Successive over-relaxation

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.
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.
Formulation.
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 previous value for x on the right hand side. Analytically, this may be written as:
where formula_7 is the "k"th approximation or iteration of formula_8 and formula_9 is the next or "k" + 1 iteration of formula_8.
However, by taking advantage of the triangular form of ("D"+"ωL"), the elements of x("k"+1) can be computed sequentially using forward substitution:
Convergence.
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 formula_12 is symmetric and positive-definite then formula_13 for formula_14. 
Thus, convergence of the iteration process follows, but we are generally interested in faster convergence rather than just convergence.
Convergence Rate.
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
Algorithm.
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
 for from 1 until do
 if ≠ then
 end if
 end (-loop)
 end (-loop)
 check if convergence is reached
 end (repeat)
Example.
We are presented the linear system
To solve the equations, we choose a relaxation factor formula_24 and an initial guess vector formula_25. 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.
(defparameter +MAXIMUM-NUMBER-OF-ITERATIONS+ 100
 "The number of iterations beyond which the algorithm should cease its
 operation, regardless of its current solution. A higher number of
 iterations might provide a more accurate result, but imposes higher
 performance requirements.")
 "For each component of the COMPUTED-SOLUTION vector, retrieves its
 error with respect to the expected EXACT-SOLUTION vector, returning a
 vector of error values.
 While both input vectors should be equal in size, this condition is
 not checked and the shortest of the twain determines the output
 vector's number of elements.
 The established formula is the following:
 Let resultVectorSize = min(computedSolution.length, exactSolution.length)
 Let resultVector = new vector of resultVectorSize
 For i from 0 to (resultVectorSize - 1)
 resultVector[i] = exactSolution[i] - computedSolution[i]
 Return resultVector"
 (declare (type (vector number *) computed-solution))
 (declare (type (vector number *) exact-solution))
 (map '(vector number *) #'- exact-solution computed-solution))
 "Checks whether the convergence is reached with respect to the
 ERRORS vector which registers the discrepancy betwixt the computed
 and the exact solution vector.
 The convergence is fulfilled if and only if each absolute error
 component is less than or equal to the ERROR-TOLERANCE, that is:
 For all e in ERRORS, it holds: abs(e) <= errorTolerance."
 (declare (type (vector number *) errors))
 (declare (type number error-tolerance))
 (flet ((error-is-acceptable (error)
 (declare (type number error))
 (<= (abs error) error-tolerance)))
 (every #'error-is-acceptable errors)))
 "Creates and returns a vector of the SIZE with all elements set to 0."
 (declare (type (integer 0 *) size))
 (make-array size :initial-element 0.0 :element-type 'number))
(defun successive-over-relaxation (A b omega
 &key (phi (make-zero-vector (length b)))
 (convergence-check
 #'(lambda (iteration phi)
 (declare (ignore phi))
 (>= iteration +MAXIMUM-NUMBER-OF-ITERATIONS+))))
 "Implements the successive over-relaxation (SOR) method, applied upon
 the linear equations defined by the matrix A and the right-hand side
 vector B, employing the relaxation factor OMEGA, returning the
 calculated solution vector.
 The first algorithm step, the choice of an initial guess PHI, is
 represented by the optional keyword parameter PHI, which defaults
 to a zero-vector of the same structure as B. If supplied, this
 vector will be destructively modified. In any case, the PHI vector
 constitutes the function's result value.
 The terminating condition is implemented by the CONVERGENCE-CHECK,
 an optional predicate
 lambda(iteration phi) => generalized-boolean
 which returns T, signifying the immediate termination, upon achieving
 convergence, or NIL, signaling continuant operation, otherwise. In
 its default configuration, the CONVERGENCE-CHECK simply abides the
 iteration's ascension to the ``+MAXIMUM-NUMBER-OF-ITERATIONS+",
 ignoring the achieved accuracy of the vector PHI."
 (declare (type (array number (* *)) A))
 (declare (type (vector number *) b))
 (declare (type number omega))
 (declare (type (vector number *) phi))
 (declare (type (function ((integer 1 *)
 (vector number *))
 convergence-check))
 (let ((n (array-dimension A 0)))
 (declare (type (integer 0 *) n))
 (loop for iteration from 1 by 1 do
 (loop for i from 0 below n by 1 do
 (let ((rho 0))
 (declare (type number rho))
 (loop for j from 0 below n by 1 do
 (when (/= j i)
 (let ((a[ij] (aref A i j))
 (phi[j] (aref phi j)))
 (incf rho (* a[ij] phi[j])))))
 (setf (aref phi i)
 (+ (* (- 1 omega)
 (aref phi i))
 (* (/ omega (aref A i i))
 (- (aref b i) rho))))))
 (format T "~&~d. solution = ~a" iteration phi)
 ;; Check if convergence is reached.
 (when (funcall convergence-check iteration phi)
 (return))))
 (the (vector number *) phi))
 :initial-contents
 '(( 4 -1 -6 0 )
 ( -5 -4 10 8 )
 ( 0 9 4 -2 )
 ( 1 0 -7 5 ))))
 (b (vector 2 21 -12 -6))
 (omega 0.5)
 (exact-solution (vector 3 -2 2 1)))
 (successive-over-relaxation
 A b omega
 :convergence-check
 #'(lambda (iteration phi)
 (declare (type (integer 0 *) iteration))
 (declare (type (vector number *) phi))
 (let ((errors (get-errors phi exact-solution)))
 (declare (type (vector number *) errors))
 (format T "~&~d. errors = ~a" iteration errors)
 (or (is-convergent errors :error-tolerance 0.0)
 (>= iteration +MAXIMUM-NUMBER-OF-ITERATIONS+))))))
A simple Python implementation of the pseudo-code provided above.
import numpy as np
def sor_solver(A, b, omega, initial_guess, convergence_criteria):
 This is an implementation of the pseudo-code provided in the Wikipedia article.
 Arguments:
 A: nxn numpy matrix.
 b: n dimensional numpy vector.
 omega: relaxation factor.
 initial_guess: An initial solution guess for the solver to start with.
 convergence_criteria: The maximum discrepancy acceptable to regard the current solution as fitting.
 Returns:
 phi: solution vector of dimension n.
 step = 0
 phi = initial_guess[:]
 residual = np.linalg.norm(np.matmul(A, phi) - b) # Initial residual
 while residual > convergence_criteria:
 for i in range(A.shape[0]):
 sigma = 0
 for j in range(A.shape[1]):
 if j != i:
 sigma += A[i, j] * phi[j]
 phi[i] = (1 - omega) * phi[i] + (omega / A[i, i]) * (b[i] - sigma)
 residual = np.linalg.norm(np.matmul(A, phi) - b)
 step += 1
 print("Step {} Residual: {:10.6g}".format(step, residual))
 return phi
residual_convergence = 1e-8
omega = 0.5 # Relaxation factor
A = np.array(4, -1, -6, 0],
 [-5, -4, 10, 8],
 [0, 9, 4, -2],
 [1, 0, -7, 5)
b = np.array([2, 21, -12, -6])
initial_guess = np.zeros(4)
phi = sor_solver(A, b, omega, initial_guess, residual_convergence)
print(phi)
Symmetric successive over-relaxation.
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..
Other applications of the method.
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 formula_32. Values of formula_33 are used to speed up convergence of a slow-converging process, while values of formula_34 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 formula_35 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.