Difference between revisions of "Scilab/C4/Linear-equations-Iterative-Methods/English-timed"
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|Dear Friends, | |Dear Friends, | ||
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| − | | 00 | + | | 00:02 |
| Welcome to the Spoken Tutorial on '''“Solving System of Linear Equations using Iterative Methods” ''' | | Welcome to the Spoken Tutorial on '''“Solving System of Linear Equations using Iterative Methods” ''' | ||
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| − | | 00 | + | | 00:10 |
| At the end of this tutorial, you will learn how to: | | At the end of this tutorial, you will learn how to: | ||
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| − | |00 | + | |00:14 |
|Solve system of '''linear equations''' using '''iterative methods''' | |Solve system of '''linear equations''' using '''iterative methods''' | ||
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| − | |00 | + | |00:18 |
|Develop '''Scilab code''' to solve '''linear equations''' | |Develop '''Scilab code''' to solve '''linear equations''' | ||
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| − | | 00 | + | | 00:22 |
|To record this tutorial, I am using | |To record this tutorial, I am using | ||
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| − | |00 | + | |00:25 |
|'''Ubuntu 12.04''' as the operating system | |'''Ubuntu 12.04''' as the operating system | ||
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| − | | 00 | + | | 00:28 |
|and '''Scilab 5.3.3''' version | |and '''Scilab 5.3.3''' version | ||
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| − | | 00 | + | | 00:33 |
| Before practising this tutorial, a learner should have basic knowledge of | | Before practising this tutorial, a learner should have basic knowledge of | ||
|- | |- | ||
| − | |00 | + | |00:38 |
|'''Scilab''' | |'''Scilab''' | ||
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| − | |00 | + | |00:39 |
| and '''Solving Linear Equations''' | | and '''Solving Linear Equations''' | ||
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| − | | 00 | + | | 00:42 |
| For '''Scilab,''' please refer to the relevant tutorials available on the '''Spoken Tutorial''' website. | | For '''Scilab,''' please refer to the relevant tutorials available on the '''Spoken Tutorial''' website. | ||
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| − | | 00 | + | | 00:50 |
| The first '''iterative method''' we will be studying is '''Jacobi method.''' | | The first '''iterative method''' we will be studying is '''Jacobi method.''' | ||
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|Given a '''system of linear equations, with n equations and n unknowns''' | |Given a '''system of linear equations, with n equations and n unknowns''' | ||
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| − | |01 | + | |01:02 |
| We rewrite the equations such that ''' x of i k plus one is equal to b i minus summation of a i j x j k from j equal to one to n divided by a i i''' where ''' i''' is from '''one to n''' | | We rewrite the equations such that ''' x of i k plus one is equal to b i minus summation of a i j x j k from j equal to one to n divided by a i i''' where ''' i''' is from '''one to n''' | ||
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| − | |01 | + | |01:24 |
|We assume values for each '''x of i''' | |We assume values for each '''x of i''' | ||
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|Then we substitute the values in the equations obtained in the previous step. | |Then we substitute the values in the equations obtained in the previous step. | ||
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|We continue the iteration until the solution converges. | |We continue the iteration until the solution converges. | ||
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|Let us solve this example using '''Jacobi Method''' | |Let us solve this example using '''Jacobi Method''' | ||
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||Let us look at the code for '''Jacobi Method.''' | ||Let us look at the code for '''Jacobi Method.''' | ||
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| − | | 01 | + | | 01:48 |
|| We use '''format''' method to specify the format of the displayed answers on the '''Scilab console. ''' | || We use '''format''' method to specify the format of the displayed answers on the '''Scilab console. ''' | ||
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|| Here '''e''' denotes the answer should be in '''scientific notation.''' | || Here '''e''' denotes the answer should be in '''scientific notation.''' | ||
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| And '''twenty''' specifies the number of digits to be displayed. | | And '''twenty''' specifies the number of digits to be displayed. | ||
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|Then we use '''input''' function to get the values for | |Then we use '''input''' function to get the values for | ||
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|'''the matrices coefficient matrix,''' | |'''the matrices coefficient matrix,''' | ||
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||'''right hand side matrix,''' | ||'''right hand side matrix,''' | ||
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| − | |02 | + | |02:14 |
|'''initial values matrix,''' | |'''initial values matrix,''' | ||
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| − | | | + | | 05:.17 |
|'''maximum number of iteration and''' | |'''maximum number of iteration and''' | ||
Revision as of 17:51, 10 July 2014
| Time | Narration |
| 00:01 | Dear Friends, |
| 00:02 | Welcome to the Spoken Tutorial on “Solving System of Linear Equations using Iterative Methods” |
| 00:10 | At the end of this tutorial, you will learn how to: |
| 00:14 | Solve system of linear equations using iterative methods |
| 00:18 | Develop Scilab code to solve linear equations |
| 00:22 | To record this tutorial, I am using |
| 00:25 | Ubuntu 12.04 as the operating system
|
| 00:28 | and Scilab 5.3.3 version |
| 00:33 | Before practising this tutorial, a learner should have basic knowledge of |
| 00:38 | Scilab |
| 00:39 | and Solving Linear Equations |
| 00:42 | For Scilab, please refer to the relevant tutorials available on the Spoken Tutorial website. |
| 00:50 | The first iterative method we will be studying is Jacobi method. |
| 00:56 | Given a system of linear equations, with n equations and n unknowns |
| 01:02 | We rewrite the equations such that x of i k plus one is equal to b i minus summation of a i j x j k from j equal to one to n divided by a i i where i is from one to n |
| 01:24 | We assume values for each x of i
|
| 01:27 | Then we substitute the values in the equations obtained in the previous step. |
| 01:34 | We continue the iteration until the solution converges. |
| 01:39 | Let us solve this example using Jacobi Method |
| 01:44 | Let us look at the code for Jacobi Method. |
| 01:48 | We use format method to specify the format of the displayed answers on the Scilab console. |
| 01:56 | Here e denotes the answer should be in scientific notation. |
| 02:01 | And twenty specifies the number of digits to be displayed.
|
| 02:06 | Then we use input function to get the values for
|
| 02:10 | the matrices coefficient matrix, |
| 02:12 | right hand side matrix, |
| 02:14 | initial values matrix,
|
| 05:.17 | maximum number of iteration and
|
| 02.19 | convergence tolerance |
| 02.22 | Then we use size function to check if A matrix is a square matrix. |
| 02.29
|
If it isn't, we use error function to display an error. |
| 02.34 | We then check if matrix A is diagonally dominant. |
| 02.40 | The first half calculates the sum of each row of the matrix. |
| 02.45 | Then it checks if twice the product of the diagonal element is greater than the sum of the elements of that row. |
| 02.54 | If it isn't, an error is displayed using error function. |
| 03.01 | Then we define the function Jacobi Iteration with input arguments |
| 03.07 | A, b , x zero,
|
| 03.09 | maximum iteration and tolerance level.
|
| 03.14 | Here x zero is the initial values matrix.
|
| 03.19 | We check if the size of A matrix and initial values matrix are compatible with each other. |
| 03.28 | We calculate the value for x k p one and then check if the relative error is lesser than tolerance level. |
| 03.38 | If it is lesser than tolerance level, we break the iteration and the solution is returned. |
| 03.45 | Finally we end the function.
|
| 03.48 | Let us save and execute the function.
|
| 03.51 | Switch to Scilab console. |
| 03.54 | Let us enter the values at each prompt.
|
| 03.57 | The coefficient matrix A is open square bracket two space one semi colon five space seven close square bracket
|
| 04.08 | Press Enter.
|
| 04.10 | Then we type open square bracket eleven semicolon thirteen close square bracket
|
| 04.17 | Press Enter.
|
| 04.20 | The initial values matrix is open square bracket one semi colon one close square bracket |
| 04.28 | Press Enter.
|
| 04.30 | The maximum number of iterations is twenty five.
|
| 04.34 | Press Enter. |
| 04.36 | Let the convergence tolerance level be zero point zero zero zero zero one |
| 04.44 | Press Enter.
|
| 04.46 | We call the function by typing |
| 04.48 | Jacobi Iteration open paranthesis A comma b comma x zero comma M a x I t e r comma t o l close paranthesis
|
| 05.04 | Press Enter.
|
| 05.06 | The values for x one and x two are shown on the console.
|
| 05.11 | The number of iterations are also shown. |
| 05.14 | Let us now study Gauss Seidel method.
|
| 05.19 | 'Given a system of linear equations, with n equations and n unknowns
|
| 05.26 | We rewrite the equations for each unknown
|
| 05.29 | by subtracting the other variables and their coefficients from the corresponding right hand side element.
|
| 05.37 | Then we divide this by the coefficient a i i of the unknown variable' for that variable.
|
| 05.45 | This is done for every given equation.
|
| 05.49 | In Jacobi method, for the computation of x of i k plus one, every element of x of i k is used except x of i k plus one
|
| 06.03 | In Gauss Seidel method, we over write the value of x of i k with x of i k plus one
|
| 06.12 | Let us solve this example using Gauss Seidel Method
|
| 06.17 | Let us look at the code for Gauss Seidel Method |
| 06.21 | The first line specifies the format of the displayed answer on the console using format function. |
| 06.29 | Then we use input function to get the values of |
| 06.32 | coefficient matrix, |
| 06.34 | right hand side matrix, |
| 06.36 | initial values of the variables matrix, |
| 06.38 | maximum number of iterations and
|
| 06.40 | tolerance level |
| 06.43 | Then we define the function Gauss Seidel with input arguments A comma b comma x zero comma max iterations and tolerance level and output argument solution |
| 06.58 | We check if matrix A is square and the sizes of initial vector and matrix A are compatible using size and length function. |
| 07.10 | Then we start the iterations. |
| 07.13 | We equate the initial values vector x zero to x k. |
| 07.19 | We create a matrix of zeros with the same size of x k and call it x k p one. |
| 07.28 | We solve for each equation to get the value of the unknown variable for that equation using x k p one. |
| 07.38 | At each iteration, the value of x k p one gets updated. |
| 07.44 | Also, we check if relative error is lesser than specified tolerance level. |
| 07.50 | If it is, we break the iteration. |
| 07.54 | Then equate x k p one to the variable solution. |
| 07.59 | Finally, we end the function. |
| 08.02 | Let us save and execute the function. |
| 08.06 | Switch to Scilab console |
| 08.09 | For the first prompt, we type matrix A. |
| 08.12 | Type open square bracket two space one semi colon five space seven close square bracket |
| 08.21 | Press Enter |
| 08.22 | For the next prompt, |
| 08.24 | type open square bracket eleven semi colon thirteen close square bracket
|
| 08.31 | Press Enter.
|
| 08.33 | We provide the values of initial value vector by typing |
| 08.38 | open square bracket one semicolon one close square bracket |
| 08.43 | Press Enter. |
| 08.45 | Then we specify the maximum number of iterations to be twenty five |
| 08.50 | Press Enter.
|
| 08.52 | Let us define 'tolerance level to be zero point zero zero zero zero one |
| 08.58 | Press Enter
|
| 09.01 | Finally we call the function by typing |
| 09.04 | G a u s s S e i d e l open paranthesis A comma b comma x zero comma M a x I t e r comma t o l close paranthesis
|
| 09.24 | Press Enter.
|
| 09.26 | The values of x one and x two are displayed.
|
| 09.30 | The number of iterations to solve the same problem are lesser than Jacobi method. |
| 09.37 | Solve this problem on your own using Jacobi and Gauss Seidel methods
|
| 09.43 | In this tutorial, we have learnt to:
|
| 09.47 | Develop Scilab code for solving system of linear equations |
| 09.52 | Find the value of the unknown variables of a system of linear equations
|
| 09.58 | Watch the video available at the following link |
| 10.01 | It summarises the Spoken Tutorial project
|
| 10.04 | If you do not have good bandwidth, you can download and watch it |
| 10.09 | The spoken tutorial project Team |
| 10.11 | Conducts workshops using spoken tutorials
|
| 10.15 | Gives certificates to those who pass an online test
|
| 10.18 | For more details, please write to contact@spoken-tutorial.org
|
| 10.25 | Spoken Tutorial Project is a part of the Talk to a Teacher project
|
| 10.30 | It is supported by the National Mission on Eduction through ICT, MHRD, Government of India. |
| 10.37 | More information on this mission is available at http://spoken-tutorial.org/NMEICT-Intro |
| 10.49 | This is Ashwini Patil signing off. |
| 10.51 | Thank you for joining. |
