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Mastering Solutions of Linear Equations: Gaussian Elimination Techniques

Explore methods like Naive Gaussian Elimination, Forward Elimination, Backward Substitution, and Scaled Partial Pivoting to solve systems of linear equations efficiently. Learn about inconsistent systems, infinite solutions, practicality of Cramer's Rule, and the importance of elementary row operations. Dive into examples, problems, determinants, and pseudo-codes for a deeper understanding of numerical methods in linear algebra.

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Mastering Solutions of Linear Equations: Gaussian Elimination Techniques

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  1. SE301: Numerical MethodsTopic 3:Solution of Systems of Linear EquationsLectures 12-17: KFUPM Read Chapter 9 of the textbook

  2. Lecture 12Vector, Matrices, andLinear Equations

  3. VECTORS

  4. MATRICES

  5. MATRICES

  6. Determinant of a MATRICES

  7. Adding and Multiplying Matrices

  8. Systems of Linear Equations

  9. Solutions of Linear Equations

  10. Solutions of Linear Equations • A set of equations is inconsistent if there exists no solution to the system of equations:

  11. Solutions of Linear Equations • Some systems of equations may have infinite number of solutions

  12. Graphical Solution of Systems ofLinear Equations Solution x1=1, x2=2

  13. Cramer’s Rule is Not Practical

  14. Lecture 13Naive Gaussian Elimination Naive Gaussian Elimination Examples

  15. Naive Gaussian Elimination • The method consists of two steps: • Forward Elimination: the system is reduced to upper triangular form. A sequence of elementary operations is used. • Backward Substitution: Solve the system starting from the last variable.

  16. Elementary Row Operations • Adding a multiple of one row to another • Multiply any row by a non-zero constant

  17. ExampleForward Elimination

  18. ExampleForward Elimination

  19. ExampleForward Elimination

  20. ExampleBackward Substitution

  21. Forward Elimination

  22. Forward Elimination

  23. Backward Substitution

  24. Lecture 14Naive Gaussian Elimination Summary of the Naive Gaussian Elimination Example Problems with Naive Gaussian Elimination Failure due to zero pivot element Error Pseudo-Code

  25. Naive Gaussian Elimination • The method consists of two steps • Forward Elimination: the system is reduced to upper triangular form. A sequence of elementary operations is used. • Backward Substitution: Solve the system starting from the last variable. Solve for xn ,xn-1,…x1.

  26. Example 1

  27. Example 1

  28. Example 1Backward Substitution

  29. Determinant

  30. How Many Solutions Does a System of Equations AX=B Have?

  31. Examples

  32. Pseudo-Code: Forward Elimination Do k = 1 to n-1 Do i = k+1 to n factor = ai,k / ak,k Do j = k+1 to n ai,j = ai,j – factor * ak,j End Do bi = bi – factor * bk End Do End Do

  33. Pseudo-Code: Back Substitution xn = bn / an,n Do i = n-1 downto 1 sum = bi Do j = i+1 to n sum = sum – ai,j * xj End Do xi = sum / ai,i End Do

  34. Lectures 15-16:Gaussian Elimination with Scaled Partial Pivoting Problems with Naive Gaussian Elimination Definitions and Initial step Forward Elimination Backward substitution Example

  35. Problems with Naive Gaussian Elimination • The Naive Gaussian Elimination may fail for very simple cases. (The pivoting element is zero). • Very small pivoting element may result in serious computation errors

  36. Example 2

  37. Example 2Initialization step Scale vector: disregard sign find largest in magnitude in each row

  38. Why Index Vector? • Index vectors are used because it is much easier to exchange a single index element compared to exchanging the values of a complete row. • In practical problems with very large N, exchanging the contents of rows may not be practical.

  39. Example 2Forward Elimination--Step 1: eliminate x1

  40. Example 2Forward Elimination--Step 1: eliminate x1 First pivot equation

  41. Example 2Forward Elimination--Step 2: eliminate x2

  42. Example 2Forward Elimination--Step 3: eliminate x3 Third pivot equation

  43. Example 2Backward Substitution

  44. Example 3

  45. Example 3Initialization step

  46. Example 3Forward Elimination--Step 1: eliminate x1

  47. Example 3Forward Elimination--Step 1: eliminate x1

  48. Example 3Forward Elimination--Step 2: eliminate x2

  49. Example 3Forward Elimination--Step 2: eliminate x2

  50. Example 3Forward Elimination--Step 3: eliminate x3

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