| Visual Cue
|
Narration
|
| 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:
- 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
|
| Slide 4-System Requirement slide
|
* To record this tutorial, I am using
- Ubuntu 12.04 as the operating system
- and Scilab 5.3.3 version
|
| Slide 5- Prerequisites slide
|
* 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.
|
| Slide 6- Numerical Integration
|
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
|
| Slide 7,8- Composite Trapezoidal Rule-I
|
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
|
| 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
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)))
|
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
|
| 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
deff ('[y]=f(x)','y=1/(2*x+1)')
Trap_composite(f, 0, 1, 10)
|
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
|
| Slide 10, 11- Composite Simpson's Rule
|
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
|
| Slide 12- Example
|
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
|
| 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)))
|
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
|
| 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
deff ('[y]=f(x)','y=1/(1+x^3)')
Simp_composite( f, 1, 2 20)
|
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.
|
| Slide 13, 14- Composite Midpoint Rule
|
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
|
| Slide 15- Example
|
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
|
| 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))
|
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
|
| 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
deff ('[y]=f(x)','y=1-x^2')
Type mid_composite(f, 0, 1.5, 20)
|
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
|
| Slide 16- Summary
|
Let us summarize this tutorial. In this tutorial we have learnt to:
- Develop Scilab code for numerical integration
- Find the value of an integral
|
| 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
|
* 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
|
| 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
|
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
|
| 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
|
* 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
|
|
|
* This is Ashwini Patil signing off. Thank you for joining.
|
