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					More on iterative solution of Ax  b


                  Iterative solution


                                  Jacobi’s method


                                       Gauss’-Seidel’s
                                         Approach

                                       SOR method



SOR method. Successive Over-Relaxation Method

Consider the basic framework for an iterative
solution of Ax  b through an example.

         4 x1  3 x2          24
         3x1  4 x2  x3  30
              x2  4 x3  24

Its solution is x  3 4  5t .

We begin with a starting solution x( 0 )  1 1 1t .

Gauss-Seidel frames the equations in the following
way:
                x1 k )  0.75 x2k 1 )  6
                 (              (


                x2k )  0.75 x1 k )  0.25 x3 k 1 )  7.5
                 (             (             (

                 (             (
                x3 k )  0.25 x2k )  6

For Gauss’-Seidel,

                   1  i 1             n
                                             ( k 1 )      
      xi( k )                  (k )
                        aij x j   aij x j         bi 
                  aii  j 1
                                    j i 1               
                                                           

In SOR, we begin right about here. We take ( 1   )
units of the previous estimate of xi and take  units
of the currently available estimate to formulate the
iterative solution:

xi( k )  ( 1   )xi( k 1 ) 
                i 1              n                             
          
                         (k )
                   aij x j               aij x(j k 1 )    bi 
      aii 
                 j 1            j i 1                         
                                                                  

The value of  is usually less than 2.0. When   1
we get the Gauss’-Seidel method; for 0    1, we
get Under-Relaxation Method. For 1    2 , we
have the Over-Relaxation Method.
If we take   1.5 , our SOR equations appear as

     x1 k )  0.5 x1 k 1 ) 
      (             (            1.5
                                  4
                                     3x2k 1 )  24
                                         (
                                                        
                          1.5
x2k )  0.5 x2k 1 ) 
 (            (
                              ( 3x1 k )  4 x2k 1 )  x3 k 1 )  30 )
                                   (          (          (
                           4

    ….

Assignment. Write a program that would iteratively
obtain SOR solutions given A and b .

				
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posted:3/8/2012
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