1. Thomas algorithm to solve tridiagonal matrices

2. Basically sets up an LU decomposition
three parts
1) decomposition
2) forward substitution
3) backward substitution

3. 1) decomposition
loop from rows 2 to n
ai = ai/bi-1
bi = bi-ai*ci-1
end loop
2) forward substitution
loop from 2 to n
ri = ri - ai*ri-1

4. 3) back substitution
xn = rn/bn
loop from n-1 to 1
xi = (ri-ci*xi+1)/bi
end loop

5. Example
First decompose T

6. loop from rows 2 to 5
ai = ai/bi-1
bi = bi-ai*ci-1
end loop

7. forward substitution
loop from 2 to n
ri = ri - ai*ri-1
end loop

8. back substitution
xn = rn/bn
loop from n-1 to 1
xi = (ri-ci*xi+1)/bi
end loop

9. Crout algorithm - alternate LU decomposition
Instead of 1’s on diagonal of L, get 1’s on diagonal of U
Operate on rows and columns sequentially, narrowing down to single element

10. Algorithm
for j=2 to n-1
For j=n

11. Cholesky decomposition
A decomposition for symmetric matrices
Symmetric matrix e.g. a covariance matrix

12. For symmetric matrices, can write
Can develop relationships for l

13. Example:
1st row
Skip this equation

14. Row 2

15. Row 3

16. Row 4

18. Iterative methods for solving matrix equations
1. Jacobi
2. Gauss-Seidel
3. Successive overrelaxation (SOR)
Idea behind iterative methods: Convert Ax=b into Cx=d, which has sequence of approximations x(1), x(2), … with

19. What are C and d?
Recall from fixed-point iteration

20. Rewrite matrix equation in same way
becomes

21. Then

22. Jacobi method is like fixed point iteration
Example: Shape of a stretched membrane

23. Shape can be described by potential function
Let us give some boundary conditions

24. Problem look likes this
Solve for u’s

25. Leads to this system of equations

26. Choose an initial u=[1 1 1 1 …1]’
Iterate using x=Cx+d

27. Matlab solution, 49 iterations

28. Gauss-Seidel method differs from Jacobi by sequential updating
- use new xi’s as they become available

29. Example:
Jacobi
Gauss-Seidel