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## Homework Statement

## Homework Equations

P

^{k}= A*X

^{k}- F

τ

_{k}= ((A*X

^{k}- F)*(A*X

^{k}- F))/(A*(A*X

^{k}- F))*(A*X

^{k}- F)

X

^{k+1}= X

^{k}- τ

_{k}* (A*X

^{k}- F)

## The Attempt at a Solution

We are given a system of equations:

AX=F, where A

_{3x3}= (2 1 0.95; 1 2 1; 0.95 1 2) and F= (3.95;4;3.95)

^{τ}

(The solution is (1;1;1)

^{τ})

We choose the first X freely: X

^{0}= (0;0;0)

^{τ}

Therefore:

**(1)**AX

^{0}-F = -F = (-3.95;-4;-3.95)

^{τ}

**(2)**(A*X

^{0}- F)*(A*X

^{0}- F) = 47.205

**(3)**A*(A*X

^{0}- F) = (-15.6525;-15.9;-15.6525)

^{τ}

**(4)**(A*(A*X

^{k}- F))*(A*X

^{k}- F) = 187.25475

So τ

_{0}= 0.252090

Therefore our new approximate X is:

**(5)**X

^{1}= X

^{0}- τ

_{0}* (A*X

^{k}- F) = (0;0;0)

^{τ}- 0.252090 * (-3.95;-4;-3.95)

^{τ}≈ (0.995754;1.008359;0.995754)

^{τ}

We repeat this process with the new approximation of X, until we reach the desired accuracy.

This is the method of steepest descent without preconditioning. Can anyone show me how

*each step*of this algorithm changes with preconditioning applied? Suppose we use a preconditioner matrix B which is equal to the inverse of A (or any matrix that you think would be the best).

I'd highly appreciate any help. I've made a program to calculate solutions of a system of equations using SDM already, and it works fine, I just don't understand how to apply preconditioning. Hopefully the equations I've written are readable, apologies.

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