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I correct please read again...

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Hello,

I have read this thesis in Mathematics and statistics about
conjugate gradients solver with incomplete Cholesky factorization:

http://repositories.tdl.org/ttu-ir/bitstream/handle/2346/47490/KENNEDY-THESIS.pdf?sequence=2

You will notice that the preconditionner with the incomplete Cholesky 
factorization will give around 3x improvement on the number of iterations...

But as you have noticed i have not used a preconditionner for
my conjugate gradiaents system solver cause it's a parallel solver:

https://sites.google.com/site/aminer68/parallel-implementation-of-conjugate-gradient-linear-system-solver


But if you want me to implement a precondtionner that's easy:

So to solve a linear system of linear equations:

A*x = b  [1]

we have to first use incomplete Cholesky factorication on A
this will give us A := R*Transpose(R)

Note: Tanspose(R) means the matrix transpose of R
       and Inverse(M) means the inverse of the matrix

[1] equal to inverse(M)*A = inverse(M)*b

M is equal to R*Transpose(R)

So after resolving, this will give:

Inverse(Transpose(R))*A*Inverse(R)*R*x = Inverse(Transpose(R))*b

So we have to resolve the follwing system of equations:

A1*x1=b1

where A1=Inverse(Transpose(R))*A*Inverse(R)

and x1=R*x
and b = Inverse(Transpose(R))*b


So we have to apply the conjugate gradients solver to the system:

A1*x1=b1


And after that we have to resolve the system R*x=x1 to find the vector x


So we have to parallelize the calculation of the determinant of
a matrix to calculate the inverse of a matrix and we have to parallelize 
the calculation of the incomplete Cholesky factorization,
and that's not so difficult for me.

That's how the systeme will be preconditionned to accelerate the 
convergence and that's will give a 3x improvement on the number of 
iterations..




Thank you,
Amine Moulay Ramdane.

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I correct please read again... aminer <aminer@toto.net> - 2014-05-21 13:21 -0700

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