1. Linear System Remark: Remark: Remark: Example: Solve:
We want to solve the following linear system
Example: Solve:
Remark:
(1) has a unique solution
A is invertable
Remark:
A is invertable
det(A)=0
Remark:
A is invertable

2. Linear System Remark: Remark:
We want to solve the following linear system
Remark:
A is invertable
Rank(A)=n
Remark:

3. Linear System 2 classes of methods
We want to solve the following linear system
2 classes of methods
Direct Methods
Iterative Methods
Gaussian Elimination
LU, Choleski
These methods generate a sequence of approximate solutions

4. Iterative Method Remark:
These methods generate a sequence of approximate solutions
Remark:

5. Example Jacobi Method Consider 4x4 case 10 -1 2 0 -1 11 -1 3

6. Jacobi Method K=1 K=2 K=3 K=4 K=5 0.6000 1.0473 0.9326 1.0152 0.9890x1 x2 x3 X4
K=6
K=7
K=8
K=9
K=10 x1 x2 x3 X4

7. Gauss Seidel Method
Note that in the Jacobi iteration one does not use the most recently available information.
K=1
K=2
K=3
K=4
K=5 x1 x2 x3 X4

8. Gauss Seidel Method Jacobi iteration for general n:
Gauss-Seidel iteration for general n:

9. Splittings and Convergence
DEF:
with eigenvalues
spectral radius of A is defined to be
DEF:
Splitting
A large family of iteration

10. Splittings and Convergence
A large family of iteration
Diagonal
Lower
Upper
Jacobi:
Gauss-Seidel:

11. Splittings and Convergence
A large family of iteration
THM:

12. Splittings and Convergence
Example: 10 11 8 -1 -1
Jacobi:
U=triu(A,1)
L=tril(A,-1)
D=diag(diag(A))
eig(inv(M)*N)
GS:
i i

13. Splittings and Convergence
A large family of iteration
Remarks:
THM:
Proof: (Golub p511)

14. Splittings and Convergence
THM:
Proof: (Golub p512) show that all eigenvalues are less than one.

15. Splittings and Convergence
DEF: IF
Example:
THM:

16. Successive over Relaxation Successive over Relaxation
The Gauss-Seidel iteration is very attractive because of its simplicity. Unfortunately, if the spectral radius is close to one, then convergence is vey slow. One solution for this
Successive over Relaxation
GS:
Jacobi:

17. Successive over Relaxation Successive over Relaxation
Example:
Successive over Relaxation

18. Successive over Relaxation Successive over Relaxation
Example:
K=1
K=2
K=3 x1 x2 x3 X4

19. MATLAB CODE Ex: Jacobi iteration for general n:
Write a Matlab function for Jacobi
Jacobi iteration for general n:
function [sol,X]=jacobi(A,b,x0)
n=length(b);
maxiter=10;
x=x0;
for k=1:maxiter
for i=1:n
sum1=0;
for j=1:i-1
sum1=sum1+A(i,j)*x(j);
end
sum2=0;
for j=i+1:n
sum2=sum2+A(i,j)*x(j);
xnew(i)=(b(i)-sum1-sum2)/A(i,i)
X(1:n,k)=xnew;
x=xnew;
sol=xnew;

20. MATLAB CODE Ex: GS iteration for general n:
Write a Matlab function for GS
GS iteration for general n:
function [sol,X]=gs(A,b,x0)
n=length(b);
maxiter=10;
x=x0;
for k=1:maxiter
for i=1:n
sum1=0;
for j=1:i-1
sum1=sum1+A(i,j)*x(j);
end
sum2=0;
for j=i+1:n
sum2=sum2+A(i,j)*x(j);
x(i)=(b(i)-sum1-sum2)/A(i,i)
X(1:n,k)=x;
sol=x;

21. Another Look Jacobi: GS: Remark: Given:
We want to improve this approximate:
Jacobi:
GS:

22. Examples of Splittings
1) Non-symmetric Matrix:
symmetric
Skew-symmetric
2) Domain Decomposition: