Solutions of Systems of Linear Equations: Direct vs Indirect Methods

1. Chem 302 - Math 252 Chapter 2Solutions of Systems of Linear Equations / Matrix Inversion

2. Solutions of Systems of Linear Equations • n linear equations, n unknowns • Three possibilities • Unique solution • No solution • Infinite solutions • Numerically systems that are almost singular cause problems • Range of solutions • Ill-conditioned problem Singular Systems

7. Solutions of Systems of Linear Equations • Direct Methods • Determine solution in finite number of steps • Usually preferred • Round-off error can cause problems • Indirect Methods • Use iteration scheme • Require infinite operations to determine exact solution • Useful when Direct Methods fail

8. Direct Methods • Cramer’s Rule • Gaussian Elimination • Gauss-Jordan Elimination • Maximum Pivot Strategy

9. Cramer’s Rule • Write coefficient matrix (A) • Evaluate |A| • If |A|=0 then singular • Form A1 • Replace column 1 of A with answer column • Compute x1 = |A1|/|A| • Repeat 3 and 4 for other variables

10. Cramer’s Rule Not singular: System has unique solution

11. Cramer’s Rule

12. Cramer’s Rule • Good for small systems • Good if only one or two variables are needed • Very slow and inefficient for large systems • n order system requires (n+1)! × & (n+1)! Additions • 2nd order 6 ×, 6 + • 10th order 3628800 ×, 3628800 + • 600th order 1.27×101408 ×, 1.27×101408 +

13. Gaussian Elimination • Form augmented matrix • Use elementary row operations to transform the augmented matrix so that the A portion is in upper triangular form • Switch rows • Multiply row by constant • Linear combination of rows • Use back substitution to find solutions • Requires n3+n2- n ×, n3+½n2- n +

14. Gaussian Elimination

15. Gauss-Jordan Elimination • Form augmented matrix • Normalize 1st row • Use elementary row operations to transform the augmented matrix so that the A portion is the identity matrix • Switch rows • Multiply row by constant • Linear combination of rows • Requires ½n3+n2- 2½n+2 ×, ½n3-1½n+1 + • Can also be used to find matrix inverse

16. Gauss-Jordan Elimination

17. Maximum Pivot Strategy • Elimination methods can run into difficulties if one or more of diagonal elements is close to (or exactly) zero • Normalize row with largest (magnitude) element.

18. Gauss-Jordan Elimination

19. Comparison of Direct Methods • Small systems (n<10) not a big deal • Large systems critical Number of floating point operations

20. Comparison of Direct Methods Time required on a 300 MFLOP computer (500 TFLOP)

21. Indirect Methods • Jacobi Method • Gauss-Seidel Method • Use iterations • Guess solution • Iterate to self consistent • Can be combined with Direct Methods

22. Jacobi Method • Rearrange system of equations to isolate the diagonal elements • Guess solution • Iterate until self-consistent

23. Jacobi Method

24. Gauss-Seidel Method • Same as Jacobi method, but use updated values as soon as they are calculated.

25. Jacobi Method Gauss-Seidel Method

26. Indirect Methods • Sufficient condition • Diagonally dominant • Large problems • Sparse matrix (many zeros)