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Iterative methods

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For solving a set of linear equations, we seek the solution to the problem:

 AX = Q

After k iterations we obtain an approaximation to the solution as:

 Ax^{(k)}  = Q - r^{(k)}

where  r^{(k)} is the residual after k iterations.
Defining:

 \varepsilon ^{(k)}  = x - x^{(k)}

as the difference between the exact and approaximate solution.
we obtain :

 A\varepsilon ^{(k)}  = r^{(k)}

the purpose of iterations is to drive this residual to zero.

Stationary Iterative Methods

Iterative methods that can be expressed in the simple form:


x^{(k+1)}  = Bx^{(k)}  + c

When neither B nor c depend upon the iteration count (k), the iterative method is called stationary iterative method. Some of the stationary iterative methods are:

  1. Jacobi method
  2. Gauss-Seidel method
  3. Successive Overrelaxation (SOR) method and
  4. Symmetric Successive Overrelaxation (SSOR) method

Nonstationary Iterative Methods

When during the iterations B and c changes during the iterations, the method is called Nonstationary Iterative Method. Typically, constants B and c are computed by taking inner products of residuals or other vectors arising from the iterative method.

Some examples are:

  1. Conjugate Gradient Method (CG)
  2. MINRES and SYMMLQ
  3. Generalized Minimal Residual (GMRES)
  4. BiConjugate Gradient (BiCG)
  5. Quasi-Minimal Residual (QMR)
  6. Conjugate Gradient Squared Method (CGS)
  7. BiConjugate Gradient Stabilized (Bi-CGSTAB)
  8. Chebyshev Iteration
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