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 29 i. e. the sequence tends to the root a

d of the polynomial (5) /(A) = A

2 - (a

d + ß

d)\ + cx

dß

d = A

2 - V

dX + B

d

. Recently, many authors have studied the connection between recurrences and iterative transformations. The main idea is to consider such sequence transformations T of the convergent sequence {A

n}^_

0 into the sequence {T

n , where {T

n}^l

0 converges more quickly to the same limit X. Thus, one can investigate the properties of these transformations or the accelerations of the convergence. We say that {T

nconverges more quickly to X than {X

nif T

n — X = o(X

n — X), i. e. if lim ((T

N - X) / {X

n - X)) = 0. n—+oo The most known four sequence transformations to accelerate the convergence of a sequence are the secant S(X

n,X

m), Newton-Raphson N(X

n), Halley H(X

n) and Aitken transformation A(X

n

, X

m, Xt), namely if {A'

n - { R^h \ and L 'J n=0 X = a

d (i. e. the root of /(A) = Ü in (5)), then (6) S(X

n,X

m)= , A

n + A

m - Vd (7) N(X

n) = A

2 - B

c 2X

r V

d (8) H(X

n) A3 - 3

B

dX

n + V

dB> 3A'

2 - ZV

dX, + K

2 Bd (9) A(X

n,X

m,X

t) = y

X

n'\

l

Y , A

n — ZA

m -t Af where we assume that division by zero does not occur. (The formulae (6)-(9) can be obtained from (5) using the known forms of the transformations S, N, H and A, or they can be found in [4] p. 366 and p. 369.) Some results from the recent past: G. M. Phillips [5] proved that if r

n = ^J^then A(r

n_

t,r

n,r

n+

t) = r

2

n. J. H. McCabe and G. M. Phillips [3] generalized this for r

n = , and they also proved that 5' (^r

n,r

mj = r

n+

m and N(r

n) = r

2

n. M. J. Jamieson [1] investigated the case r

n = 1 for d > 1. J. B. Muskat [4], using the notations r

n = l and R

n — Vy

+

d (d > 1), proved that (a) S(r

n, r

m) — r

n_f.

m, ,b

(R

n, Rm) — ?

>

n-j.

m, (10) (6) N(r

n) = r

2

n, N(R

n) = r

2

n,