By I. S. Berezin and N. P. Zhidkov (Auth.)

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**Example text**

If h > 0, the process is non-stationary. I t is always convergent by virtue of inequalities (26) and (34). This process, first proposed by Richardson, is naturally known as the Richardson method. (3) The inversion of matrices by the method of successive approximation We briefly consider the inversion of matrices by successive approximation. Suppose that we have in some way found the approximate value of B0 for the matrix A'1. I t is also presupposed that a certain norm of the matrix (35) O0 = I-AB0 is less than unity, ||C 0 || =s= q < 1.

Xk. We shall say t h a t it is a first-order method if Fk only depends on xk a n d is independent of x0, xv . . , xk__v Weshall call t h e method stationary if Fk is independent of Jc. 48 COMPUTING METHODS The simplest functions Fk are linear functions. The most general linear first-order method of successive approximation is of t h e following t y p e 5 *Ä+I = Ä + cfe, (2) where Bk is a square matrix and ck a vector. I t is naturally required of methods of successive approximation t h a t t h e substitution of the exact solution of t h e set A~xl· for xk on t h e right-hand sides of (1) and (2) should reproduce A~xl· on t h e left.

Note also t h e following proper- THE S O L U T I O N OF SETS OF L I N E A R A L G E B R A I C E Q U A T I O N S 27 ties of r (2) : (22) (23) We have t o find t h e normal vector r (2) in the hyperplane (vector p ) t o the ellipsoid f(x) = /(# (2) ), provided x = # (2) , in t h e form i2) (24) I t is required t h a t p(2) be orthogonal t o Ar^0) a n d Ap^K But, (25) {2) The condition on which p and Ap {1) are orthogonal is (26) We again have t o find the value of a for which f(x{2) + ocp(2)) is at a minimum.