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**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

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**Linear System Remark: Remark:**

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

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**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

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**Iterative Method Remark:**

These methods generate a sequence of approximate solutions Remark:

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

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**Jacobi Method K=1 K=2 K=3 K=4 K=5 0.6000 1.0473 0.9326 1.0152 0.9890**

x1 x2 x3 X4 K=6 K=7 K=8 K=9 K=10 x1 x2 x3 X4

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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

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**Gauss Seidel Method Jacobi iteration for general n:**

Gauss-Seidel iteration for general n:

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**Splittings and Convergence**

DEF: with eigenvalues spectral radius of A is defined to be DEF: Splitting A large family of iteration

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**Splittings and Convergence**

A large family of iteration Diagonal Lower Upper Jacobi: Gauss-Seidel:

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**Splittings and Convergence**

A large family of iteration THM:

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**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

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**Splittings and Convergence**

A large family of iteration Remarks: THM: Proof: (Golub p511)

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**Splittings and Convergence**

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

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**Splittings and Convergence**

DEF: IF Example: THM:

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**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:

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**Successive over Relaxation Successive over Relaxation**

Example: Successive over Relaxation

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**Successive over Relaxation Successive over Relaxation**

Example: K=1 K=2 K=3 x1 x2 x3 X4

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**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;

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**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;

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**Another Look Jacobi: GS: Remark: Given:**

We want to improve this approximate: Jacobi: GS:

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**Examples of Splittings**

1) Non-symmetric Matrix: symmetric Skew-symmetric 2) Domain Decomposition:

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Linear Algebra and Complexity Chris Dickson CAS 746 - Advanced Topics in Combinatorial Optimization McMaster University, January 23, 2006.

Linear Algebra and Complexity Chris Dickson CAS 746 - Advanced Topics in Combinatorial Optimization McMaster University, January 23, 2006.

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