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Anonymous

summary about conditioner I want someone write to me papers about preconditioner for solving a linear system from three to five pages. The topic (preconditioner of the iterative method for solving a linear system)

1- introduction about preconditioner

Talk about the two strategies

a- splitting matrices technique (Jacobi, Gauss-Seidel, and SOR method)

b- Incomplete Cholesky and ILU preconditioners (conjugate gradient and GMRES methods)

Hello here we go please go through it and let me know in case of anything, I hope it is to your expectation. Goodbye

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Summary on Numerical Methods

The term preconditioning is often used in a numerical method as an extra step

designed to accelerate the convergence. Preconditioning is an important technique in iterative

methods for solving large systems of linear equations. Preconditioning for eigenvalue

problems is not as straightforward and has been slower to catch on. There are preconditioners

which are based on simple iterative solvers, these include: Jacobi, Gauss-Seidel and SOR

preconditioners. The second category is the one based on direct solvers and have been

modified for fast but approximatively solving such as the incomplete Cholesky.

Splitting the matrix

The splitting technique was the first class of iterative solvers for linear systems of

equations. The splits were such that [55,127,135], where the matrix A was split in a sum of

two matrices for example, B and A-B we therefore come up with the equation Bx = (B-A) x

+b instead of writing Ax = b. we then obtain the fixed point formulation if B is not singular.

X = B-1(B-A)x + B-1b

We the substitute the left side x with xk+1 and R.H.S with xk to get the iteration scheme:

Xk+1 = B-1 (B-A) xk + B-1b = xk –B-1Axk +B-1b

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This then becomes the convergent if and only if B-1(B-A) which is the spectral radius

is less than one. So as to use the iteration scheme used as a preconditioner, we come up with

...

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