MATHEMATICS

A. B. NAISHUL

IMPROVING THE CONVERGENCE OF METHODS OF SUCCESSIVE APPROXIMATIONS FOR LINEAR EQUATIONS

(Presented by Academician A. Yu. Ishlinskii, December 30, 1963)

The number of approximations required in solving a system of linear equations

\[ \bar{x}=B\bar{x}+\bar{b} \tag{1} \]

by the method of successive approximations

\[ \bar{x}_{n+1}=B\bar{x}_{n}+\bar{b} \tag{2} \]

may be large if the norm of the matrix \(B\) is close to 1, the initial approximation \(\bar{x}_0\) is far from the solution, and the result must be obtained with high accuracy. Below a method is proposed for improving convergence, in which \(s\) approximations are equivalent to \(n=2^{s-1}\) ordinary approximations.

Solving equation (1) by successive approximations, we obtain:

\[ \begin{aligned} \bar{x}_1&=B\bar{x}_0+\bar{b},\\ \bar{x}_2&=B^2\bar{x}_0+(E+B)\bar{b},\\ &\ldots\\ \bar{x}_n&=B^n\bar{x}_0+(E+B+B^2+\ldots+B^{n-1})\bar{b}. \end{aligned} \tag{3} \]

Transform the expression

\[ \begin{aligned} U_n&=E+B+B^2+\ldots+B^{n-1},\\ U_n&=(E+B)+B^2(E+B)+B^4(E+B)+\ldots+B^{n-2}(E+B)=\\ &=(E+B^2+B^4+\ldots+B^{n-2})(E+B)=\\ &=((E+B^2)+B^4(E+B^2)+\ldots+B^{n-4}(E+B^2))(E+B)=\\ &=(E+B^4+B^8+\ldots+B^{n-4})(E+B^2)(E+B). \end{aligned} \tag{4} \]

Continuing the process, we obtain

\[ U_n=(E+B^{2^{s-2}})(E+B^{2^{s-3}})\ldots(E+B^2)(E+B). \tag{5} \]

Of course, in this case

\[ n=2^{s-1}. \tag{6} \]

Using relations (3), we obtain

\[ \bar{x}_{2^{s-1}} = B^{2^{(s-1)}}\bar{x}_0 + (E+B^{2^{s-2}})(E+B^{2^{s-3}})\ldots(E+B^4)(E+B^2)(E+B)\bar{b} = \]

\[ = B^{2^{s-1}}\bar{x}_0+U_{2^{s-1}}\bar{b}. \tag{7} \]

The sequence of matrices \(U_{2^s-1}\) is easily computed, since

\[ U_{2^s-1}=\left(E+B^{2^{s-2}}\right)U_{2^{s-2}},\qquad U_1=E; \tag{8} \]

\[ B^{2^s-1}=\left(B^{2^{s-2}}\right)^2. \tag{9} \]

The same method gives a way of determining the inverse matrix

\[ G=(E-B)^{-1}, \]

\[ G_{2^s-1}=B^{2^s-1}G_0+U_{2^s-1}. \]

It should be noted that the method can also be applied to functional equations, provided it is possible to construct an effective process for computing the operators \(B^{2^s-1}\) and \(U_{2^s-1}\).

Received
14 X 1963