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Gaussian, Oddeven reduction Algorithms

Gaussian, Oddeven reduction Algorithms . Ayaz Khaliq. Odd even reduction algorithm illustration. Gaussian Elimination. One of the most popular techniques for solving simultaneous linear equations of the form Consists of 2 steps 1. Forward Elimination of Unknowns. 2. Back Substitution.

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Gaussian, Oddeven reduction Algorithms

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  1. Gaussian, Oddeven reduction Algorithms Ayaz Khaliq

  2. Odd even reduction algorithm illustration

  3. Gaussian Elimination One of the most popular techniques for solving simultaneous linear equations of the form Consists of 2 steps 1. Forward Elimination of Unknowns. 2. Back Substitution

  4. Forward Elimination The goal of Forward Elimination is to transform the coefficient matrix into an Upper Triangular Matrix

  5. Forward Elimination Linear Equations A set of n equations and n unknowns . . . . . .

  6. Forward Elimination Transform to an Upper Triangular Matrix Step 1: Eliminate x1 in 2nd equation using equation 1 as the pivot equation Which will yield

  7. Forward Elimination Step 2: Eliminate x2 in the 3rd equation. Equivalent to eliminating x1 in the 2nd equation using equation 2 as the pivot equation.

  8. Example: Assume Results in a matrix template of the form: Using date from the time / velocity table, the matrix becomes:

  9. Example---- Forward Elimination: Step 1 Yields

  10. Example: Forward Elimination: Step 1 Yields

  11. Example: Rocket Velocity Forward Elimination: Step 2 Yields This is now ready for Back Substitution

  12. Example: Back Substitution: Solve for a3 using the third equation

  13. Partial Pivoting Gaussian Elimination with partial pivoting applies row switching to normal Gaussian Elimination. How? At the beginning of the kth step of forward elimination, find the maximum of If the maximum of the values is In the pth row, then switch rows p and k.

  14. Partial Pivoting What does it Mean? Gaussian Elimination with Partial Pivoting ensures that each step of Forward Elimination is performed with the pivoting element |akk| having the largest absolute value.

  15. Partial Pivoting: Example Consider the system of equations In matrix form = Solve using Gaussian Elimination with Partial Pivoting using five significant digits with chopping

  16. Partial Pivoting: Example Forward Elimination: Step 1 Examining the values of the first column |10|, |-3|, and |5| or 10, 3, and 5 The largest absolute value is 10, which means, to follow the rules of Partial Pivoting, we switch row1 with row1. Performing Forward Elimination

  17. Partial Pivoting: Example Forward Elimination: Step 2 Examining the values of the first column |-0.001| and |2.5| or 0.0001 and 2.5 The largest absolute value is 2.5, so row 2 is switched with row 3 Performing the row swap

  18. Partial Pivoting: Example Forward Elimination: Step 2 Performing the Forward Elimination results in:

  19. Partial Pivoting: Example Back Substitution Solving the equations through back substitution

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