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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 .
In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method used to solve a system of linear equations. It is named after the German mathematicians Carl Friedrich Gauss and Philipp Ludwig von Seidel .
Relaxation methods are used to solve the linear equations resulting from a discretization of the differential equation, for example by finite differences. [ 2 ] [ 3 ] [ 4 ] Iterative relaxation of solutions is commonly dubbed smoothing because with certain equations, such as Laplace's equation , it resembles repeated application of a local ...
If an equation can be put into the form f(x) = x, and a solution x is an attractive fixed point of the function f, then one may begin with a point x 1 in the basin of attraction of x, and let x n+1 = f(x n) for n ≥ 1, and the sequence {x n} n ≥ 1 will converge to the solution x.
In applied mathematics, symmetric successive over-relaxation (SSOR), [1] is a preconditioner. If the original matrix can be split into diagonal, lower and upper triangular as = + + then the SSOR preconditioner matrix is defined as = (+) (+)
In some cases, Newton's method can be stabilized by using successive over-relaxation, or the speed of convergence can be increased by using the same method. In a robust implementation of Newton's method, it is common to place limits on the number of iterations, bound the solution to an interval known to contain the root, and combine the method ...
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011.
Stone's method: also known as the strongly implicit procedure or SIP, is an algorithm for solving a sparse linear system of equations; Successive over-relaxation (SOR): method used to speed up convergence of the Gauss–Seidel method; Tridiagonal matrix algorithm (Thomas algorithm): solves systems of tridiagonal equations; Sparse matrix algorithms