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In numerical linear algebra, the method of successive over-relaxation (SOR) is a variant of the Gauss–Seidel method for solving a linear system of equations, resulting in faster convergence. A similar method can be used for any slowly converging iterative process .
Matlab/Octave built-in Preconditioners: Direct preconditioner, Krylov, SOR, SSOR, SORU, SOR line, SOR gauge, SOR vector, Jacobi, incomplete and hierarchical LU, SAI, SCGS, Vanka, AMS Algebraic, Geometric, and p-multigrid. Block ILU preconditioning. Support for hypre's AMS and ADS preconditioners for H(curl) and H(div). Basic ones (ILU, ILUT)
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See, in particular, the successive over-relaxation (SOR) and symmetric successive over-relaxation (SSOR) methods. [2] When David Young first began his research on iterative methods in the late 1940s, there was some skepticism with the idea of using iterative methods on the new computing machines to solve industrial-size problems. Ever since ...
While the method converges under general conditions, it typically makes slower progress than competing methods. Nonetheless, the study of relaxation methods remains a core part of linear algebra, because the transformations of relaxation theory provide excellent preconditioners for new methods. Indeed, the choice of preconditioner is often more ...
The conjugate gradient method can be derived from several different perspectives, including specialization of the conjugate direction method for optimization, and variation of the Arnoldi/Lanczos iteration for eigenvalue problems. Despite differences in their approaches, these derivations share a common topic—proving the orthogonality of the ...
One such method is the famous Babylonian method, which is given by x k+1 = (x k + 2/x k)/2. Another method, called "method X", is given by x k+1 = (x k 2 − 2) 2 + x k. [note 1] A few iterations of each scheme are calculated in table form below, with initial guesses x 0 = 1.4 and x 0 = 1.42.
In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations.