| Visual Cue
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Narration
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| Slide 1
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Dear Friends,
Welcome to the Spoken Tutorial on “ Composite Numerical Integration”
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| Slide 2,3 -Learning Objective Slide
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At the end of this tutorial, you will learn how to:
- Develop Scilab code for different Composite Numerical Integration algorithms
- Divide the integral into equal intervals
- Apply the algorithm to each interval
- Calculate the composite value of the integral
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| Slide 4-System Requirement slide
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* To record this tutorial, I am using Ubuntu 12.04 as the operating system with Scilab 5.3.3 version
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| Slide 5- Prerequisites slide
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* 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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| Slide 6- Numerical Integration
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Numerical Integration is the:
- Study of how the numerical value of an integral can be found
- It is used when exact mathematical integration is not available
- It approximates a definite integral from values of the
integrand
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| Slide 7,8- Composite Trapezoidal Rule-I
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Let us study Composite Trapezoidal Rule. This rule is
- The extension of trapezoidal rule
- We divide the interval a comma b into n equal intervals
- Then,
- h equal to b minus a divided by n is the common length of the intervals
- 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
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| Slide 9- Example
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* Let us solve an example using composite trapezoidal rule:
- Assume the number of intervals n is equal to ten.
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| Switch to Scilab editor
Highlight
function [I1] = Trap_composite(f, a, b, n)
x = linspace(a, b, n+1)
I1 = (h/2)*(2*sum(f(x)) - f(x(1)) - f(x(n+1)))
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* 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.
- 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
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| Click on Execute on Scilab editor and choose Save and Execute the code
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* Click on Execute on Scilab editor and choose Save and Execute the code
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| Switch to Scilab Console
deff ('[y]=f(x)','y=1/(2*x+1)')
Trap_composite(f, 0, 1, 10)
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* 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
- Press enter
- Type
- Trap underscore composite open paranthesis f comma zero comma one comma ten close paranthesis
- Press enter
- The answer is displayed on the console
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| Slide 10, 11- Composite Simpson's Rule
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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
- Apply Simpson's rule to each interval
- 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
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| Slide 12- Example
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* 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
- Let the number of intervals be twenty
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| Switch to Scilab Editor and show the code for Simp_composite.sci
Highlight
function I = Simp_composite(f, a, b, n)
for i = 1:(n/2)-1
x1(i) = x(2*i)
end
for j = 2:n/2
x2(i) = x(2*i-1)
end
I = (h/3)*(f(x(1)) + 2*sum(f(x1)) + 4*sum(f(x2)) + f(x(n)))
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* Let us look at the code for Composite simpson's 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 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
- Let us execute the code
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| Click on Execute and choose
Save and execute the file
Simp_composite.sci
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* Save and execute the file
- Simp underscore composite dot s c i
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| Switch to Scilab Console
Type
clc
deff ('[y]=f(x)','y=1/(1+x^3)')
Simp_composite( f, 1, 2 20)
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* 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
- Press enter
- Type Simp underscore composite open paranthesis f comma one comma two comma twenty close paranthesis
- Press enter
- The answer is displayed on the console
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| Slide 13, 14- Composite Midpoint Rule
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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
- Finds the midpoint of each interval indicated by x i
- We find the sum of the values of the integral at each midpoint
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| Slide 15- Example
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Let us solve this problem using Composite Midpoint Rule
- 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
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| Switch to Scilab Editor
Show the file mid_composite.sci
Highlight
function I = mid_composite(f, a, b, n)
x = linspace(a + h/2, b - h/2, n)
I = h*sum(f(x))
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* 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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| Click on Execute and choose
Save and execute the file mid_composite.sci
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* Save and execute the file mid underscore composite dot s c i
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| On the Scilab Console type:
clc
deff ('[y]=f(x)','y=1-x^2')
Type mid_composite(f, 0, 1.5, 20)
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* 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
- Press enter
- Then type
mid underscore composite open paranthesis f comma zero comma one point five comma twenty close paranthesis
- Press enter
- The answer is displayed on the console
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| Slide 16- Summary
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Let us summarize this tutorial. In this tutorial we have learnt to:
- Develop Scilab code for numerical integration
- Find the value of an integral
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| Show Slide 17
Title: About the Spoken Tutorial Project
- It summarises the Spoken Tutorial project
- If you do not have good bandwidth, you can download and watch it
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* Watch the video available at the following link
- It summarises the Spoken Tutorial project
- If you do not have good bandwidth, you can download and watch it
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| Show Slide 18
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
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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
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| Show Slide 12
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
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* 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
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* This is Ashwini Patil signing off. Thank you for joining.
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