Scilab/C4/Interpolation/English - Revision history

Nancyvarkey at 17:04, 1 February 2014

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Nancyvarkey at 04:50, 26 December 2013

2013-12-26T04:50:46Z

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Nancyvarkey at 04:30, 26 December 2013

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Nancyvarkey at 03:05, 26 December 2013

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Lavitha Pereira: Created page with ''''Title of script''': '''Numerical Interpolation''' '''Author: Shamika''' '''Keywords: Interpolation, Lagrange method, Newton divided difference method''' {| style="border-s…'

2013-12-24T11:56:17Z

Created page with ''''Title of script''': '''Numerical Interpolation''' '''Author: Shamika''' '''Keywords: Interpolation, Lagrange method, Newton divided difference method''' {| style="border-s…'

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'''Title of script''': '''Numerical Interpolation'''

'''Author: Shamika'''

'''Keywords: Interpolation, Lagrange method, Newton divided difference method'''

{| style="border-spacing:0;"
! <center>Visual Cue</center>
! <center>Narration</center>

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 1
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Dear Friends,

Welcome to the Spoken Tutorial on “'''Numerical Interpolation'''”

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 2 -Learning Objective Slide
| 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:

* Develop Scilab code for different Numerical Interpolation algorithms
* Calculate new value of function from given data points

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 3-System Requirement slide
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| To record this tutorial, I am using

* '''Ubuntu 12.04''' as the operating system
* and '''Scilab 5.3.3''' version

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 4- Prerequisites slide
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| 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.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 5- Numerical Interpolation
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| '''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.

|-
| 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.

Then we find the unique '''N''' '''order polynomial y of x''' which '''interpolates''' the '''data''' samples.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 7- Example
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| We are given the natural '''logarithm''' values for nine, nine point five and eleven.

We have to find the value ofIs this correct? Pls check.''Reply to nancy (10/06/2013, 16:20): "..."''

Change made. natural logarithm of''' nine point two'''.

Let us solve this problem using '''Lagrange interpolation method.'''

|-
| 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: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:none;padding:0.097cm;"| Highlight

Lagrange(x0, x,f, n)

| 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 '''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'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight

m <nowiki>=</nowiki> n + 1<nowiki>;</nowiki>

N <nowiki>=</nowiki> ones(1,m)<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 use '''n''' to initialize '''m''' and '''vector N.'''

The order of the '''interpolating polynomail '''determines the number of '''nodes '''created.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight

for j <nowiki>=</nowiki> 1:m

for k <nowiki>=</nowiki> 1:m

if (k<nowiki><></nowiki>j) then

N(j) <nowiki>=</nowiki> N(j)<nowiki>*</nowiki>(x0 - x(k))

D(j) <nowiki>=</nowiki> D(j)<nowiki>*</nowiki>(x(j) - x(k))

end

end

| 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.'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight

L(j) <nowiki>=</nowiki> N(j)/D(j)<nowiki>;</nowiki>

y <nowiki>=</nowiki> y + L(j)<nowiki>*</nowiki>f(j)<nowiki>;</nowiki>

end

disp(L','L')

disp(f,'f(x)')

| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| 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.'''

|-
| 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;"| Let us save and execute the file.

|-
| 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''' to solve the example problem.

|-
| 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=[9.0,9.5,11.0]</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 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'''

|-
| 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=[2.1972,2.2513,2.3979]</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

'''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'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Type on console

x0=9.2
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Then type

'''x zero equal to nine point two'''

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

n=2
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us use a '''quadratic polynomial interpolating polynomial. '''

Type''' n equal to two'''

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

y = Lagrange(x0, x,f, n)
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| To call the '''function, '''type

'''y equal to Lagrange open paranthesis x zero comma x comma f comma n 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 the function '''y''' at '''x''' '''equal to nine point two''' is displayed.

|-
| 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;"| Let us look at '''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, '''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.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 9- Example

| 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'''.

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'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Switch to Scilab editor

| 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 divided dot sci''' on '''Scilab Editor.'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight

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;"| 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'''.

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight

n <nowiki>=</nowiki> length(x)<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 find the length of '''vector''' and then equate it to''' n.'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight

a(1) <nowiki>=</nowiki> f(1)<nowiki>;</nowiki>

| 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.'''

|-
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Highlight

for k <nowiki>=</nowiki> 1 : n - 1

D(k, 1) <nowiki>=</nowiki> (f(k+1) - f(k))/(x(k+1) - x(k))<nowiki>;</nowiki>

end

for j <nowiki>=</nowiki> 2:n-1

for k <nowiki>=</nowiki> 1:n-j

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')

| 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'''

|-
| 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>

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

| 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'''

|-
| 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.

|-
| 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.

|}

@@ Line 3: / Line 3: @@
 '''Author: Shamika'''
-'''Keywords: Interpolation, Lagrange method, Newton divided difference method'''
+'''Keywords: Interpolation, Lagrange method, Newton Divided Difference method'''
@@ Line 97: / Line 97: @@
 | 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 polynomail '''determines the number of '''nodes '''created.
+The order of the '''interpolating polynomial '''determines the number of '''nodes '''created.
 |-
@@ Line 118: / Line 118: @@
-| 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,
 *to find the value of the '''numerator''' and '''denominator.'''
@@ Line 158: / Line 159: @@
 | 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.'''
@@ Line 216: / Line 217: @@
 |-
 | 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, '''divided differences recursive method '''is used.
+| 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.
@@ Line 229: / Line 230: @@
-| 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'''.
+| 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'''.
@@ Line 245: / Line 246: @@
-| 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'''.
+| 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 divided dot sci''' on '''Scilab Editor.'''
+Open the file '''Newton underscore Divided dot sci''' on '''Scilab Editor.'''
 |-
@@ Line 339: / Line 340: @@
 |-
 | 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:1pt solid #000000;padding:0.097cm;"| Save and execute the file '''Newton underscore Divided dot sci. '''
 |-
@@ Line 410: / Line 411: @@
 |-
 | 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'''.
+| 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'''.
 |-

@@ Line 57: / Line 57: @@
 |-
 | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 7- Example
-| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| We are given the natural '''logarithm''' values for nine, nine point five and eleven.
+| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| 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.
+We have to find the value of '''natural logarithm''' of nine point two.
@@ Line 76: / Line 76: @@
-| 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 '''Lagrange''' with '''arguments x zero, x, f '''and''' n'''.
+| 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 Lagrange''' with '''arguments x zero, x, f '''and''' n'''.
 '''X zero''' is the unknown '''interpolation point.'''
@@ Line 82: / Line 82: @@
 '''x''' is the '''vector '''containing the '''data points.'''
-'''f''' is the '''vector '''containing the values of the '''function '''at correspoding '''data points.'''
+'''f''' is the '''vector '''containing the values of the '''function '''at corresponding '''data points.'''
 And '''n '''is the '''order '''of the '''interpolating polynomial'''.
@@ Line 118: / Line 118: @@
-| 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.'''
+| 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.'''
 |-
@@ Line 139: / Line 139: @@
-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.'''
+We use '''L''' to find the value of the function '''y''' at the given data point.
 |-
@@ Line 147: / Line 150: @@
 |-
 | 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''' to solve the example problem.
+| 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''' to solve the example problem.
 |-
@@ Line 155: / Line 158: @@
 | 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.'''
@@ Line 170: / Line 173: @@
 '''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'''
+Press '''Enter'''.
 |-
@@ Line 180: / Line 183: @@
 '''x zero equal to nine point two'''
-Press '''Enter'''
+Press '''Enter'''.
 |-
@@ Line 190: / Line 193: @@
 Type''' n equal to two'''
-Press '''Enter'''
+Press '''Enter'''.
 |-
@@ Line 201: / Line 204: @@
-Press '''Enter'''
+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 the function '''y''' at '''x''' '''equal to nine point two''' is displayed.
+| 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 the function '''y''' at '''x equal to nine point two''' is displayed.
 |-
@@ Line 219: / Line 222: @@
-In spite of this, the same '''interpolating polynomial '''as in '''Lagrange method''' is generated.
+In spite of this, the same '''interpolating polynomial''', as in '''Lagrange method''', is generated.
 |-