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Pitfalls of elimination methods

Pitfalls of elimination methods. Division by zero: during row normalization. Round of errors: more significant figure gives less errors.(Important when dealing with 100 or more equations) 3. III-conditioned systems: small change in coefficients results large changes in solution

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Pitfalls of elimination methods

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  1. Pitfalls of elimination methods • Division by zero: during row normalization. • Round of errors: more significant figure gives less errors.(Important when dealing with 100 or more equations) 3. III-conditioned systems: small change in coefficients results large changes in solution 4. Singular systems: when two equations are identical there are more unknowns than equations

  2. Techniques for improving solutions • Use more significant figures. • Pivoting: switching the rows so that the largest is that the pivot element (when the pivot element a11 is zero) avoid division by zero during normalization.

  3. Gauss-Jordan • It is a variation of gauss elimination. • The elimination step results in an identity matrix …………. than a triangular matrix

  4. Example Solve:

  5. Example Solution: Put the equation in augmented matrix form:

  6. Example Solution: 2. Exchange R1 with R2 (pivoting)

  7. Example Solution: 3. Replace R3 with R3 -2R2

  8. Example • Solution: • 4. Replace R1with R1 –(1/2)R2 • 5. Replace R3 with R3 +(1/2)R2

  9. Example • Solution: • 6. Replace R1with R1+(3/5)R3 • 7. Replace R2 with R2 +(2/5)R3

  10. Example • Solution: • 8. Replace R2 with (1/2)R2 • 9. Replace R3 with -(2/5)R3

  11. Example Thus x1=11/5 x2=7/5 x3=6/5

  12. Chapter 12 Special matrices and Gauss-Seidel

  13. Special matrices and Gauss-Seidel • Banded and symmetric matrices are solved by efficient methods. • Gauss-Seidel is an alternative to elimination methods. • It is an approximate and iterative method. • It employs initial gausses. • It is well suited for large number of equations. • It is the most commonly used iterative method.

  14. Special matrices and Gauss-Seidel • If [A][X]=[B] take 3X3 matrix.

  15. Special matrices and Gauss-Seidel Then:

  16. Special matrices and Gauss-Seidel • Assume all (x)’s are zeros (initial guess). • Substitute in the equation of x1 to get a new value: • Substitute this in the equation of x2 to get a new value for x2 . • Then x3, until we get the solution.

  17. Special matrices and Gauss-Seidel • Convergence criterion if we have (εs) is: where: i: is the number of variables. j: is the number of iteration. • The problem with Gauss-Seidel is the slow convergence

  18. Convergence Criterion: It is sufficient but not ………….. • It means that the diagonal element must be greater than off-diagonal element for each row. • Gauss-Seidel avoids round-off errors existing in element ……………… • It is particularly good for large sparse matrices (most elements ………….) because it doesn’t store zeros.

  19. Note: • Let the number of equations is (m) and the number of unknowns is (n), then: • If m<n: undetermined system (no solution or ………………….) • If m>n: overdetermined system (generally there is no exact solution).

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