  Anonymous

### Question Description

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)

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

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

AlhandroK
School: Rice University  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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