1. Chapter 2 Solutions of Systems of Linear Equations / Matrix Inversion
Chem Math 252
Chapter 2 Solutions 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| Form A1
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 ×,
600th order 1.27× ×, 1.27×

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 n
Cramer’s
Gaussian Elimination
Gauss-Jordan Elimination 2 12 9 7 3 48 28 27 4
240 62 67 5
1440
115
133 10
805
1063 20
1.0×1020
5910
8323
100
1.9×10160
681550
1000
4.0×102570
6.7×108
1.0×109

20. Comparison of Direct Methods
Time required on a 300 MFLOP computer (500 TFLOP) n
Cramer’s
Gaussian Elimination
Gauss-Jordan Elimination 2
2.4×10-8s
1.8×10-8s
1.4×10-8s 3
9.6×10-8s
5.6×10-8s
5.4×10-8s 4
4.8×10-7s
1.2×10-7s
1.3×10-7s 5
2.9×10-6s
2.3×10-7s
2.7×10-7s 10
0.16s
1.6×10-6s
2.1×10-6s 20
6475 years (2.4 days)
1.2×10-5s
1.7×10-5s
100
1×10144 (1×10138) years
1.4×10-3s
2.0×10-3s
1000
(102548) years
1.3

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 iteration x1 x2 x3 1 0.571429 1.333333 2 1.0952381 2 3 4 5 6 7 8 9 10 11 12 13

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

25. Jacobi Method Gauss-Seidel Method iteration x1 x2 x3 1 0.5714291 2 3 4 5 6 7 8 9 10 11 12 13
iteration x1 x2 x3 1 2 3 4 5 6 7 8 9

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