Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2001. Sectio Mathematicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 28)
MÁTYÁS, F., Linea r recurrences and rootfinding methods
Linear recurrences and rootfinding methods 31 (a) N «>) = «SS, (b, H (<>) = *£»>. The following theorem implies that the transformations S, N, H and A produce such sequences from the sequence \ Rn dj wllich ten d very quickly to a . I 'J n=0 Theorem 2. Let I > k > 1 be fixed integers. Then Corollary. Theorem 1 and (11) show that the transformations S,N,A and H transform R^ d into R„ kJ and into R^ kJ , respectively, thus Theorem 2 implies that all of the mentioned transformations give accelerations of the convergence. 3. Proofs of Lemmas and Theorems Proof of Lemma 1. Because of the similarity of the proofs we present only the proof of part (a). Using the explicit form (3) of we write w{k) w(k) w(k) w(k) R d _ (a ka nk+ d - b kß» k+ d)(a ka m k+ d - b k^ k+ d) (a - ß) (a ka n k - b k ß n k)(a k a m k - b kß m k)a dß d a d - ß d a ß)' 2 k ^2k+d _ b2k ß^2k+d ^ = UdW^l a - 0 -±r-> d Proof of Lemma 2. Here we also give only the proof of part (a). By (3) wW wm w(k) w i2k) R d _ (a k<x nk+ d - b kß" k+ d)(a* k a*" k+ d - b* kß*» k+ d) n,d n.d ~ v vn,0 v vn,0 / J — : (a-ß) (< a ka n k - b kß n k)(a 2 ka 2n k - b 2 kß 2n k)a dß d _ _ a d - ß d {a-ß) 2 a-ß
