Difference between revisions of "Scilab/C4/Interpolation/English"
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'''Author: Shamika''' | '''Author: Shamika''' | ||
− | '''Keywords: Interpolation, Lagrange method, Newton | + | '''Keywords: Interpolation, Lagrange method, Newton Divided Difference method''' |
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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 use '''n''' to initialize '''m''' and '''vector N.''' | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| We use '''n''' to initialize '''m''' and '''vector N.''' | ||
− | The order of the '''interpolating | + | The order of the '''interpolating polynomial '''determines the number of '''nodes '''created. |
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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 '''Lagrange interpolation formula''' | + | | 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 '''Lagrange interpolation formula''' | ||
*to find the value of the '''numerator''' and '''denominator.''' | *to find the value of the '''numerator''' and '''denominator.''' | ||
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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 define the '''data points vector'''. | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us define the '''data points vector'''. | ||
− | On the console type, | + | On the '''console''' type, |
'''x equal to open square bracket nine point zero comma nine point five comma eleven point zero close square bracket.''' | '''x equal to open square bracket nine point zero comma nine point five comma eleven point zero close square bracket.''' | ||
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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 8- Newton's Divided Difference Method | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 8- Newton's Divided Difference Method | ||
− | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| In this method, ''' | + | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| In this method, '''Divided Differences recursive method '''is used. |
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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 solve this example using ''' | + | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us solve this example using '''Divided Difference method'''. |
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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 '''Newton Divided | + | | 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 '''Newton Divided Difference method'''. |
− | Open the file '''Newton underscore | + | Open the file '''Newton underscore Divided dot sci''' on '''Scilab Editor.''' |
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| 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: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 | + | | 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. ''' |
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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 10- Assignment | | 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 | + | | 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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Latest revision as of 22:34, 1 February 2014
Title of script: Numerical Interpolation
Author: Shamika
Keywords: Interpolation, Lagrange method, Newton Divided Difference method
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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:
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Slide 3-System Requirement slide | To record this tutorial, I am using
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Slide 4- Prerequisites slide | To practise this tutorial, a learner should have
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
We can solve interpolation problems using numerical methods. |
Slide 6- Lagrange Interpolation | In Lagrange interpolation,
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Slide 7- Example | We are given the natural logarithm values for nine, nine point five and eleven.
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Show Lagrange.sci code on Scilab editor | Let us look at the code for Lagrange interpolation. |
Highlight
Lagrange(x0, x,f, n)
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We define the function Lagrange with arguments x zero, x, f and n.
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Highlight
m = n + 1; N = ones(1,m);
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We use n to initialize m and vector N.
The order of the interpolating polynomial 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
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Then,
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Highlight
L(j) = N(j)/D(j);
end disp(L','L') disp(f,'f(x)')
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Then we divide the numerator and denominator to get the value of L.
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.
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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 |
Let us use a quadratic polynomial interpolating polynomial.
Type n equal to two Press Enter |
Type on console
y = Lagrange(x0, x,f, n) |
To call the function, type
y equal to Lagrange open paranthesis x zero comma x comma f comma n close paranthesis
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Show console | The value of the function y at x equal to nine point two is displayed. |
Let us look at Newton's Divided Difference Method. | |
Slide 8- Newton's Divided Difference Method | In this method, Divided Differences recursive method is used.
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Slide 9- Example
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Let us solve this example using Divided Difference method.
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Switch to Scilab editor
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Let us look at the code for Newton Divided Difference method.
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Highlight
Newton_Divided(x,f,x0)
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We define the function Newton underscore Divided with arguments x, f and x zero.
f is the corresponding function *value and
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Highlight
n = length(x); |
We find the length of vector and then equate it to n. |
Highlight
a(1) = f(1); |
The first value of vector is equated to a of one. |
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for k = 1 : n - 1 D(k, 1) = (f(k+1) - f(k))/(x(k+1) - x(k)); end for j = 2:n-1 for k = 1:n-j D(k, j) = (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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Then we apply divided difference algorithm and compute the divided difference table. |
Highlight
Df(1) = 1; c(1) = a(1);
Df(j)=(x0 - x(j-1)).*Df(j-1); c(j) = a(j).*Df(j); end
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Then we find the coefficient list of the Newton polynomial |
Highlight
IP = sum(c); |
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Click on Execute and select Save and Execute | Save and execute the file Newton underscore Divided dot sci. |
Switch to Scilab console | Switch to Scilab console |
Type clc | Clear the screen by typing c l c.
Press Enter. |
Type on console
x=[2,2.5,3.25,4] |
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
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Type on console
f=[0.5,0.4,0.3077,0.25] |
Then type values of the function
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. |
Type on console
x0=3 |
Type x zero equal to three
Press Enter. |
Type on console
IP = Newton_Divided(x,f,x0) |
Then call the function by typing
i p equal to Newton underscore Divided open parenthesis x comma f comma x zero close parenthesis Press Enter. |
Show console | The value of y at x equal to three is shown. |
Slide 9- Summary | Let us summarize this tutorial.
we have learnt to develop Scilab code for interpolation methods.
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Slide 10- Assignment | Solve this problem on your own using Lagrange method and Newton's Divided Difference method. |
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* About the Spoken Tutorial Project
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Show Slide 12
Title: Spoken Tutorial Workshops The Spoken Tutorial Project Team
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The Spoken Tutorial Project Team
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Show Slide 13
Title: Acknowledgement
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* Spoken Tutorial Project is a part of the Talk to a Teacher project
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This is Ashwini Patil signing off. Thanks for joining. |