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

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| 00:01
 
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|Dear Friends,  
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|Dear Friends, Welcome to the spoken tutorial on '''Solving System of Linear Equations using Gauss Elimination and Gauss-Jordan Methods'''.
  
 
|-
 
|-
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| 00:02
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| 00:12
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|Welcome to the spoken tutorial on  ''' “Solving System of Linear Equations using Gauss Elimination and Gauss-Jordan Methods”  '''
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−
 
+
−
 
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|-
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| 00:12.
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| 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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|-
 
|-
 
|00:20
 
|00:20
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|Develop '''Scilab''' code to solve linear equations  
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|Develop '''Scilab''' code to solve linear equations.
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+
  
 
|-
 
|-
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|00:27
 
|00:27
 
|'''Ubuntu 12.04''' as the operating system  
 
|'''Ubuntu 12.04''' as the operating system  
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|-
 
|-
 
| 00:31
 
| 00:31
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|with '''Scilab 5.3.3''' version  
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|with '''Scilab 5.3.3''' version.
  
 
|-
 
|-
 
| 00:36
 
| 00:36
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| To practise this tutorial, a learner should have basic knowledge of '''Scilab'''  
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| To practice this tutorial, a learner should have basic knowledge of '''Scilab'''  
  
 
|-
 
|-
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|-
 
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| 00:55
 
| 00:55
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| Finite collection of '''linear equations''' of the same set of  '''variables'''  
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| finite collection of '''linear equations''' of the same set of  '''variables'''.
  
 
|-
 
|-
 
|01:00
 
|01:00
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|Let us study  '''Gauss elimination method'''
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|Let us study  '''Gauss elimination method'''.
  
 
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|-
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|01:06
 
|01:06
 
|'''A x equal to b'''  
 
|'''A x equal to b'''  
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|01:10
 
|01:10
  
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|'''n''' unknowns  
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|'''n''' unknowns.
  
 
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| 01:16
 
| 01:16
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||along with the '''constants b one''' to  '''b n ''' of the system of equations  
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||along with the '''constants b one''' to  '''b m ''' of the system of equations  
  
 
|-
 
|-
  
 
| 01:22
 
| 01:22
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|| in one '''matrix''' called the '''augmented matrix'''  
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|| in one '''matrix''' called the '''augmented matrix'''.
  
 
|-
 
|-
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| We do so by performing row wise manipulation of the '''matrix.'''  
 
| We do so by performing row wise manipulation of the '''matrix.'''  
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|-
 
|-
 
|01:40
 
|01:40
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|Let us solve this system of equations using '''Gaussian elimination method'''
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|Let us solve this system of equations using '''Gaussian elimination method'''.
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+
 
   
 
   
 
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|-
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|01:58
 
|01:58
 
|This defines how many digits should be displayed in the answer.  
 
|This defines how many digits should be displayed in the answer.  
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|-
 
|-
 
| 02:04
 
| 02:04
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|The letter 'e' within single quotes denotes that the answer should be displayed in '''scientific notation'''  
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|The letter 'e' within single quotes denotes that the answer should be displayed in '''scientific notation'''.
−
 
+
  
 
|-
 
|-
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|02:26
 
|02:26
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| The argument '''zero''' specifies that '''Scilab''' need not do anything if the variables are redefined.  
 
| The argument '''zero''' specifies that '''Scilab''' need not do anything if the variables are redefined.  
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|-
 
|-
 
| 02:43
 
| 02:43
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| It will display a message to the user and get the values of '''A''' and '''B''' matrices.   
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| It will display a message to the user and get the values of '''A''' and '''b''' matrices.   
  
 
|-
 
|-
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| 03:11
 
| 03:11
 
|Then we define the function '''naive gaussian elimination.'''
 
|Then we define the function '''naive gaussian elimination.'''
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−
 
  
 
|-
 
|-
 
| 03:15
 
| 03:15
 
|And we state that '''A''' and '''b''' are the '''arguments''' of the function '''naive gaussian elimination.'''  
 
|And we state that '''A''' and '''b''' are the '''arguments''' of the function '''naive gaussian elimination.'''  
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|-
 
|-
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|Similarly we can use '''m one''' and '''p''' for matrix '''b.'''  
 
|Similarly we can use '''m one''' and '''p''' for matrix '''b.'''  
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−
 
  
 
|-
 
|-
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| 03:48
 
| 03:48
 
|| Then we have to determine if the matrices are compatible with each other and  
 
|| Then we have to determine if the matrices are compatible with each other and  
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|-
 
|-
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| 03:57
 
| 03:57
  
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| If '''n''' and '''n one''' are not equal, then we display a message that '''Matrix A must be square.'''  
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| If '''n''' and '''n one''' are not equal then we display a message that '''Matrix A must be square.'''  
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+
  
 
|-
 
|-
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| 04:05
 
| 04:05
  
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| If '''n''' and '''m''' one are not equal, we display a message  
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| If '''n''' and '''m one''' are not equal, we display a message  
−
 
+
  
 
|-
 
|-
 
|04:10
 
|04:10
 
| '''incompatible dimension of A and b.'''  
 
| '''incompatible dimension of A and b.'''  
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−
 
  
 
|-
 
|-
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| If the matrices are compatible, we place matrices '''A''' and '''b''' in one matrix, '''C.'''  
 
| If the matrices are compatible, we place matrices '''A''' and '''b''' in one matrix, '''C.'''  
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|-
 
|-
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||This matrix '''C''' is called '''augmented matrix. '''
 
||This matrix '''C''' is called '''augmented matrix. '''
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|-
 
|-
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| This code converts the '''augmented matrix to upper triangular matrix''' form.  
 
| This code converts the '''augmented matrix to upper triangular matrix''' form.  
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−
 
  
 
|-
 
|-
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|Finally, we perform '''back substitution.'''  
 
|Finally, we perform '''back substitution.'''  
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|-
 
|-
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| Then once one variable is solved, we take this variable to solve the other variables.  
 
| Then once one variable is solved, we take this variable to solve the other variables.  
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|-
 
|-
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||Thus the system of '''linear equations''' is solved.  
 
||Thus the system of '''linear equations''' is solved.  
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−
 
  
 
|-
 
|-
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|On the '''console,''' we have a prompt to enter the value of the '''coefficient matrix.'''  
 
|On the '''console,''' we have a prompt to enter the value of the '''coefficient matrix.'''  
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|-
 
|-
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|So we enter the values of '''matrix A.'''  
 
|So we enter the values of '''matrix A.'''  
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|-
 
|-
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|05:20
 
|05:20
  
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|Type '''square bracket three point four one space one point two three space minus one point zero nine semi colon '''
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|Type: '''square bracket three point four one space one point two three space minus one point zero nine semi colon '''
−
 
+
−
 
+
  
 
|-
 
|-
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| '''two point seven one space two point one four space one point two nine semi colon '''
 
| '''two point seven one space two point one four space one point two nine semi colon '''
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−
 
−
 
  
 
|-
 
|-
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| '''one point eight nine space minus one point nine one space minus one point eight nine close square bracket.'''  
 
| '''one point eight nine space minus one point nine one space minus one point eight nine close square bracket.'''  
−
 
−
 
−
 
  
 
|-
 
|-
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|05:53
 
|05:53
  
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||Press '''Enter'''  
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||Press '''Enter'''.  The next prompt is for '''matrix b.'''  
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+
−
 
+
−
 
+
−
|-
+
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+
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| 05:54
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−
 
+
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| The next prompt is for '''matrix b.'''  
+
  
 
|-
 
|-
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| 05:57
 
| 05:57
  
−
|So we type  
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|So we type '''open square bracket four point seven two semi colon three point one semi colon two point nine one close square bracket.'''
−
 
+
−
|-
+
−
 
+
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| 05:58
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−
 
+
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| '''open square bracket four point seven two semi colon three point one semi colon two point nine one close square bracket. '''
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−
 
+
  
 
|-
 
|-
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| 06:10
 
| 06:10
  
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|Press '''Enter'''  
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|Press '''Enter'''.
−
 
+
  
 
|-
 
|-
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| Then we call the function by typing  
 
| Then we call the function by typing  
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|-
 
|-
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| 06:16
 
| 06:16
  
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| '''naive gaussian elimination open paranthesis A comma b close paranthesis '''
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| '''naive gaussian elimination open parenthesis A comma b close parenthesis '''
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+
−
 
+
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+
  
 
|-
 
|-
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| 06:24
 
| 06:24
  
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| Press '''Enter'''
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| Press '''Enter'''.
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+
−
 
+
  
 
|-
 
|-
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|-
 
|-
 
| 06:32
 
| 06:32
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|Next we shall study the '''Gauss- Jordan method.'''  
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|Next we shall study the '''Gauss-Jordan method.'''  
  
 
|-
 
|-
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| 06:36
 
| 06:36
  
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| In '''Gauss – Jordan Method'''
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| In '''Gauss–Jordan Method''',
  
 
|-
 
|-
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| 06:38
 
| 06:38
  
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| The first step is to form the '''augmented matrix.'''  
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| the first step is to form the '''augmented matrix.'''  
  
 
|-
 
|-
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| 06:42
 
| 06:42
  
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| To do this place the coefficient '''matrix A''' and the right hand side '''matrix b''' together in one '''matrix.'''  
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| To do this, place the coefficient '''matrix A''' and the right hand side '''matrix b''' together in one '''matrix.'''  
  
 
|-
 
|-
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| In diagonal form, only the elements '''a i i ''' are non-zero. Rest of the elements are zero.  
 
| In diagonal form, only the elements '''a i i ''' are non-zero. Rest of the elements are zero.  
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|-
 
|-
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| 07:27
 
| 07:27
  
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|Let us solve this example using '''Gauss- Jordan Method.'''  
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|Let us solve this example using '''Gauss-Jordan Method.'''  
  
 
|-
 
|-
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| 07:55
 
| 07:55
  
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|Then we get the '''A''' and '''b matrix''' using the '''input function. '''
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|Then we get the '''A''' and '''b matrix''' using the '''input''' function.
  
 
|-
 
|-
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| 08:11
 
| 08:11
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|We get the size of '''matrix A''' and store it in '''m''' and '''n'''
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|We get the size of '''matrix A''' and store it in '''m''' and '''n'''.
  
 
|-
 
|-
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| 08:17
 
| 08:17
  
−
|Similarly, we get the size of '''matrix b''' and store it in '''r''' and '''s'''
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|Similarly, we get the size of '''matrix b''' and store it in '''r''' and '''s'''.
  
 
|-
 
|-
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| 08:45
 
| 08:45
  
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|Then we create a '''matrix''' of zeros called '''x''' with '''m''' rows and ''' s columns.'''  
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|Then we create a '''matrix''' of '''zeros''' called '''x''' with '''m''' rows and '''s columns.'''  
  
 
|-
 
|-
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| 08:54
 
| 08:54
  
−
|we divide the right hand side part of '''augmented matrix'''  by the corresponding '''diagonal element to get the value of each variable.'''
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|we divide the right hand side part of '''augmented matrix'''  by the corresponding '''diagonal element''' to get the value of each variable.  
−
 
+
  
 
|-
 
|-
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| 09:22
 
| 09:22
  
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|So we type  
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|So we type '''open square bracket zero point seven comma one seven two five semi colon'''
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+
−
|-
+
−
 
+
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| 09:23
+
−
 
+
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|'''open square bracket zero point seven comma one seven two five semi colon '''
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+
  
 
|-
 
|-
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| 09:41
 
| 09:41
  
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|Press '''Enter'''  
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|Press '''Enter'''.
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+
  
 
|-
 
|-
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| 09:45
 
| 09:45
  
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|So we type '''open squre bracket one seven three nine semi colon '''
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|So we type '''open square bracket one seven three nine semi colon'''
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+
−
 
+
  
 
|-
 
|-
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| 09:51
 
| 09:51
  
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|'''three point two seven one close square bracket'''  
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|'''three point two seven one close square bracket'''.
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+
  
 
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|Press '''Enter '''
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|Press '''Enter'''.
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+
  
 
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| 10:01
  
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|'''Gauss Jordan Elimination open paranthesis A comma b close paranthesis '''
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|'''Gauss Jordan Elimination open parenthesis A comma b close parenthesis '''
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+
  
 
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|-
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| 10:08
  
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|Press '''Enter'''  
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|Press '''Enter'''.
−
 
+
  
 
|-
 
|-
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|In this tutorial, we have learnt to:  
 
|In this tutorial, we have learnt to:  
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| 10:21
 
| 10:21
  
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|Develop '''Scilab''' code for solving system of '''linear equations'''  
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|Develop '''Scilab''' code for solving system of '''linear equations'''.
  
 
|-
 
|-
 
|10:25
 
|10:25
  
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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'''.
  
 
|-
 
|-
 
|10:32
 
|10:32
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| Watch the video available at the link shown below  
+
| Watch the video available at the link shown below.
  
 
|-
 
|-
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| 10:35
  
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| It summarises the Spoken Tutorial project  
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| It summarizes the Spoken Tutorial project.
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+
−
 
+
  
 
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|-
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|10:38
 
|10:38
  
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||If you do not have good bandwidth, you can download and watch it  
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||If you do not have good bandwidth, you can download and watch it.
  
 
|-
 
|-
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|10:43
  
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||The spoken tutorial project Team
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||The spoken tutorial project Team:
  
 
|-
 
|-
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|10:45
  
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||Conducts workshops using spoken tutorials  
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||Conducts workshops using spoken tutorials.
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+
  
 
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||Gives certificates to those who pass an online test  
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||Gives certificates to those who pass an online test.
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+
  
 
|-
 
|-
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|10:52
 
|10:52
  
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||For more details, please write to conatct@spoken-tutorial.org  
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||For more details, please write to conatct@spoken-tutorial.org.
−
 
+
  
 
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|10:59
  
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|Spoken Tutorial Project is a part of the Talk to a Teacher project  
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|Spoken Tutorial Project is a part of the Talk to a Teacher project.
−
 
+
−
 
+
  
 
|-
 
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| 11:10
  
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|More information on this mission is available at  http://spoken-tutorial.org/NMEICT-Intro
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|More information on this mission is available at  http://spoken-tutorial.org/NMEICT-Intro.
  
 
|-
 
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| 11:21
  
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|This is Ashwini Patil signing off.
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|This is Ashwini Patil, signing off.
  
 
|-
 
|-

Latest revision as of 15:14, 20 November 2017

Time Narration
00:01 Dear Friends, Welcome to the spoken tutorial on Solving System of Linear Equations using Gauss Elimination and Gauss-Jordan Methods.
00:12 At the end of this tutorial, you will learn how to:
00:15 Solve system of linear equations using Scilab
00:20 Develop Scilab code to solve linear equations.
00:25 To record this tutorial, I am using
00:27 Ubuntu 12.04 as the operating system
00:31 with Scilab 5.3.3 version.
00:36 To practice this tutorial, a learner should have basic knowledge of Scilab
00:40 and should know how to solve Linear Equations.
00:45 To learn Scilab, please refer to the relevant tutorials available on the Spoken Tutorial website.
00:52 A system of linear equations is a
00:55 finite collection of linear equations of the same set of variables.
01:00 Let us study Gauss elimination method.
01:04 Given a system of equations
01:06 A x equal to b
01:08 with m equations and
01:10 n unknowns.
01:12 We write the coefficients of the variables a one to a n
01:16 along with the constants b one to b m of the system of equations
01:22 in one matrix called the augmented matrix.
01:27 How do we convert the augmented matrix to an upper triangular form matrix?
01:33 We do so by performing row wise manipulation of the matrix.
01:40 Let us solve this system of equations using Gaussian elimination method.
01:45 Before we solve the system, let us go through the code for Gaussian elimination method.
01:52 The first line of the code is format e comma twenty.
01:58 This defines how many digits should be displayed in the answer.
02:04 The letter 'e' within single quotes denotes that the answer should be displayed in scientific notation.
02:12 The number twenty is the number of digits that should be displayed.
02:17 The command funcprot is used to let Scilab know what to do when variables are redefined.
02:26 The argument zero specifies that Scilab need not do anything if the variables are redefined.
02:33 Other arguments are used to issue warnings or errors if the variables are redefined.
02:40 Next we use the input function.
02:43 It will display a message to the user and get the values of A and b matrices.
02:51 The message should be placed within double quotes.
02:55 The matrices that the user enters, will be stored in the variables A and b.
03:02 Here A is the coefficient matrix and b is the right-hand-side matrix or the constants matrix.
03:11 Then we define the function naive gaussian elimination.
03:15 And we state that A and b are the arguments of the function naive gaussian elimination.
03:22 We store the output in variable x.
03:27 Then we find the size of matrices A and b using the size command.
03:34 Since they are two dimensional matrices, we use n and n one to store the size of matrix A.
03:42 Similarly we can use m one and p for matrix b.
03:48 Then we have to determine if the matrices are compatible with each other and
03:53 if A is a square matrix.
03:57 If n and n one are not equal then we display a message that Matrix A must be square.
04:05 If n and m one are not equal, we display a message
04:10 incompatible dimension of A and b.
04:15 If the matrices are compatible, we place matrices A and b in one matrix, C.
04:23 This matrix C is called augmented matrix.
04:28 The next block of code performs forward elimination.
04:32 This code converts the augmented matrix to upper triangular matrix form.
04:39 Finally, we perform back substitution.
04:42 Once the upper triangular matrix is obtained, we take the last row and find the value of the variable in that row.
04:52 Then once one variable is solved, we take this variable to solve the other variables.
04:59 Thus the system of linear equations is solved.
05:03 Let us save and execute the file.
05:06 Switch to Scilab console to solve the example.
05:10 On the console, we have a prompt to enter the value of the coefficient matrix.
05:17 So we enter the values of matrix A.
05:20 Type: square bracket three point four one space one point two three space minus one point zero nine semi colon
05:33 two point seven one space two point one four space one point two nine semi colon
05:41 one point eight nine space minus one point nine one space minus one point eight nine close square bracket.
05:53 Press Enter. The next prompt is for matrix b.
05:57 So we type open square bracket four point seven two semi colon three point one semi colon two point nine one close square bracket.
06:10 Press Enter.
06:13 Then we call the function by typing
06:16 naive gaussian elimination open parenthesis A comma b close parenthesis
06:24 Press Enter.
06:26 The solution to the system of linear equations is shown on Scilab console.
06:32 Next we shall study the Gauss-Jordan method.
06:36 In Gauss–Jordan Method,
06:38 the first step is to form the augmented matrix.
06:42 To do this, place the coefficient matrix A and the right hand side matrix b together in one matrix.
06:50 Then we perform row operations to convert matrix A to diagonal form.
06:56 In diagonal form, only the elements a i i are non-zero. Rest of the elements are zero.
07:05 Then we divide the diagonal element and corresponding element of right hand side element, by the diagonal element.
07:14 We do this to get diagonal element equal to one.
07:19 The resulting value of the elements of each row of the right hand side matrix gives the value of each variable.
07:27 Let us solve this example using Gauss-Jordan Method.
07:33 Let us look at the code first.
07:36 The first line of the code uses format function to specify the format of the displayed answers.
07:44 The parameter e specifies the answer should be in scientific notation.
07:49 Twenty (20) denotes that only twenty digits should be displayed.
07:55 Then we get the A and b matrix using the input function.
08:00 We define the function Gauss Jordan Elimination with input arguments A and b and output argument x.
08:11 We get the size of matrix A and store it in m and n.
08:17 Similarly, we get the size of matrix b and store it in r and s.
08:23 If the sizes of A and b are not compatible, we display an error on the console using error function.
08:33 Then we perform row operations to get diagonal form of the matrix.
08:38 Here pivot refers to the first non-zero element of a column.
08:45 Then we create a matrix of zeros called x with m rows and s columns.
08:52 Once we have the diagonal form,
08:54 we divide the right hand side part of augmented matrix by the corresponding diagonal element to get the value of each variable.
09:04 We store the value of each variable in x.
09:08 Then we return the value of x.
09:11 Finally, we end the function.
09:13 Now let us save and execute the function.
09:18 The prompt requires us to enter the value of matrix A.
09:22 So we type open square bracket zero point seven comma one seven two five semi colon
09:31 zero point four three five two comma minus five point four three three close square bracket.
09:41 Press Enter.
09:43 The next prompt is for vector b.
09:45 So we type open square bracket one seven three nine semi colon
09:51 three point two seven one close square bracket.
09:55 Press Enter.
09:58 Then we call the function by typing
10:01 Gauss Jordan Elimination open parenthesis A comma b close parenthesis
10:08 Press Enter.
10:10 The values of x one and x two are shown on the console.
10:15 Let us summarize this tutorial.
10:18 In this tutorial, we have learnt to:
10:21 Develop Scilab code for solving system of linear equations.
10:25 Find the value of the unknown variables of a system of linear equations.
10:32 Watch the video available at the link shown below.
10:35 It summarizes the Spoken Tutorial project.
10:38 If you do not have good bandwidth, you can download and watch it.
10:43 The spoken tutorial project Team:
10:45 Conducts workshops using spoken tutorials.
10:48 Gives certificates to those who pass an online test.
10:52 For more details, please write to conatct@spoken-tutorial.org.
10:59 Spoken Tutorial Project is a part of the Talk to a Teacher project.
11:03 It is supported by the National Mission on Eduction through ICT, MHRD, Government of India.
11:10 More information on this mission is available at http://spoken-tutorial.org/NMEICT-Intro.
11:21 This is Ashwini Patil, signing off.
11:23 Thank you for joining.

Contributors and Content Editors

Gaurav, PoojaMoolya, Sandhya.np14

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