1 / 19

Pitfalls of Elimination Methods and Techniques for Improving Solutions

This guide explores common pitfalls in elimination methods like division by zero, round-off errors, ill-conditioned and singular systems, and offers techniques to enhance solutions with more significant figures and pivoting. It also introduces the Gauss-Jordan method and discusses special matrices and the Gauss-Seidel iterative approach.

jroesch
Download Presentation

Pitfalls of Elimination Methods and Techniques for Improving Solutions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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).

More Related