F 9 The solution of transcendental equations

O u r c h a p t e r c o n c e r n s t h e a p p r o x i m a t i o n of a zero off(x) o r , equ iva len t ly , t h e a p p r o x i m a t i o n pf a r o o t o f t h e e q u a t i o n f(x) = 0 b y i te ra t ive m e t h o d s . W e shal l t a k e / as t r a n s c e n d e n t a l a l t h o u g h t h e m e t h o d s w e discuss c a n a lso b e u s e d if / is a p o l y n o m i a l . I f w e use these m e t h o d s in th is l a t t e r case , we a r e n o t t a k i n g full a d v a n t a g e of t h e fact t h a t we a r e dea l ing w i th t h e special case o f a p o l y n o m i a l . ! W e res t r ic t ourse lves t o f(x) w h i c h a r e rea l s ingle-valued func t ions of a rea l va r i ab le , possess ing a ce r t a in n u m b e r of c o n t i n u o u s der iva t ives i n t h e n e i g h b o r h o o d of a rea l ze ro a. I n Sec t ion 2g, / is a vec to r func t ion of a vec to r va r i ab le . T h e n u m b e r of c o n t i n u o u s der iva t ives a s s u m e d var ies u p w a r d s f r o m ze ro . A z e r o a is o f mul t ip l ic i ty m if