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

Acta Acad. Paed. Agriensis, Sectio Mathematicae 28 (2001) 27-34 LINEAR RECURRENCES AND ROOTFINDING METHODS Ferenc Mátyás (Eger, Hungary) Abstract. Let A,B,G

0 and G, Le fixed complex numbers, where v4Z?(jG

01 + |Gi Denote by a and ß the roots of the equation X'

!-AX+B=0 and suppose that |a|>|/3|. The sequence { W

{k

)

d } ^ is defined by W

(k

)

d = (a

ka

n

k+

d -b

kß

n

k+

d)l{a-ß), where k> 1 and d>0 are fixed integers, a—Gi-ßG and b=G i -ctG

0. In this paper, using new identities of the sequence { } °° ' a

n °tb

e

i' proof is presented for the Newton-Raphson and Halley transformations (accelerations) of the sequence {j /W^o

} • ft is also shown that the (transformed) sequences obtained by the secant, Newton-Raphson, Halley and Aitken transformations of the sequence { W^/W^ltend to a

d in order of o (wjf l/w^-a*) . AMS Classification Number: 11B39, 65B05. Keywords: linear recursive sequences, rootfinding methods, accelerations of convergence. 1. Introduction Let the n

t

h (n > 2) term of the sequence {G

n

}be defined by the recursion G

N = AG

N-\ — BG

N- 2, where A,B,Go and G\ are fixed complex numbers and AB(\Gq\ + |G'i|) / 0. If it is needed then the notation G

N(A, B,Gq,G\) is also used. For example, the n

t

h term of the Fibonacci sequence is F

n = G

n(l, — 1,0,1). The abbreviations U

N = G

N(A, B, 0,1) and V

N = (A, B, 2, A) will also be very useful for us. Let a and ß be the roots of the equation A

2 — A\ + B = 0 (a + ß = A

, aß — B) and suppose that |a| > \ß\. By the well known Binet formula we get that the explicit form of the term G

n{A

, B, Go, Gi) is aa

n - !)3

n (1) G

N(A,B,G

0,G

1) = — -f- (n> 0), Research supported by the Hungarian OTKA Foundation No. T 032898.