Relaxation method

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weakening method

A method for the iterative solution of a system of linear algebraic equations , the elementary step of which consists of varying only one component of the vector of unknowns, the number of variable components being chosen in a specific cyclic order. The relaxation method is most often used for solving systems with a positive-definite matrix .

If one component of the vector of unknowns is varied such that for the new approximation the quadratic form is minimized, then the relaxation method is called a complete relaxation method. If, however, after one elementary step the value of the quadratic form is only reduced and not minimized, the relaxation method is called an incomplete relaxation method.

The best investigated method is that of successive upper relaxation, where the matrix possesses the so-called property and is ordered accordingly. A matrix is called a matrix possessing property if there is a permutation matrix such that the matrix has the form

where and are square diagonal matrices.

The iteration scheme of the relaxation method is as follows:

where is the relaxation parameter, is the diagonal, is the lower-triangular and is the upper-triangular matrix in the decomposition . If , then the method is called an upper relaxation method (over-relaxation), and if , a lower relaxation method. The parameter is chosen from the condition of minimization of the spectral radius of the matrix of transfer from iteration to iteration:

If is a symmetric matrix with positive diagonal elements and are the roots of the determinant equation , then the optimum value of the parameter is given by the formula

where . For the spectral radius of is equal to

Cases are examined where some are complex. Block relation methods have been developed.


[1] D.M. Young, "Iterative methods for solving partial differential equations of elliptic type" Trans. Amer. Math. Soc. , 76 : 1 (1954) pp. 92–111
[2] D.M. Young, "Iterative solution of large linear systems" , Acad. Press (1971)
[3] W. Wasow, J. Forsyth, "Finite-difference methods for partial differential equations" , Wiley (1960)
[4] D.K. Faddeev, V.N. Faddeeva, "Computational methods of linear algebra" , Freeman (1963) (Translated from Russian)
[5] L.A. Hageman, D.M. Young, "Applied iterative methods" , Acad. Press (1981)
How to Cite This Entry:
Relaxation method. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.S. Nikolaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article