Computing Methods in Applied Sciences and Engineering: by Alan George (auth.), Prof. R. Glowinski, Prof. J. L. Lions

By Alan George (auth.), Prof. R. Glowinski, Prof. J. L. Lions (eds.)

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Additional resources for Computing Methods in Applied Sciences and Engineering: Second International Symposium December 15–19, 1975

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E. lIu** 11= u*m~n lIu*l\. e W(A,f) (5) It is known that the generalized normal solution of system (1) always exists, is unique and is caleulated by the formula u"* = A+:f, (6) where A + is the generalized inverse matrix for the matrix (see t for example, [6] ). Defini tion 2. The n x n matrix S is ealled R A -matrix, if ~ ~T statement 1. m ~ = im f, Introduce the matrix where H (10) T = I - HA , is a certain n xn matrix satisfying the requirement ke<. mA :- {9) • ( 11 ) where 9 is the zero element of E.

Eoineides with the projeetor P defined Definition 3. The iterative method (16)-(18) is referred to as symmetrized provided the following eonditions are satisfied: 1. ke'Z D 1. ,,:rn A 2. D~/2 A H [J)+) ~/2 is symmetrie. The iterative methods whieh are symmetrized represent an important subelass of iterative methods of (16) - (18) type. Their speeifie feature is a eonvenient realization with the aid of formulae of the method of eonjugate gradients or minimal iterations method, simple neeessary and suffieient eonditions of eonvergenee, eonvenient estimates of the rate of eonvergenee.

H = k,e'Z. A'I' • Then the iterative sequenee {uk converges to some generalized solution u* of system (1). m H = tm A'f' then the sequence { U k } converges to the generalized normal solution u. * It of this system. 1 Remark 1. The conditions of theorem 4 will be satisfied. for example, in two cases when either ke'l. B or 'I' H "" B~A BI!. , keoz. m A'I' = {8 \ 47 im A and Theorem 5. , ••• converges to the generalized solution oi system (1). {u k Proof. Let us represent the vectors u~ E. and } im A f in the form and u ~ and 72.

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