Linear System

1. Linear System We want to solve the following linear system Example: Solve: 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8 Remark: (1) has a unique solution A is invertable Remark: A is invertable det(A)=0 Remark: A is invertable

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

3. Linear System 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 These methods generate a sequence of approximate solutions Remark:

5. Jacobi Method Consider 4x4 case Example 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8

6. Jacobi Method 0.6000 1.0473 0.9326 1.0152 0.9890 2.2727 1.7159 2.0533 1.9537 2.0114 -1.1000 -0.8052 -1.0493 -0.9681 -1.0103 1.8750 0.8852 1.1309 0.9738 1.0214 11.3537 4.9910 2.0299 0.8911 0.3686 1.0032 0.9981 1.0006 0.9997 1.0001 1.9922 2.0023 1.9987 2.0004 1.9998 -0.9945 -1.0020 -0.9990 -1.0004 -0.9998 0.9944 1.0036 0.9989 1.0006 0.9998 0.1605 0.0671 0.0290 0.0122 0.0053

7. Gauss Seidel Method Note that in the Jacobi iteration one does not use the most recently available information. 0.6000 1.0302 1.0066 1.0009 1.0001 2.3273 2.0369 2.0036 2.0003 2.0000 -0.9873 -1.0145 -1.0025 -1.0003 -1.0000 0.8789 0.9843 0.9984 0.9998 1.0000 5.6930 0.4300 0.0662 0.0082 0.0009

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 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8 10 11 10 8 -1 2 -1 0 3 -1 -1 2 0 -1 3 -1 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8 Jacobi: U=triu(A,1) L=tril(A,-1) D=diag(diag(A)) eig(inv(M)*N) -0.4264 -0.1040 0.1860 0.3445 GS: 0 0 0.0839 - 0.0322i 0.0839 + 0.0322i

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 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 Example: Successive over Relaxation 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8

18. Successive over Relaxation Successive over Relaxation Example: 10 -1 2 0 -1 11 -1 3 2 -1 10 -1 0 3 -1 8 0.6180 1.0553 1.0055 2.3988 2.0273 2.0004 -1.0132 -1.0211 -1.0014 0.8743 0.9905 1.0000

19. MATLAB CODE Ex: 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); end xnew(i)=(b(i)-sum1-sum2)/A(i,i) end X(1:n,k)=xnew; x=xnew; end sol=xnew;

20. MATLAB CODE Ex: 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); end x(i)=(b(i)-sum1-sum2)/A(i,i) end X(1:n,k)=x; end sol=x;

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

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