Successive over-relaxation

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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.

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, and they required 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.[1]

Formulation[edit]

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 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(k+1) can be computed sequentially using forward substitution:

Convergence[edit]

Spectral radius of the iteration matrix for the SOR method . The plot shows the dependence on the spectral radius of the Jacobi iteration 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.

Convergence Rate[edit]

The convergence rate for the SOR method can be analytically derived. One needs to assume the following

  • the relaxation parameter is appropriate:
  • Jacobi's iteration matrix has only real eigenvalues
  • Jacobi's method is convergent:
  • a unique solution exists: .

Then the convergence rate can be expressed as[2]

where the optimal relaxation parameter is given by

Algorithm[edit]

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: A, b, ω
Output: 
Choose an initial guess  to the solution
repeat until convergence
  for i from 1 until n do
    
    for j from 1 until n do
      if ji then
        
      end if
    end (j-loop)
    
  end (i-loop)
  check if convergence is reached
end (repeat)
Note
can also be written , thus saving one multiplication in each iteration of the outer for-loop.


Example[edit]

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, following table is obtained, representing an exemplary iteration with approximations, which ideally, but not necessarily, finds the exact solution, (3, −2, 2, 1), in 38 steps.

Iteration
1 0.25 -2.78125 1.6289062 0.5152344
2 1.2490234 -2.2448974 1.9687712 0.9108547
3 2.070478 -1.6696789 1.5904881 0.76172125
... ... ... ... ...
37 2.9999998 -2.0 2.0 1.0
38 3.0 -2.0 2.0 1.0

A simple implementation of the algorithm in Common Lisp is offered below. Beware its proclivity towards floating-point overflows in the general case.

(defparameter +MAXIMUM-NUMBER-OF-ITERATIONS+ 100
  "The number of iterations beyond which the algorithm should
   cease its operation, regardless of its current solution.")

(defun get-errors (computed-solution exact-solution)
  "For each component of the COMPUTED-SOLUTION vector, retrieves its
   error with respect to the expected EXACT-SOLUTION, returning a
   vector of error values."
  (map 'vector #'- exact-solution computed-solution))

(defun is-convergent (errors &key (error-tolerance 0.001))
  "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 iff each absolute error component is
   less than or equal to the ERROR-TOLERANCE."
  (flet ((error-is-acceptable (error)
          (<= (abs error) error-tolerance)))
    (every #'error-is-acceptable errors)))

(defun make-zero-vector (size)
  "Creates and returns a vector of the SIZE with all elements set to 0."
  (make-array size :initial-element 0.0 :element-type 'number))

(defun sor (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."
  (let ((n (array-dimension A 0)))
    (loop for iteration from 1 by 1 do
      (loop for i from 0 below n by 1 do
        (let ((rho 0))
          (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))))
  phi)

;; Summon the function with the exemplary parameters.
(let ((A              (make-array (list 4 4)
                        :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)))
  (sor
    A b omega
    :convergence-check
    #'(lambda (iteration phi)
        (let ((errors (get-errors phi exact-solution)))
          (format T "~&~d. errors   = ~a" iteration errors)
          (or (is-convergent errors :error-tolerance 0.0)
              (>= iteration +MAXIMUM-NUMBER-OF-ITERATIONS+))))))

Symmetric successive over-relaxation[edit]

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[edit]

A similar technique can be used for any iterative method. If the original iteration had the form

then the modified version would use

Note however that the formulation presented above, used for solving systems of linear equations, is not a special case of this formulation if x 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.

See also[edit]

Notes[edit]

  1. ^ Young, David M. (May 1, 1950), Iterative methods for solving partial difference equations of elliptical type (PDF), PhD thesis, Harvard University, retrieved 2009-06-15
  2. ^ Hackbusch, Wolfgang (2016). "4.6.2". Iterative Solution of Large Sparse Systems of Equations | SpringerLink. Applied Mathematical Sciences. 95. doi:10.1007/978-3-319-28483-5. ISBN 978-3-319-28481-1.

References[edit]

External links[edit]