TY - JOUR

T1 - Generalized fibonacci and lucas sequences and rootfinding methods

AU - Muskat, Joseph B.

PY - 1993/7

Y1 - 1993/7

N2 - Consider the sequences (Un) and (vn) generated by un+x = pun - qun-X and vn+x = pvn - qvn-X, n > 1, where Uq = 0, ux = 1, v0 = 2, vx = p, with p and q real and nonzero. The Fibonacci sequence and the Lucas sequence are special cases of (um) and (vm), respectively. Define rn = un+j/un, Rn = vn+d/vn, where d is a positive integer. McCabe and Phillips showed that for d = 1, applying one step of Aitken acceleration to any appropriate triple of elements of (rn) yields another element of (un). They also proved for d = 1 that if a step of the Newton-Raphson method or the secant method is applied to elements of (r„) in solving the characteristic equation x2 - px + q = 0, then the result is an element of (rn). The above results are obtained for d > 1. It is shown that if any of the above methods is applied to elements of (Rn), then the result is an element of (vn). The application of certain higher-order iterative procedures, such as Halley’s method, to elements of (rn) and (Rn) is also investigated.

AB - Consider the sequences (Un) and (vn) generated by un+x = pun - qun-X and vn+x = pvn - qvn-X, n > 1, where Uq = 0, ux = 1, v0 = 2, vx = p, with p and q real and nonzero. The Fibonacci sequence and the Lucas sequence are special cases of (um) and (vm), respectively. Define rn = un+j/un, Rn = vn+d/vn, where d is a positive integer. McCabe and Phillips showed that for d = 1, applying one step of Aitken acceleration to any appropriate triple of elements of (rn) yields another element of (un). They also proved for d = 1 that if a step of the Newton-Raphson method or the secant method is applied to elements of (r„) in solving the characteristic equation x2 - px + q = 0, then the result is an element of (rn). The above results are obtained for d > 1. It is shown that if any of the above methods is applied to elements of (Rn), then the result is an element of (vn). The application of certain higher-order iterative procedures, such as Halley’s method, to elements of (rn) and (Rn) is also investigated.

UR - http://www.scopus.com/inward/record.url?scp=84966259110&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-1993-1192974-3

DO - 10.1090/S0025-5718-1993-1192974-3

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AN - SCOPUS:84966259110

SN - 0025-5718

VL - 61

SP - 365

EP - 372

JO - Mathematics of Computation

JF - Mathematics of Computation

IS - 203

ER -