Difference between revisions of "Scilab/C4/Linear-equations-Iterative-Methods/English-timed"

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|'''Time'''
 
|'''Time'''
−
 
 
|'''Narration'''
 
|'''Narration'''
  
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|-
 
|-
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|'''maximum number of iteration and'''  
 
|'''maximum number of iteration and'''  
  
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|-
 
|-
  
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||'''convergence tolerance'''
 
||'''convergence tolerance'''
  
 
|-
 
|-
  
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||Then we use '''size''' function to check if '''A matrix''' is a '''square matrix.'''
 
||Then we use '''size''' function to check if '''A matrix''' is a '''square matrix.'''
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|-
 
|-
  
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|-
 
|-
  
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| We then check if '''matrix A''' is '''diagonally dominant.'''  
 
| We then check if '''matrix A''' is '''diagonally dominant.'''  
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|-
 
|-
  
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|| The first half calculates the sum of each row of the '''matrix.'''  
 
|| The first half calculates the sum of each row of the '''matrix.'''  
  
 
|-
 
|-
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| 02.45
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| Then it checks if twice the product of the '''diagonal element'''  is greater than the sum of the elements of that row.  
 
| Then it checks if twice the product of the '''diagonal element'''  is greater than the sum of the elements of that row.  
  
 
|-
 
|-
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| If it isn't, an error is displayed using ''' error function. '''
 
| If it isn't, an error is displayed using ''' error function. '''
  
 
|-
 
|-
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|03:01
 
| Then we define the function '''Jacobi Iteration''' with input arguments  
 
| Then we define the function '''Jacobi Iteration''' with input arguments  
  
 
|-
 
|-
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| '''A, b , x zero, '''
 
| '''A, b , x zero, '''
 
   
 
   
  
 
|-
 
|-
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|'''maximum iteration and tolerance level. '''
 
|'''maximum iteration and tolerance level. '''
  
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|-
 
|-
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|Here '''x zero''' is the '''initial values matrix. '''
 
|Here '''x zero''' is the '''initial values matrix. '''
  
  
 
|-
 
|-
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| 03:19
 
|We check if the size of '''A matrix''' and '''initial values matrix''' are compatible with each other.  
 
|We check if the size of '''A matrix''' and '''initial values matrix''' are compatible with each other.  
  
 
|-
 
|-
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|03:28
 
| We calculate the value for '''x k p one''' and then check if the '''relative error''' is lesser than '''tolerance level.'''
 
| We calculate the value for '''x k p one''' and then check if the '''relative error''' is lesser than '''tolerance level.'''
  
 
|-
 
|-
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| If it is lesser than '''tolerance level''', we break the iteration and the solution is returned.  
 
| If it is lesser than '''tolerance level''', we break the iteration and the solution is returned.  
 
|-
 
|-
  
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|Finally we end the function.  
 
|Finally we end the function.  
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|-
 
|-
  
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| 03:48
 
|| Let us save and execute the function.  
 
|| Let us save and execute the function.  
  
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|-
 
|-
  
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|03:51
  
 
||Switch to '''Scilab console. '''
 
||Switch to '''Scilab console. '''
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|-
 
|-
  
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| Let us enter the values at each prompt.  
 
| Let us enter the values at each prompt.  
  
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|-
 
|-
  
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| The '''coefficient matrix A is open square bracket two space one semi colon five space seven close square bracket '''
 
| The '''coefficient matrix A is open square bracket two space one semi colon five space seven close square bracket '''
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|-
 
|-
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|04:08
 
| Press '''Enter. '''
 
| Press '''Enter. '''
  
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|-
 
|-
  
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| 04:10
  
 
| Then we type '''open square bracket eleven semicolon thirteen close square bracket'''
 
| Then we type '''open square bracket eleven semicolon thirteen close square bracket'''
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|-
 
|-
  
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|04:17
  
 
||Press '''Enter. '''
 
||Press '''Enter. '''
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|-
 
|-
  
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|04:20
  
 
|The '''initial values matrix is open square bracket one semi colon one close square bracket'''
 
|The '''initial values matrix is open square bracket one semi colon one close square bracket'''
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|-
 
|-
  
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| Press '''Enter.'''  
 
| Press '''Enter.'''  
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|-
 
|-
  
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| 04:30
  
 
|The '''maximum number of iterations''' is twenty five.  
 
|The '''maximum number of iterations''' is twenty five.  
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|-
 
|-
  
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| 04:34
  
 
| Press '''Enter. '''
 
| Press '''Enter. '''
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|-
 
|-
  
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| Let the '''convergence tolerance level be zero point zero zero zero zero one '''
 
| Let the '''convergence tolerance level be zero point zero zero zero zero one '''
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|-
 
|-
  
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| 04:44
  
 
||Press '''Enter. '''
 
||Press '''Enter. '''
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|-
 
|-
  
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| 04:46
  
 
||We call the function by typing  
 
||We call the function by typing  
 
|-
 
|-
  
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| 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 '''
 
||'''Jacobi Iteration open paranthesis A comma b comma x zero comma M a x I t e r comma t o l close paranthesis '''
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|-
 
|-
  
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|Press '''Enter. '''
 
|Press '''Enter. '''
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|-
 
|-
  
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|The values for '''x one''' and '''x two''' are shown on the '''console. '''
 
|The values for '''x one''' and '''x two''' are shown on the '''console. '''
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|-
 
|-
  
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|The number of iterations are also shown.  
 
|The number of iterations are also shown.  
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|-
 
|-
  
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|05:14
  
 
| Let us now study '''Gauss Seidel method. '''
 
| Let us now study '''Gauss Seidel method. '''
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|-
 
|-
  
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| 05.19
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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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|-
 
|-
  
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|05:26
  
 
||We rewrite the equations for each unknown  
 
||We rewrite the equations for each unknown  
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|-
 
|-
  
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| by subtracting the other variables and their coefficients from the corresponding right hand side element.  
 
| by subtracting the other variables and their coefficients from the corresponding right hand side element.  
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|-
 
|-
  
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| Then we divide this by the '''coefficient a i i of the''' unknown variable' for that variable.  '''
 
| Then we divide this by the '''coefficient a i i of the''' unknown variable' for that variable.  '''
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|-
 
|-
  
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|This is done for every given equation.  
 
|This is done for every given equation.  
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|-
 
|-
  
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|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 '''
 
|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 '''
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|-
 
|-
  
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| In '''Gauss Seidel method,''' we over write the value of '''x of i k''' with '''x of i k plus one'''
 
| In '''Gauss Seidel method,''' we over write the value of '''x of i k''' with '''x of i k plus one'''
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|-
 
|-
  
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|Let us solve this example using '''Gauss Seidel Method'''
 
|Let us solve this example using '''Gauss Seidel Method'''
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|-
 
|-
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| Let us look at the code for '''Gauss Seidel Method'''
 
| Let us look at the code for '''Gauss Seidel Method'''
  
 
|-
 
|-
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|The first line specifies the '''format''' of the displayed answer on the '''console''' using '''format function.'''  
 
|The first line specifies the '''format''' of the displayed answer on the '''console''' using '''format function.'''  
  
 
|-
 
|-
  
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| Then we use '''input function''' to get the values of  
 
| Then we use '''input function''' to get the values of  
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|-
 
|-
  
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| '''coefficient matrix, '''
 
| '''coefficient matrix, '''
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|-
 
|-
  
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| 06:34
  
 
| '''right hand side matrix,'''
 
| '''right hand side matrix,'''
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|-
 
|-
  
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| 06:36
  
 
| '''initial values of the variables matrix, '''
 
| '''initial values of the variables matrix, '''
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|-
 
|-
  
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| 06.38
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| 06:38
  
 
| '''maximum number of iterations'''  and  
 
| '''maximum number of iterations'''  and  
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|-
 
|-
  
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| 06.40
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| 06:40
  
 
| '''tolerance level'''  
 
| '''tolerance level'''  
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|-
 
|-
  
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| 06.43
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| 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  
 
| 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  
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|-
 
|-
  
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| 06.58
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| We check if '''matrix A is square''' and the sizes of '''initial vector and matrix A''' are compatible using '''size and length function.'''   
 
| We check if '''matrix A is square''' and the sizes of '''initial vector and matrix A''' are compatible using '''size and length function.'''   
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|-
 
|-
  
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|Then we start the iterations.  
 
|Then we start the iterations.  
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|-
 
|-
  
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|We equate the '''initial values vector x zero to x k. '''
 
|We equate the '''initial values vector x zero to x k. '''
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|-
 
|-
  
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|We create a '''matrix of zeros''' with the same size of ''' x k'''  and call it '''x k p one.'''   
 
|We create a '''matrix of zeros''' with the same size of ''' x k'''  and call it '''x k p one.'''   
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|-
 
|-
  
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|We solve for each equation to get the value of the '''unknown variable''' for that equation using '''x k p one. '''
 
|We solve for each equation to get the value of the '''unknown variable''' for that equation using '''x k p one. '''
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|-
 
|-
  
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|At each iteration, the value of '''x k p one''' gets updated.  
 
|At each iteration, the value of '''x k p one''' gets updated.  
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|-
 
|-
  
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|Also, we check if '''relative error''' is lesser than specified '''tolerance level.'''  
 
|Also, we check if '''relative error''' is lesser than specified '''tolerance level.'''  
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|-
 
|-
  
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|If it is, we break the iteration.  
 
|If it is, we break the iteration.  
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|-
 
|-
  
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| 07.54
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| 07:54
 
|Then equate '''x k p one''' to the ''' variable solution.'''  
 
|Then equate '''x k p one''' to the ''' variable solution.'''  
 
|-
 
|-
  
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|Finally, we end the function.  
 
|Finally, we end the function.  
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|-
 
|-
  
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|Let us save and execute the function.  
 
|Let us save and execute the function.  
 
|-
 
|-
  
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|Switch to '''Scilab console'''
 
|Switch to '''Scilab console'''
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|-
 
|-
  
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|For the first prompt, we type ''' matrix A.'''
 
|For the first prompt, we type ''' matrix A.'''
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|-
 
|-
  
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|Type '''open square bracket two space one semi colon five space seven close square bracket'''
 
|Type '''open square bracket two space one semi colon five space seven close square bracket'''
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|-
 
|-
  
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|Press '''Enter'''
 
|Press '''Enter'''
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|-
 
|-
  
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|For the next prompt,  
 
|For the next prompt,  
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|-
 
|-
  
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|type '''open square bracket eleven semi colon thirteen close square bracket'''
 
|type '''open square bracket eleven semi colon thirteen close square bracket'''
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|-
 
|-
  
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|Press '''Enter. '''
 
|Press '''Enter. '''
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|-
 
|-
  
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|We provide the values of '''initial value vector'''  by typing  
 
|We provide the values of '''initial value vector'''  by typing  
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|-
 
|-
  
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|'''open square bracket one semicolon one close square bracket'''
 
|'''open square bracket one semicolon one close square bracket'''
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|-
 
|-
  
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| 08:43
  
 
|Press '''Enter. '''
 
|Press '''Enter. '''
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|-
 
|-
  
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|Then we specify the ''' maximum number of iterations''' to be twenty five  
 
|Then we specify the ''' maximum number of iterations''' to be twenty five  
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|-
 
|-
  
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|Press '''Enter. '''
 
|Press '''Enter. '''
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|-
 
|-
  
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|Let us define '''tolerance level'' to be zero point zero zero zero zero one  
 
|Let us define '''tolerance level'' to be zero point zero zero zero zero one  
 
|-
 
|-
  
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|Press '''Enter'''  
 
|Press '''Enter'''  
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|-
 
|-
  
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|Finally we call the function by typing  
 
|Finally we call the function by typing  
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|-
 
|-
  
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|'''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'''
 
|'''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'''
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|-
 
|-
  
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| 09:24
  
 
|Press '''Enter'''.  
 
|Press '''Enter'''.  
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|-
 
|-
  
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| 09:26
  
 
|The values of '''x one''' and '''x two''' are displayed.  
 
|The values of '''x one''' and '''x two''' are displayed.  
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|-
 
|-
  
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|The number of iterations to solve the same problem are lesser than '''Jacobi method.'''  
 
|The number of iterations to solve the same problem are lesser than '''Jacobi method.'''  
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|-
 
|-
  
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|Solve this problem on your own using '''Jacobi''' and '''Gauss Seidel methods'''
 
|Solve this problem on your own using '''Jacobi''' and '''Gauss Seidel methods'''
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|-
 
|-
  
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|In this tutorial, we have learnt to:  
 
|In this tutorial, we have learnt to:  
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|-
 
|-
  
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|Develop '''Scilab code''' for solving system of linear equations  
 
|Develop '''Scilab code''' for solving system of linear equations  
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|-
 
|-
  
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|Find the value of the '''unknown variables''' of a system of '''linear equations'''
 
|Find the value of the '''unknown variables''' of a system of '''linear equations'''
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|-
 
|-
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| Watch the video available at the following link
 
| Watch the video available at the following link
  
 
|-
 
|-
  
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| It summarises the Spoken Tutorial project  
 
| It summarises the Spoken Tutorial project  
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|-
 
|-
  
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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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|-
 
|-
  
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||The spoken tutorial project Team
 
||The spoken tutorial project Team
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|-
 
|-
  
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||Conducts workshops using spoken tutorials  
 
||Conducts workshops using spoken tutorials  
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|-
  
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||Gives certificates to those who pass an online test  
 
||Gives certificates to those who pass an online test  
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|-
 
|-
  
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||For more details, please write to contact@spoken-tutorial.org  
 
||For more details, please write to contact@spoken-tutorial.org  
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|-
 
|-
  
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|Spoken Tutorial Project is a part of the Talk to a Teacher project  
 
|Spoken Tutorial Project is a part of the Talk to a Teacher project  
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|-
 
|-
  
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| It is supported by the National Mission on Eduction through ICT, MHRD, Government of India.  
 
| It is supported by the National Mission on Eduction through ICT, MHRD, Government of India.  
 
|-
 
|-
  
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|More information on this mission is available at  http://spoken-tutorial.org/NMEICT-Intro
 
|More information on this mission is available at  http://spoken-tutorial.org/NMEICT-Intro
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|-
 
|-
  
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|This is Ashwini Patil signing off.
 
|This is Ashwini Patil signing off.
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|-
 
|-
  
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| Thank you for joining.
 
| Thank you for joining.

Revision as of 18:00, 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,


02:17 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.

Contributors and Content Editors

Gaurav, PoojaMoolya, Pratik kamble, Sandhya.np14

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