We sum the '''coefficient list '''to find the value of the '''function''' at given '''data point.'''
| + | |
| − | | + | |
| − | |-
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Click on Execute and select Save and Execute
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Save and execute the file '''Newton underscore divided dot sci. '''
| + | |
| − | | + | |
| − | |-
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Switch to Scilab console
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Switch to '''Scilab console'''
| + | |
| − | | + | |
| − | |-
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type clc
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Clear the screen by typing '''c l c'''.
| + | |
| − | | + | |
| − | Press '''Enter'''
| + | |
| − | | + | |
| − | |-
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type on console
| + | |
| − | | + | |
| − | <nowiki>x=[2,2.5,3.25,4]</nowiki>
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us enter the '''data points vector'''
| + | |
| − | | + | |
| − | Type
| + | |
| − | | + | |
| − | '''x equal to open square bracket two comma two point five comma three point two five comma four close square bracket'''
| + | |
| − | | + | |
| − | | + | |
| − | Press '''Enter'''
| + | |
| − | | + | |
| − | |-
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type on console
| + | |
| − | | + | |
| − | <nowiki>f=[0.5,0.4,0.3077,0.25]</nowiki>
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Then type values of the '''function'''
| + | |
| − | | + | |
| − | Type '''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'''
| + | |
| − | | + | |
| − | Press '''Enter'''
| + | |
| − | | + | |
| − | |-
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type on console
| + | |
| − | | + | |
| − | x0=3
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Type '''x zero equal to three'''
| + | |
| − | | + | |
| − | Press '''Enter'''
| + | |
| − | | + | |
| − | |-
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type on console
| + | |
| − | | + | |
| − | IP = Newton_Divided(x,f,x0)
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Then call the '''function '''by typing
| + | |
| − | | + | |
| − | '''i p equal to Newton underscore divided open paranthesis x comma f comma x zero close paranthesis'''
| + | |
| − | | + | |
| − | Press '''Enter'''
| + | |
| − | | + | |
| − | |-
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Show console
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The value of '''y''' '''at x equal to three '''is shown.
| + | |
| − | | + | |
| − | |-
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 9- Summary
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us summarize this tutorial.
| + | |
| − | | + | |
| − | | + | |
| − | In this tutorial we have learnt to develop '''Scilab '''code for '''interpolation methods. '''
| + | |
| − | | + | |
| − | | + | |
| − | We have also learnt to find the value of a '''function '''at new '''data point. '''
| + | |
| − | | + | |
| − | |-
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 10- Assignment
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Solve this problem on your own using Lagrange method and Newton's divided difference method.
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| − | |-
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| '''Show Slide 11'''
| + | |
| − | | + | |
| − | '''Title: About the Spoken Tutorial Project'''
| + | |
| − | | + | |
| − | * Watch the video available at [http://spoken-tutorial.org/What_is_a_Spoken_Tutorial http://spoken-tutorial.org/What_is_a_Spoken_Tutorial]
| + | |
| − | | + | |
| − | * It summarises the Spoken Tutorial project
| + | |
| − | | + | |
| − | * If you do not have good bandwidth, you can download and watch it
| + | |
| − | | + | |
| − | | + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * About the Spoken Tutorial Project<br/>
| + | |
| − | | + | |
| − | * Watch the video available at [http://spoken-tutorial.org/ http://spoken-tutorial.org]/What_is_a_Spoken_Tutorial <br/>
| + | |
| − | | + | |
| − | * It summarises the Spoken Tutorial project<br/>
| + | |
| − | | + | |
| − | * If you do not have good bandwidth, you can download and watch it
| + | |
| − | | + | |
| − | | + | |
| − | | + | |
| − | |-
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| '''Show Slide 12'''
| + | |
| − | | + | |
| − | '''Title: Spoken Tutorial Workshops'''
| + | |
| − | | + | |
| − | The Spoken Tutorial Project Team
| + | |
| − | | + | |
| − | * Conducts workshops using spoken tutorials
| + | |
| − | | + | |
| − | * Gives certificates for those who pass an online test
| + | |
| − | | + | |
| − | * For more details, please write to contact@spoken-tutorial.org
| + | |
| − | | + | |
| − | | + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| The Spoken Tutorial Project Team
| + | |
| − | | + | |
| − | * Conducts workshops using spoken tutorials
| + | |
| − | | + | |
| − | * Gives certificates for those who pass an online test
| + | |
| − | | + | |
| − | * For more details, please write to contact at spoken hyphen tutorial dot org
| + | |
| − | | + | |
| − | | + | |
| − | | + | |
| − | |-
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| '''Show Slide 13'''
| + | |
| − | | + | |
| − | '''Title: Acknowledgement'''
| + | |
| − | | + | |
| − | * Spoken Tutorial Project is a part of the Talk to a Teacher project
| + | |
| − | | + | |
| − | * It is supported by the National Mission on Education through ICT, MHRD, Government of India
| + | |
| − | | + | |
| − | * More information on this Mission is available at
| + | |
| − | | + | |
| − | * [http://spoken-tutorial.org/NMEICT-Intro http://spoken-tutorial.org/NMEICT-][http://spoken-tutorial.org/NMEICT-Intro Intro]
| + | |
| − | | + | |
| − | | + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Spoken Tutorial Project is a part of the Talk to a Teacher project
| + | |
| − | * It is supported by the National Mission on Education through ICT, MHRD, Government of India
| + | |
| − | * More information on this Mission is available at
| + | |
| − | * spoken hyphen tutorial dot org slash NMEICT hyphen Intro <br/>
| + | |
| − | | + | |
| − | | + | |
| − | | + | |
| − | | + | |
| − | |-
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"|
| + | |
| − | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| This is Ashwini Patil signing off. Thanks for joining.
| + | |
| − | | + | |
| − | |}
| + | |
| Visual Cue
|
Narration
|
| Slide 1
|
Dear Friends,
Welcome to the Spoken Tutorial on “Numerical Interpolation”
|
| Slide 2 -Learning Objective Slide
|
At the end of this tutorial, you will learn how to:
- Develop Scilab code for different Numerical Interpolation algorithms
- Calculate new value of function from given data points
|
| Slide 3-System Requirement slide
|
To record this tutorial, I am using
- Ubuntu 12.04 as the operating system
- and Scilab 5.3.3 version
|
| Slide 4- Prerequisites slide
|
To practise this tutorial, a learner should have
- basic knowledge of Scilab
- and should know Numerical Interpolation
To learn Scilab, please refer to the relevant tutorials available on the Spoken Tutorial website.
|
| Slide 5- Numerical Interpolation
|
Numerical interpolation is a method of
- constructing new data points
- within the range of
- a discrete set of known data points.
We can solve interpolation problems using numerical methods.
|
| Slide 6- Lagrange Interpolation
|
In Lagrange interpolation,
- We pass a polynomial of degree N – 1 through N points.
- Then we find the unique N order polynomial y of x which interpolates the data samples.
|
| Slide 7- Example
|
We are given the natural logarithm values for nine, nine point five and eleven.
We have to find the value of natural logarithm of nine point two.
Let us solve this problem using Lagrange interpolation method.
|
| Show Lagrange.sci code on Scilab editor
|
Let us look at the code for Lagrange interpolation.
|
| Highlight
Lagrange(x0, x,f, n)
|
We define the function Lagrange with arguments x zero, x, f and n.
X zero is the unknown interpolation point.
x is the vector containing the data points.
f is the vector containing the values of the function at correspoding data points.
And n is the order of the interpolating polynomial.
|
| Highlight
m = n + 1;
N = ones(1,m);
|
We use n to initialize m and vector N.
The order of the interpolating polynomail determines the number of nodes created.
|
| Highlight
for j = 1:m
for k = 1:m
if (k<>j) then
N(j) = N(j)*(x0 - x(k))
D(j) = D(j)*(x(j) - x(k))
end
end
|
Then we apply Lagrange interpolation formula to find the value of the numerator and denominator.
|
| Highlight
L(j) = N(j)/D(j);
y = y + L(j)*f(j);
end
disp(L','L')
disp(f,'f(x)')
|
Then we divide the numerator and denominator to get the value of L.
We use L to find the value of the function y at the given data point. Finally we display the value of L and f of x.
|
| Click on Execute and select Save and Execute
|
Let us save and execute the file.
|
| Switch to Scilab console
|
Switch to Scilab console to solve the example problem.
|
| Type on console
x=[9.0,9.5,11.0]
|
Let us define the data points vector.
On the console type,
x equal to open square bracket nine point zero comma nine point five comma eleven point zero close square bracket.
Press Enter
|
| Type on console
f=[2.1972,2.2513,2.3979]
|
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
Press Enter
|
| Type on console
x0=9.2
|
Then type
x zero equal to nine point two
Press Enter
|
| Type on console
n=2
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Let us use a quadratic polynomial interpolating polynomial.
Type n equal to two
Press Enter
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| Type on console
y = Lagrange(x0, x,f, n)
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To call the function, type
y equal to Lagrange open paranthesis x zero comma x comma f comma n close paranthesis
Press Enter
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| Show console
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The value of the function y at x equal to nine point two is displayed.
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Let us look at Newton's Divided Difference Method.
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| Slide 8- Newton's Divided Difference Method
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In this method, divided differences recursive method is used.
It uses lesser number of computation than Lagrange method.
In spite of this, the same interpolating polynomial as in Lagrange method is generated.
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| Slide 9- Example
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Let us solve this example using divided difference method.
We are given the data points and the corresponding values of the function at those data points.
We have to find the value of the function at x equal to three.
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| Switch to Scilab editor
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Let us look at the code for Newton Divided difference method.
Open the file Newton underscore divided dot sci on Scilab Editor.
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