Relaxation method
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
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where and
are square diagonal matrices.
The iteration scheme of the relaxation method is as follows:
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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:
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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
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where . For
the spectral radius of
is equal to
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Cases are examined where some are complex. Block relation methods have been developed.
References
[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) |
Relaxation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Relaxation_method&oldid=33055