Iterative Methods for Solving Linear Systems of Equations

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Iterative Methods for Linear Systems of Equations
An iterative technique to solve the n × n linear system Ax = b starts with an initial approximation x^((0) ) to the solution x and generates a sequence of vectors {x^((k) ) }_(k=0)^? that converges to x.

Jacobi Iterative Method
The Jacobi iterative method is obtained by solving the ith equation in Ax = b for x_i to obtain (provided a_ii?0)

x_i=?_?(j=1@j?i)^n??(-(a_ij x_j)/a_ii )+b_i/a_ii ? for i=1,2,…,n.
For each k?1, generate the components x_i^((k) ) of x^((k) ) from the components of x^((k-1) ) by
x_i^((k) )=1/a_ii [?_?(j=1@j?i)^n??(-a_ij x_i^((k-1) ) )+b_i ?] for i=1,2,…,n.

Example 1: The linear system Ax = b given by
E_1: 10x_1- x_2+2x_3 =0
E_2: -x_1+11x_2-x_3+3x_4=25
E_3: 2x_1- x_2+10 x_3-x_4= -11
E_4: 3x_2 - x_3+8x_4=15
Use Jacobi s iterative method to find approximations x^((k) ) to x starting with x^((0) )=(0,0,0,0)^t until
?x^((k) )-x^((k-1) ) ?_?/?x^((k) ) ?_? < ?10?^(-3).
Solution: We first solve the equation E_i for x_i, for each i=1,2,3,4 to obtain: