1. Special Matrices Banded matrices Solutions to problems that depend on their neighbours eg 1D T i = f(T i-1,T i+1 ) 2D T i,j = f(T i-1,j,T i+1,j,T i,j -1,T i,j +1 )

2. Tridiagonal Matrices

3. Tridiagonal Matrix Algorithm

4. TDMA procedure

5. Iterative Methods Often much better on sparse matrices than direct solvers Idea: –guess {x} –use in an approximation of [A] {x}= {b} to get new values of {x} –repeat until {x} is not changing much Pros: –much less effort / faster –less problems with roundoff

6. Point-Jacobi rearrange equation set so that you get series of x i = fn(other x’s) pick order so that using the x i with the largest coefficient in each equation guess value for x i, then iterate 6 x 1 - 2 x 2 + x 3 = 11 x 1 + 2 x 2 - 5 x 3 = -1 -2 x 1 + 7 x 2 + 2 x 3 = 5

7. Jacobi Iteration Rearrange x 1 = 1/6 (11 + 2 x 2 - x 3 ) x 2 = 1/7 ( 5 + 2 x 1 - 2 x 3 ) x 3 = 1/5 (1 + x 1 + 2 x 2 ) Iterate x 1 n+1 = 1/6 (11 + 2 x 2 n - x 3 n ) x 2 n+1 = 1/7 ( 5 + 2 x 1 n - 2 x 3 n ) x 3 n+1 = 1/5 (1 + x 1 n + 2 x 2 n ) Guess, plugin, repeat n0x10x20x30n0x10x20x30

8. Importance of diagonal dominance Simple example

9. Convergence Formally

10. Stability criteria

11. Convergence criteria How to judge whether solution is “close enough”

12. Gauss-Seidal In Point-Jacobi technique, we use “old” values of x i throughout each iteration but we are calculating “new” values all the way through the procedure if we use these “new” values on the RHS’s, this is Gauss-Seidal

13. Speed-up

14. Relaxation

15. Under-relaxation and Over-relaxation x i n = x i n-1 + (x i n - x i n-1 ) New value depends on 0 < < 1 > 1 = 1

16. Engineering examples Concentrations in a series of connected tanks (steady state) mass balances 1, V1 3, V3 2, V2 Q2in, C2in Q13, C1 Q12, C1 Q21, C2 Q23, C2 Q1in, C1in Q3out, C3

17. Engineering examples - Forces on truss FcextV a b c Fab Fbc Fac FbextV FbextH FaextV Sum of forces at each node = 0 (both vertical & horizontal)

18. Trusses Forces on trusses loads: –dead weight –live (train) weight –wind loads –seismic