In numerical linear algebra, the method of successive over-relaxation (SOR) is a variant of the GaussâÂÂSeidel method for solving a linear system of equations, resulting in faster convergence. A similar method can be used for any slowly converging iterative process.
It was devised simultaneously by David M. Young Jr. and by Stanley P. Frankel in 1950 for the purpose of automatically solving linear systems on digital computers. Over-relaxation methods had been used before the work of Young and Frankel. An example is the method of Lewis Fry Richardson, and the methods developed by R. V. Southwell. However, these methods were designed for computation by human calculators, requiring some expertise to ensure convergence to the solution which made them inapplicable for programming on digital computers. These aspects are discussed in the thesis of David M. Young Jr.
Given a square system of n linear equations with unknown x:
where:
Then A can be decomposed into a diagonal component D, and strictly lower and upper triangular components L and U:
where
The system of linear equations may be rewritten as:
for a constant ω > 1, called the relaxation factor.
The method of successive over-relaxation is an iterative technique that solves the left hand side of this expression for x, using the previous value for x on the right hand side. Analytically, this may be written as:
where is the kth approximation or iteration of and is the next or k + 1 iteration of . However, by taking advantage of the triangular form of (D+ωL), the elements of x<sup>(k+1)</sup> can be computed sequentially using forward substitution:
This can again be written analytically in matrix-vector form without the need of inverting the matrix :
The choice of relaxation factor ω is not necessarily easy, and depends upon the properties of the coefficient matrix. In 1947, Ostrowski proved that if is symmetric and positive-definite then for . Thus, convergence of the iteration process follows, but we are generally interested in faster convergence rather than just convergence.
The convergence rate for the SOR method can be analytically derived. One needs to assume the following
Then the convergence rate can be expressed as
where the optimal relaxation parameter is given by
In particular, for (Gauss-Seidel) it holds that . For the optimal we get , which shows SOR is roughly four times more efficient than GaussâÂÂSeidel.
The last assumption is satisfied for tridiagonal matrices since for diagonal with entries and .
Since elements can be overwritten as they are computed in this algorithm, only one storage vector is needed, and vector indexing is omitted. The algorithm goes as follows:
Inputs: , , Output:
Choose an initial guess to the solution repeat until convergence for from 1 until do set to 0 for from 1 until do if ≠ then set to end if end (-loop) set to end (-loop) check if convergence is reached end (repeat)
We are presented the linear system
To solve the equations, we choose a relaxation factor and an initial guess vector . According to the successive over-relaxation algorithm, the following table is obtained, representing an exemplary iteration with approximations, which ideally, but not necessarily, finds the exact solution, , in 38 steps.
A simple implementation of the algorithm in Common Lisp is offered below.
A simple Python implementation of the pseudo-code provided above.
The version for symmetric matrices A, in which
is referred to as Symmetric Successive Over-Relaxation, or (SSOR), in which
and the iterative method is
The SOR and SSOR methods are credited to David M. Young Jr.
A similar technique can be used for any iterative method. If the original iteration had the form
then the modified version would use
However, the formulation presented above, used for solving systems of linear equations, is not a special case of this formulation if is considered to be the complete vector. If this formulation is used instead, the equation for calculating the next vector will look like
where . Values of are used to speed up convergence of a slow-converging process, while values of are often used to help establish convergence of a diverging iterative process or speed up the convergence of an overshooting process.
There are various methods that adaptively set the relaxation parameter based on the observed behavior of the converging process. Usually they help to reach a super-linear convergence for some problems but fail for the others.