WINSEM2012 13 CP0854 06 Mar 2013 RM01 7 Numerical Methods

7/28/2019 WINSEM2012 13 CP0854 06 Mar 2013 RM01 7 Numerical Methods

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Numerical MethodsNumerical Methods

Computational Fluid Dynamics


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OutlineOutlineSuccessive ApproximationsMatrix and Linear Equations Pivotal Condensation Method System of Linear Equations Gauss Elimination Method Partial and Complete Pivoting Gauss Jordan Elimination Method Eigen values and Eigen vectorsDifferentiation and Integration

3/5/2013 2Numerical Methods


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OutlineOutlineDifferential Equations Eulers Methods Taylors Method Runge-KuttaMethods Simultaneous Differential Equations Higher Order Equations

Boundary Value Problems Partial Differential EquationsAlgebraic Equations



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Successive ApproximationsSuccessive Approximations

Roots of any algebraic equation of degree greater

than 4 can not be evaluated exactly using finite

number of operations on its coefficients

Method of successive approximation is used to

find an approximate solution for such equations



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Successive ApproximationsSuccessive ApproximationsWrite the given eq in the form, x = f (x)

Consider the formula, xk+1 = f (xk), which is called iterative equation

Start with an initial guess, x0 = c

Evaluate, x1 = f (x0)

x2 = f (x1)

x3 = f (x2) ..

Continue the procedure until, xk+ 1 = xk

If it doesnt converge, try with another iterative equation



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Successive ApproximationsSuccessive ApproximationsALGORITHM

Define f (x)

Read x1

k =1

xk+1 = f (xk)

Print k +1, xk+1 , f (xk+1 )

If |xk+1 xk| 5e -5,

end if

Print final root, xk+1

Stop



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Successive ApproximationsSuccessive Approximations

Exercise

Find the root of an equation, 32 x5 64 x + 31 = 0

Find the root of an equation, x5 3 x2 + 3 x 1= 0



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Matrix and Linear EquationsMatrix and Linear Equations

Pivotal Condensation Method


11 12 1

21 22 2

1 2

.......

.......

..... ...... ....... ....

.......

n

n

n n nn

a a aa a a

A

a a a

Do the operation 1 111

iij ij j

aa a a

a

Then it becomes

11 12 1 22 23 2

22 2 32 33 311

2 2 3

....... .......

0 ....... .......

..... ...... ....... .... ..... ...... ....... ....

0 ....... .......

n n

n n

n nn n n nn

a a a a a aa a a a a

A a

a a a a a


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Repeat the procedure to get


33 32 3

43 44 411 22

3 4

.......

.......

..... ...... ....... ....

.......

n

n

n n nn

a a a

a a aA a a

a a a

Repeat until weget

11 22 33 ...... nnA a a a a

Pivotal Condensation Method


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Pivotal Condensation MethodPivotal Condensation MethodALGORITHM

Read nFor i =1 to nFor j =1 to nRead a ijNext jNext i

For k =1 to n 1For i =k +1 to nRatio =

For j =1 to naij =a ij ratio * a kjNext jNext iNext k


ik kka aALGORITHM (Contd.,)

Det =1

For i =1 to nDet =Det * a iiNext iPrint DetStop


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Pivotal Condensation MethodPivotal Condensation Method

Exercise

Find the determinant of,


1.2 2.1 3.2 4.3

1.4 2.6 3 4.1

2.2 1.7 4 1.2

1.1 3.6 5 4.6

A

Write a programming code and validateyour analytical

result with it.


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Gauss Elimination Method

Gauss Jordan Elimination Method


Consider a system of equations

11 1 12 2 1 1, 1

21 1 22 2 2 2, 1

1 1 2 2 1 , 1

.........

.........

...........................................

...........................................

.........

n n n

n n n

n n nn n n

a x a x a x a

a x a x a x a

a x a x a x a

Systems of Linear Equations


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Coefficient Matrix


Right hand side vector

11 12 1

21 22 2

1 2

.......

.......

..... ...... ....... ....

.......

n

n

n n nn

a a aa a a

a a a

1, 1

2, 1

, 1

........

n

n

n n

aa

a


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Bringing down the coefficient matrix to DIAGONAL MATRIX

Bringing down the coefficient matrix to LOWER TRI-DIAGONAL

MATRIX

Bringing down the coefficient matrix to UPPER TRI-DIAGONAL

MATRIX




GaussSeidal Iteration Method

Solution Techniques


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Systems of Linear Equations: Solution Techniques

Bringing down the coefficient matrix to DIAGONAL MATRIX


Consider asystem equations


1 2 3

1 2 3

1 2 3

4 3 6

1

3 5 3 4

x x xx x xx x x

Make some row operationsso that the matrix reduces to

DIAGONAL MATRIX

1

0.5

1 0 0

0 1 0

0 0 1 0.5


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Bringing down the coefficient matrix to UPPER TRI-DIAGONAL

MATRIX




1 2 3

1 2 3

1 2 3

4 3 6

1

3 5 3 4

x x xx x xx x x

Make some row operationsso that the matrix reduces to

UPPER TRIDIAGONAL MATRIX


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Gauss Elimination MethodGauss Elimination MethodALGORITHM

Read nFor i =1 to nFor j =1 to n+1Read a ijNext jNext i

For k =1 to n1For i =k+1 to nRatio =

For j =1 to n+1aij =a ij ratio * a kjNext jNext iNext k


ik kka a

ALGORITHM (Contd.,)

For k =n1 to 1, 1xk =a k,n+1For j =k+1 to nxk =x k akj x jNext jxk =x k / a kkNext kStop


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Gauss Elimination MethodGauss Elimination Method

Exercise

Solve the following system of equsby Gauss Elimination Method,


1

2

3

4

1.2 2.1 1.1 4 6

1.1 2 3.1 3.9 3.9

2.1 2.2 3.7 16 12.2

1 2.3 4.7 12 4

x

xxx


result with it.


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Gauss Seidel Iteration Method

1 1, 1 1211

2 2, 1 2122

, 11

1

1, 2

....................................................

1,

n

n j j j

n

n j j

n

n n n nj jnn

x a a xa

x a a x ja

x a a x j na

Convert them in the form

11 1 12 2 1 1, 1

21 1 22 2 2 2, 1

1 1 2 2 1 , 1

.........

.........

...........................................

....................................................

n n n

n n n

n n nn n n

a x a x a x a

a x a x a x a

a x a x a x a


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Start with an initial guess, x 1 =x 2 =x 3 .. =x n =0

Store the initial guess in another variable, y 1 =y 2 .. =y n =0

Substitute the values of x n in RHS of eq (1) and find x 1

Substitute the new x 1 along with x 3, x 4,..x n , in eq(2) and find x 2

Substitute the new x 1, x 2 along with x 4, x 5,..x n , in eq(3) and find x 3

Continue to find x 1, x 2, x 3,..x n.

Find x n yn, if it is zero stop iteration

Otherwise, assign x n to y n and repeatethe procedure until x n yn becomes

zero or within a residual value specified



Gauss Seidel Iteration Method


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Gauss Seidel Iteration MethodGauss Seidel Iteration MethodALGORITHM

Read nFor i =1 to nFor j =1 to n+1Read a ijNext jNext i

For i =1 to nxi =0, y i =0Next iIteration =112 For i =1 to n

xi =a i,n+1For j =1 to nIf i =j, go to step 17xi =x i aij x j17 Next j


ALGORITHM (Contd.,)

xi =x i / a iiNext iFor k =1 to nIf |x k yk| >0.00005, thenPrint IterationFor i =1 to nyi =x iPrint x iNext iIteration =Iteration +1Go to step 12

End if Next kPrint Final result reachedStop


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Gauss Seidel Iteration MethodGauss Seidel Iteration MethodExercise

Solve the following system of Equsby Gauss Seidel Iteration

Method,



result with it.

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

10 2 3

2 10 15

10 2 27

2 10 9

x x x xx x x x

x x x xx x x x


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M M OHANM MOHAN J AGADEESH J AGADEESH KUMARKUMAR

ASST PROFESSOR (SG)ASST PROFESSOR (SG)

SMBSSMBS,, R.NOR.NO: MB 133G: MB 133G

Thank you

3/5/2013 Numerical Methods 23

WINSEM2012 13 CP0854 06 Mar 2013 RM01 7 Numerical Methods

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