Inversionsfreie Verfahren zur Lösung nichtlinearer Operatorgleichungen

Abstract

1) Es bezeichnet L( Y , B ) den Raum aller linearen Operatoren, die Y in B sbbilden. 8* ~ ~~ ~ ____ ~ ~~~~

An error analysis for the secant method

“…In the above mentioned papers one has used the notion of divided difference of an operator. The possibility of using the more general notion of consistent approximation of the derivative was realized later (see [1] and [16]). …”

“…We shall consider the notion of consistent approximation of the derivative in the sense given in [1], which is more particular than the original sense given in [4]. …”

“…(ii) the sequence (x,),e 1 generated by (1) converges to a root x* of the equation f(x)=0. [] Having an iterative procedure of type (p.1) for the class ~ it is important to find a function ~: ~+-*IR+ and a function fl: RP+--*IR+ such that the following inequalities be satisfied…”

“…The above scheme will actually yield a sequence (x,),e o, if u o =(x_p+ I ..... Xo) is an admissible system of starting points in the sense given by the following definition: Any UoE9 ~ = ~ 9, will be called an admissible system of starting points for n>0 the recurrent scheme (1). [] If uo=(x p+ a ..... Xo) is an admissible system of starting points for the recurrent scheme (1) we shall also say that the iterative algorithm (1) is well defined.…”

On Ulm’s method using divided differences of order one

“…A convergence analysis for Ulm's and Ulm's-type methods has been given in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. As it turns out although different approaches are used in the above references the crucial sufficient convergence hypothesis (see hypothesis (2.6)) remains the same.…”