Difference between revisions of "Scilab/C4/Interpolation/Gujarati"

|Dear Friends,  

| Welcome to the Spoken Tutorial on '''Numerical Interpolation'''.

| 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'''.

|To record this tutorial, I am using

|'''Ubuntu 12.04''' as the operating system

|and '''Scilab 5.3.3''' version.

| To practice 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.

| '''Numerical interpolation''' is a method of

|We can solve '''interpolation''' problems using '''numerical methods.'''  

|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.  

|| 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.'''

|Let us look at the code for '''Lagrange interpolation.'''

||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 corresponding '''data points.'''

||And '''n''' is the '''order''' of the '''interpolating polynomial.'''  

| We use '''n''' to initialize '''m''' and '''vector N.'''

| The order of the '''interpolating polynomial''' determines the number of '''nodes''' created.  

|| Then, we apply '''Lagrange interpolation formula'''  

| 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''' f(x).

|Let us '''Save and execute''' the file.

|Switch to '''Scilab console''' to solve the example problem.  

|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'''.

||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 '''.

| Then type '''x zero equal to nine point two'''  

| Press '''Enter'''.

| Let us use a '''quadratic polynomial interpolating polynomial.'''  

||Type '''n equal to two'''.  

|Press '''Enter'''.  

| To call the '''function,''' type:  

|'''y equal to Lagrange open parenthesis x zero comma x comma f comma n close parenthesis'''  

| Press '''Enter. '''

| The value of the function '''y at x equal to nine point two''' is displayed.  

||Let us look at '''Newton's Divided Difference Method.'''  

||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.  

|Let us solve this example using '''Divided Difference method.'''  

|We are given  the '''data points''' and

| We have to find the value of the '''function''' at '''x equal to three.'''  

| Let us look at the code for '''Newton Divided Difference method. '''

|Open the file '''Newton underscore Divided dot sci''' on '''Scilab Editor.'''

|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.'''  

|We find the '''length''' of '''vector''' and then equate it to '''n.'''

| The first value of '''vector''' is equated to '''a of one''' a(1).

| Then we apply '''divided difference algorithm''' and compute the '''divided difference table.'''  

| Then we find the '''coefficient list''' of the '''Newton polynomial'''.

| We '''sum''' the '''coefficient list''' to find the value of the '''function'''  at given '''data point.'''

| '''Save and execute''' the file '''Newton underscore Divided dot sci.'''  

| Switch to '''Scilab console'''.

|Clear the screen by typing '''c l c'''.

|Press '''Enter.'''  

|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.'''  

|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 '''x zero equal to three'''

|Press '''Enter.'''  

|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.'''  

|The value of '''y at x equal to three''' is shown.  

|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.'''

|Solve this problem on your own using '''Lagrange method and Newton's Divided Difference method.'''  

| Watch the video available at the  link shown below.

| It summarizes the Spoken Tutorial project.  

||If you do not have good bandwidth, you can download and watch it.  

||The spoken tutorial project Team:

||Conducts workshops using spoken tutorials

||Gives certificates to those who pass an online test.

||For more details, please write to contact@spoken-tutorial.org  

|Spoken Tutorial Project is a part of the Talk to a Teacher project.  

| It is supported by the National Mission on Eduction through ICT, MHRD, Government of India.  

|More information on this mission is available at http://spoken-tutorial.org/NMEICT-Intro.

|This is Ashwini Patil, signing off.

| Thank you for joining.