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Simultaneous Linear Equations Gaussian Elimination with Partial Pivoting http://nm.mathforcollege.com.

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Presentation on theme: "Simultaneous Linear Equations Gaussian Elimination with Partial Pivoting http://nm.mathforcollege.com."— Presentation transcript:

1 Simultaneous Linear Equations Gaussian Elimination with Partial Pivoting

2 Naïve Gauss Elimination Pitfalls

3 Pitfall#1. Division by zero

4 Is division by zero an issue here?

5 Is division by zero an issue here? YES
Division by zero is a possibility at any step of forward elimination

6 Pitfall#2. Large Round-off Errors
Exact Solution

7 Solve it on a computer using 6 significant digits with chopping
Pitfall#2. Large Round-off Errors Solve it on a computer using 6 significant digits with chopping

8 Pitfall#2. Large Round-off Errors
Solve it on a computer using 5 significant digits with chopping Is there a way to reduce the round off error?

9 Increase the number of significant digits
Avoiding Pitfalls Increase the number of significant digits Decreases round-off error Does not avoid division by zero

10 Gaussian Elimination with Partial Pivoting
Avoiding Pitfalls Gaussian Elimination with Partial Pivoting Avoids division by zero Reduces round off error

11 Gauss Elimination with Partial Pivoting http://nm.MathForCollege.com

12 What is Different About Partial Pivoting?
At the beginning of the kth step of forward elimination, find the maximum of If the maximum of the values is in the p th row, then switch rows p and k.

13 Which two rows would you switch?
Example (2nd step of FE) Which two rows would you switch?

14 Example (2nd step of FE) Switched Rows

15 Gaussian Elimination with Partial Pivoting
A method to solve simultaneous linear equations of the form [A][X]=[C] Two steps 1. Forward Elimination 2. Back Substitution

16 Gauss Elimination with Partial Pivoting Example

17 Example 2 Solve the following set of equations by Gaussian elimination with partial pivoting 17

18 Example 2 Cont. Forward Elimination Back Substitution 18

19 Forward Elimination 19

20 Number of Steps of Forward Elimination
Number of steps of forward elimination is (n-1)=(3-1)=2

21 Forward Elimination: Step 1
Examine absolute values of first column, first row and below. Largest absolute value is 144 and exists in row 3. Switch row 1 and row 3.

22 Forward Elimination: Step 2
Examine absolute values of second column, second row and below. Largest absolute value is and exists in row 3. Switch row 2 and row 3.

23 Back Substitution 23

24 Gaussian Elimination with Partial Pivoting Solution
24

25 THE END


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