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Gaussian Elimination

Learn how to solve simultaneous linear equations using Naïve Gaussian Elimination, including the pitfalls to avoid and the use of partial pivoting. Also, explore how to calculate the determinant of a square matrix.

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Gaussian Elimination

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  1. Gaussian Elimination Major: All Engineering Majors Author(s): Autar Kaw http://nm.MathForCollege.com Transforming Numerical Methods Education for STEM Undergraduates

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

  3. Forward Elimination The goal of forward elimination is to transform the coefficient matrix into an upper triangular matrix

  4. Forward Elimination Exercise: Show the steps for this slide (10 minutes).

  5. Back Substitution Solve each equation starting from the last equation Example of a system of 3 equations

  6. THE END

  7. Naïve Gauss EliminationPitfalls

  8. Pitfall#1. Division by zero

  9. Is division by zero an issue here?

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

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

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

  13. 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?

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

  15. Avoiding Pitfalls Gaussian Elimination with Partial Pivoting • Avoids division by zero • Reduces round off error

  16. THE END

  17. Gauss Elimination with Partial Pivotinghttp://nm.MathForCollege.com

  18. 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.

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

  20. Example (2nd step of FE) Switched Rows

  21. 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

  22. THE END

  23. Gauss Elimination with Partial PivotingExample

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

  25. Example 2 Cont. • Forward Elimination • Back Substitution

  26. Forward Elimination

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

  28. 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.

  29. Forward Elimination: Step 1 (cont.) Divide Equation 1 by 144 and multiply it by 64, . . Subtract the result from Equation 2 Substitute new equation for Equation 2

  30. Forward Elimination: Step 1 (cont.) Divide Equation 1 by 144 and multiply it by 25, . . Subtract the result from Equation 3 Substitute new equation for Equation 3

  31. Forward Elimination: Step 2 Examine absolute values of second column, second row and below. • Largest absolute value is 2.917 and exists in row 3. • Switch row 2 and row 3.

  32. Forward Elimination: Step 2 (cont.) Divide Equation 2 by 2.917 and multiply it by 2.667, . Subtract the result from Equation 3 Substitute new equation for Equation 3

  33. Back Substitution

  34. Back Substitution Solving for a3

  35. Back Substitution (cont.) Solving for a2

  36. Back Substitution (cont.) Solving for a1

  37. Gaussian Elimination with Partial Pivoting Solution

  38. THE END

  39. Determinant of a Square MatrixUsing Naïve Gauss EliminationExamplehttp://nm.MathForCollege.com

  40. Theorem of Determinants If a multiple of one row of [A]nxn is added or subtracted to another row of [A]nxn to result in [B]nxn then det(A)=det(B)

  41. Theorem of Determinants The determinant of an upper triangular, lower triangular or diagonal matrix [A]nxn is given by

  42. Forward Elimination of a Square Matrix Using forward elimination to transform [A]nxn to an upper triangular matrix, [U]nxn.

  43. Example Using Naive Gaussian Elimination method, find the determinant of the following square matrix.

  44. Finding the Determinant After forward elimination steps .

  45. THE END

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