Difference between revisions of "Scilab/C4/Integration/English"
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* Apply the algorithm to each '''interval''' | * Apply the algorithm to each '''interval''' | ||
* Calculate the '''composite value of the integral''' | * Calculate the '''composite value of the integral''' | ||
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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 4-System Requirement slide | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 4-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 | + | | 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 5- Prerequisites slide | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 5- Prerequisites slide | ||
− | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Before practising this tutorial, a learner should have basic knowledge of '''Scilab | + | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| * Before practising this tutorial, a learner should have basic knowledge of |
− | + | *'''Scilab''' and | |
− | * | + | *'''Integration using Numerical Methods''' |
− | + | ||
+ | * For '''Scilab''', please refer to the relevant tutorials available on the '''Spoken Tutorial '''website. | ||
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* Study of how the numerical value of an '''integral '''can be found | * Study of how the numerical value of an '''integral '''can be found | ||
* It is used when exact mathematical integration is not available | * It is used when exact mathematical integration is not available | ||
− | * It approximates a definite '''integral '''from values of the | + | * It approximates a definite '''integral '''from values of the '''integrand ''' |
− | + | ||
− | + | ||
|- | |- | ||
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 7,8- Composite Trapezoidal Rule-I | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 7,8- Composite Trapezoidal Rule-I | ||
− | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;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 study '''Composite Trapezoidal Rule'''. This rule is |
* The extension of '''trapezoidal rule''' | * The extension of '''trapezoidal rule''' | ||
* We divide the interval '''a comma b '''into '''n''' equal intervals | * We divide the interval '''a comma b '''into '''n''' equal intervals | ||
+ | |||
+ | |||
* Then, | * Then, | ||
* '''h equal to b minus a divided by n''' is the common length of the intervals | * '''h equal to b minus a divided by n''' is the common length of the intervals | ||
+ | |||
+ | |||
* Then '''composite trapezoidal rule '''is given by | * Then '''composite trapezoidal rule '''is given by | ||
* '''The integral of the function F of x in the interval a to b is approximately equal to h multiplied by the sum of the values of the function at x zero to x n''' | * '''The integral of the function F of x in the interval a to b is approximately equal to h multiplied by the sum of the values of the function at x zero to x 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;"| Slide 9- Example | | 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 an example using '''composite trapezoidal rule'''. | ||
− | + | Assume the number of intervals '''n''' is equal to ten. | |
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'''I1 = (h/2)*(2*sum(f(x)) - f(x(1)) - f(x(n+1)))''' | '''I1 = (h/2)*(2*sum(f(x)) - f(x(1)) - f(x(n+1)))''' | ||
− | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;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 the code for '''Composite Trapezoidal Rule '''on''' Scilab Editor''' |
− | * We first define the function with parameters''' f , a , b , n. f refers to the function we have to solve, a is the lower limit of the integral, b is the upper limit of the integral and n is the number of intervals. | + | |
+ | * We first define the function with parameters''' f , a , b , n.''' | ||
+ | **''' f''' refers to the function we have to solve, | ||
+ | **'''a''' is the lower limit of the integral, | ||
+ | **'''b''' is the upper limit of the integral and | ||
+ | **'''n''' is the number of intervals. | ||
* '''linspace''' function is used to create ten equal intervals between zero and one | * '''linspace''' function is used to create ten equal intervals between zero and one | ||
− | * | + | * We find the value of the integral and store it in '''I one''' |
− | + | ||
− | + | ||
|- | |- | ||
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Click on Execute on Scilab editor and choose Save and Execute the code | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Click on Execute on Scilab editor and choose Save and Execute the code | ||
− | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;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;"| Click on '''Execute''' on '''Scilab editor''' and choose '''Save and Execute''' the code. |
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'''Trap_composite(f, 0, 1, 10)''' | '''Trap_composite(f, 0, 1, 10)''' | ||
− | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;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;"|Define the example function by typing: |
* '''d e f f open paranthesis open single quote open square bracket y close square bracket is equal to f of x close quote comma open quote y is equal to one by open paranthesis two asterisk x plus one close paranthesis close quote close paranthesis''' | * '''d e f f open paranthesis open single quote open square bracket y close square bracket is equal to f of x close quote comma open quote y is equal to one by open paranthesis two asterisk x plus one close paranthesis close quote close paranthesis''' | ||
− | * Press ''' | + | * Press '''Enter''' |
* Type | * Type | ||
* '''Trap underscore composite open paranthesis f comma zero comma one comma ten close paranthesis''' | * '''Trap underscore composite open paranthesis f comma zero comma one comma ten close paranthesis''' | ||
− | * Press ''' | + | * Press '''Enter''' |
− | * The answer is displayed on the console | + | * The answer is displayed on the '''console''' |
− | + | ||
− | + | ||
|- | |- | ||
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 10, 11- Composite Simpson's Rule | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 10, 11- Composite Simpson's Rule | ||
− | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Next we shall study '''Composite simpson's rule. In this rule | + | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Next we shall study '''Composite simpson's rule'''. In this rule, we |
* decompose the interval''' a comma b''' into '''''n is greater than 1 '''''subintervals of equal length | * decompose the interval''' a comma b''' into '''''n is greater than 1 '''''subintervals of equal length | ||
* Apply '''Simpson's rule''' to each interval | * Apply '''Simpson's rule''' to each interval | ||
* We get the value of the integral to be | * We get the value of the integral to be | ||
− | * '''h by three multiplied by the sum of f zero, four into f one , two into f two to f n''' | + | ** '''h by three multiplied by the sum of f zero, four into f one , two into f two to f n''' |
− | + | ||
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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;"| | + | | 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 an example using '''Composite Simpson's rule'''. |
* We are given a '''function one by one plus x cube d x in the interval one to two''' | * We are given a '''function one by one plus x cube d x in the interval one to two''' | ||
* Let the number of intervals be '''twenty''' | * Let the number of intervals be '''twenty''' | ||
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'''I = (h/3)*(f(x(1)) + 2*sum(f(x1)) + 4*sum(f(x2)) + f(x(n)))''' | '''I = (h/3)*(f(x(1)) + 2*sum(f(x1)) + 4*sum(f(x2)) + f(x(n)))''' | ||
− | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;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 the code for '''Composite simpson's rule''' |
− | * | + | * We first define the function with parameters '''f , a , b , n. ''' |
− | * We find two sets of points | + | **'''f''' refers to the function we have to solve, |
− | * We find the value of the function with one set and multiply it with | + | **'''a''' is the lower limit of the integral, |
− | * With the other set we find the value and multiply it with | + | **'''b''' is the upper limit of the integral and |
+ | **'''n''' is the number of intervals. | ||
+ | * We find two sets of points. | ||
+ | * We find the value of the function with one set and multiply it with two | ||
+ | * With the other set, we find the value and multiply it with four | ||
* We sum these values and multiply it with '''h by three and store the final value in I''' | * We sum these values and multiply it with '''h by three and store the final value in I''' | ||
− | |||
+ | Let us execute the code | ||
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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 choose Save and execute the file Simp_composite.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 '''Simp underscore composite dot s c i''' | ||
|- | |- | ||
| 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:none;padding:0.097cm;"| Switch to Scilab Console | ||
− | + | Type '''clc''' | |
− | + | ||
− | + | ||
− | '''clc''' | + | |
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'''Simp_composite( f, 1, 2 20)''' | '''Simp_composite( f, 1, 2 20)''' | ||
− | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;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 me clear the screen first. |
− | + | ||
+ | Define the function given in the example by typing | ||
* '''d e f f open paranthesis open single quote open square bracket y close square bracket is equal to f of x close quote comma open quote y is equal to one divided by open paranthesis one plus x cube close paranthesis close quote close paranthesis''' | * '''d e f f open paranthesis open single quote open square bracket y close square bracket is equal to f of x close quote comma open quote y is equal to one divided by open paranthesis one plus x cube close paranthesis close quote close paranthesis''' | ||
− | * Press ''' | + | * Press '''Enter''' |
− | * ''' | + | * Type '''Simp underscore composite open paranthesis f comma one comma two comma twenty close paranthesis''' |
− | * Press ''' | + | * Press '''Enter''' |
− | + | ||
+ | The answer is displayed on the console. | ||
|- | |- | ||
| style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 13, 14- Composite Midpoint Rule | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| Slide 13, 14- Composite Midpoint Rule | ||
− | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us now look at '''Composite Midpoint Rule. | + | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;padding:0.097cm;"| Let us now look at '''Composite Midpoint Rule.''' |
− | * | + | * It integrates polynomials of degree one or less |
* Divides the interval '''a comma b into n subintervals''' of equal width | * Divides the interval '''a comma b into n subintervals''' of equal width | ||
− | * Finds the | + | * Finds the midpoint of each interval indicated by '''x i ''' |
* We find the sum of the values of the integral at each midpoint | * We find the sum of the values of the integral at each midpoint | ||
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* '''We are given a function one minus x square d x in the interval zero to one point five''' | * '''We are given a function one minus x square d x in the interval zero to one point five''' | ||
* We assume '''n''' is equal to''' twenty''' | * We assume '''n''' is equal to''' twenty''' | ||
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'''I = h*sum(f(x))''' | '''I = h*sum(f(x))''' | ||
+ | | 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 '''Composite Midpoint rule''' | ||
+ | * We first define the function with parameters '''f , a , b , n. ''' | ||
+ | **'''f''' refers to the function we have to solve, | ||
+ | **'''a''' is the lower limit of the integral, | ||
+ | **'''b''' is the upper limit of the integral and | ||
+ | **'''n''' is the number of intervals.''' | ||
+ | * We find the midpoint of each interval | ||
+ | * Find the value of integral at each midpoint and then find the sum and store it in '''I'''. | ||
− | + | Let us now solve the example | |
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Save and execute the file mid_composite.sci | Save and execute the file mid_composite.sci | ||
− | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;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;"|Save and execute the file '''mid underscore composite dot s c i ''' |
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|- | |- | ||
− | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| On the Scilab Console type: | + | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:none;padding:0.097cm;"| On the Scilab Console type: '''clc''' |
− | + | ||
− | + | ||
− | '''clc''' | + | |
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Type '''mid_composite(f, 0, 1.5, 20)''' | Type '''mid_composite(f, 0, 1.5, 20)''' | ||
− | | style="border-top:none;border-bottom:1pt solid #000000;border-left:1pt solid #000000;border-right:1pt solid #000000;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 me clear the screen |
− | + | ||
+ | We define the function given in the example by typing | ||
* '''d e f f open paranthesis open single quote open square bracket y close square bracket is equal to f of x close quote comma open quote y is equal to one minus x square close quote close paranthesis''' | * '''d e f f open paranthesis open single quote open square bracket y close square bracket is equal to f of x close quote comma open quote y is equal to one minus x square close quote close paranthesis''' | ||
− | * Press ''' | + | * Press '''Enter''' |
− | * Then type | + | * Then type '''mid underscore composite open paranthesis f comma zero comma one point five comma twenty close paranthesis''' |
− | * Press ''' | + | * Press '''Enter''' |
− | + | ||
+ | The answer is displayed on the '''console''' | ||
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* Develop '''Scilab''' code for '''numerical integration''' | * Develop '''Scilab''' code for '''numerical integration''' | ||
* Find the value of an '''integral ''' | * Find the value of an '''integral ''' | ||
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* If you do not have good bandwidth, you can download and watch it | * If you do not have good bandwidth, you can download and watch it | ||
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* For more details, please write to contact at spoken hyphen tutorial dot org | * For more details, please write to contact at spoken hyphen tutorial dot org | ||
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* It is supported by the National Mission on Education through ICT, MHRD, Government of India | * It is supported by the National Mission on Education through ICT, MHRD, Government of India | ||
* More information on this Mission is available at | * More information on this Mission is available at | ||
− | * spoken hyphen tutorial dot org slash NMEICT hyphen Intro | + | * spoken hyphen tutorial dot org slash NMEICT hyphen Intro |
− | + | ||
− | + | ||
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| 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: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. Thank you for joining. | | 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. Thank you for joining. | ||
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|} | |} |
Revision as of 08:04, 22 December 2013
Title of script: Numerical Methods for Integration
Author: Shamika
Keywords: Integration, Numerical Methods, integral
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Slide 1 | Dear Friends,
Welcome to the Spoken Tutorial on “ Composite Numerical Integration” |
Slide 2,3 -Learning Objective Slide | At the end of this tutorial, you will learn how to:
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Slide 4-System Requirement slide | * To record this tutorial, I am using
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Slide 5- Prerequisites slide | * Before practising this tutorial, a learner should have basic knowledge of
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Slide 6- Numerical Integration | Numerical Integration is the:
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Slide 7,8- Composite Trapezoidal Rule-I | Let us study Composite Trapezoidal Rule. This rule is
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Slide 9- Example | Let us solve an example using composite trapezoidal rule.
Assume the number of intervals n is equal to ten. |
Switch to Scilab editor
function [I1] = Trap_composite(f, a, b, n) x = linspace(a, b, n+1)
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Let us look at the code for Composite Trapezoidal Rule on Scilab Editor
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Click on Execute on Scilab editor and choose Save and Execute the code | Click on Execute on Scilab editor and choose Save and Execute the code. |
Switch to Scilab Console
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Define the example function by typing:
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Slide 10, 11- Composite Simpson's Rule | Next we shall study Composite simpson's rule. In this rule, we
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Slide 12- Example
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Let us solve an example using Composite Simpson's rule.
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Switch to Scilab Editor and show the code for Simp_composite.sci
function I = Simp_composite(f, a, b, n)
x1(i) = x(2*i) end for j = 2:n/2 x2(i) = x(2*i-1) end
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Let us look at the code for Composite simpson's rule
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Click on Execute and choose Save and execute the file Simp_composite.sci | Save and execute the file Simp underscore composite dot s c i |
Switch to Scilab Console
Type clc
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Let me clear the screen first.
Define the function given in the example by typing
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Slide 13, 14- Composite Midpoint Rule | Let us now look at Composite Midpoint Rule.
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Slide 15- Example
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Let us solve this problem using Composite Midpoint Rule
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Switch to Scilab Editor
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Let us look at the code for Composite Midpoint rule
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Click on Execute and choose
Save and execute the file mid_composite.sci |
Save and execute the file mid underscore composite dot s c i |
On the Scilab Console type: clc
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Let me clear the screen
We define the function given in the example by typing
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Slide 16- Summary | Let us summarize this tutorial. In this tutorial we have learnt to:
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Show Slide 17
Title: About the Spoken Tutorial Project
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* Watch the video available at the following link
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Show Slide 18
Title: Spoken Tutorial Workshops The Spoken Tutorial Project Team
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The Spoken Tutorial Project Team
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Show Slide 12
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. Thank you for joining. |