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ECIV 301

ECIV 301. Programming & Graphics Numerical Methods for Engineers Lecture 14 Elimination Methods. Objectives. Introduction to Matrix Algebra Express System of Equations in Matrix Form Introduce Methods for Solving Systems of Equations Advantages and Disadvantages of each Method.

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ECIV 301

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  1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 14 Elimination Methods

  2. Objectives • Introduction to Matrix Algebra • Express System of Equations in Matrix Form • Introduce Methods for Solving Systems of Equations • Advantages and Disadvantages of each Method

  3. Last Time Matrix Algebra Rectangular Array of Elements Represented by a single symbol [A]

  4. Row 1 Row 3 Column m Column 2 Last Time Matrix Algebra n x m Matrix

  5. 3rd Row 2nd Column Last Time Matrix Algebra

  6. Last Time Matrix Algebra 1 Row, m Columns Row Vector

  7. Last Time Matrix Algebra n Rows, 1 Column Column Vector

  8. Main Diagonal Last Time Matrix Algebra If n = m Square Matrix e.g. n=m=5

  9. Last Time Matrix Algebra Special Types of Square Matrices Symmetric: aij = aji

  10. Last Time Matrix Algebra Special Types of Square Matrices Diagonal: aij = 0, ij

  11. Last Time Matrix Algebra Special Types of Square Matrices Identity: aii=1.0 aij = 0, ij

  12. Last Time Matrix Algebra Special Types of Square Matrices Upper Triangular

  13. Last Time Matrix Algebra Special Types of Square Matrices Lower Triangular

  14. Last Time Matrix Algebra Special Types of Square Matrices Banded

  15. Last Time Matrix Operating Rules - Equality [A]mxn=[B]pxq n=p m=q aij=bij

  16. Last Time Matrix Operating Rules - Addition [C]mxn= [A]mxn+[B]pxq n=p cij = aij+bij m=q

  17. Last Time Multiplication by Scalar

  18. m=p Last Time Matrix Multiplication [A] n x m . [B] p x q = [C] n x q

  19. Last Time Matrix Multiplication

  20. Last Time Matrix Multiplication

  21. Last Time Operations - Transpose

  22. Last Time Operations - Inverse [A] [A]-1 [A] [A]-1=[I] If [A]-1 does not exist [A] is singular

  23. Last Time Operations - Trace Square Matrix tr[A] = Saii

  24. Equations in Matrix Form Consider

  25. Linear Equations in Matrix Form

  26. Linear Equations in Matrix Form

  27. Linear Equations in Matrix Form

  28. Linear Equations in Matrix Form

  29. Linear Equations in Matrix Form

  30. Linear Equations in Matrix Form

  31. # Equations = # Unknowns = n Square Matrix n x n

  32. Solution of Linear Equations Consider the system

  33. Solution of Linear Equations

  34. Solution of Linear Equations

  35. Solution of Linear Equations

  36. Solution of Linear Equations What is the characteristic? Express In Matrix Form Upper Triangular Solution by Back Substitution

  37. 0 Solution of Linear Equations Objective Can we express any system of equations in a form

  38. Background Consider (Eq 1) 2*(Eq 1) (Eq 2) (Eq 2) Solution Solution !!!!!! Scaling Does Not Change the Solution

  39. Background Consider (Eq 1) (Eq 1) (Eq 2) (Eq 2)-(Eq 1) Solution Solution !!!!!! Operations Do Not Change the Solution

  40. Gauss Elimination Example Forward Elimination

  41. Gauss Elimination -

  42. Gauss Elimination Substitute 2nd eq with new

  43. Gauss Elimination -

  44. Gauss Elimination Substitute 3rd eq with new

  45. Gauss Elimination -

  46. Gauss Elimination Substitute 3rd eq with new

  47. Gauss Elimination

  48. Gauss Elimination

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