Acceleration of extended fibonacci sequences

Applied Numerical Mathematics 2 (1986) l-8 North-Holland

ACCELERATION OF EXTENDED FIBONACCI SEQUENCES * C. BREZINSKI

and A. LEMBARKI

Luboratoire d’Analyse Numhique France

The repeated applicatton

al

et d’optimisation,

UER IEEA, Univers:th de Lille I, 59655 Villeneuve d’Ascq,

of Aitken’s A2 process and of Shanks’ transformation

to the convergents of

al

1+,+,+are studied. It is proved that they both produce subsequences of the initial sequence. Their rates of convergence are studied showing the superiority of Aitken’s process. The iterates of various methods for computing the dominant zero of x2 - x - a are related to the previous continued fraction.

1. Introduction

In [4], Phillips considered the Fibonacci sequence C,=l, It

c,+,=1+1/c,,

n=O,l,...

.

is well known that C,, = u,, + i/u,, ,

n 2 0

where the u,, are the Fibonacci numbers given by U t1+ 1 =

u,, + u,,+

I1 >

1

with u0 = u, = 1 and that the sequence (C,,) tends to the ‘golden section’ i( 1 + 6) positive zero of the polynomial x2 - x - 1. The C,, are also the successive convergents of the continued fraction

which is the

Phillips considered the repeated application of Airken’s A” process to the sequence (C,,) in order to accelerate its convergence and he proved that the new sequences obtained are subsequences of the initial one. Thus the question arises to know whether Shanks’ transformation (that is the E-algorithm) possesses a similar property. The same questions can also be raised for the extended Fibonacci sequence formed by the converqents of the continued fraction

The aim of this paper is to answer these questions. One can, of course, think that such questions are irrelevant since the limit of these continued fractions is already known and they don’t need to * Work performed under the Nato Research Grant 027.81. 0168-9274/86/$3.50

0 1986, Elsevier Science Publishers B.V. (North-Holland)

C. Bre:inski,

2

A. Lmttbarki

/ Exrende-

Fihonacci sequences

be accelerated. However, apart from the curious results obtained, the interest of the paper is to show that sequence transformations can transform a logarithmic sequence into a linear one and a linear one into a sequence converging super-quadratically. Moreover, we hope that some of the results on the rate of convergence proved herein could be extended to limit periodic continued fractions. Thus, let us consider the continued fraction

where a is a nonzero complex number. It satisfies C = 1 + a/C

and its convergents are given by C,=l,

C,,+,=l

+-a/C,,.

We have C,, = u,,+ , /u,, where the 140 = u, =

1,

II,,

n=O,l,...

are given by

u t1+ 2 =u ,,+,+au,,,

Let xi and x1 be the zeros of the polynomial u,=

(x,

)I+

1 -

.

17=0.1,...

.

x2 - x - a. It is easy to check that if a # - $,

-x;+‘)/(X,-x2)

and

or, setting r = x,/x, c,, = _y,(l -

r”‘Z)/(l

- I-“+‘).

(2)

If C:= - i. we have u,, = (11 -I- 1)/2”

(3)

C,#= ( 17+ 2)/2( 12+ 1).

(4)

and Ifast+ c where c is a real nonpositive number, then the continued fraction converges to ~1 the zero of greatest modulus of x2 - x - a. If a = - ! + c where c is a real strictly negative number. x, = X2 and X, # x2. Thus, by (2) the continuid fraction does not converge. 19 fact all the C,, exist and are real and x, and x2 are complex. In Section 2 we shall study the application of the iterated A2 process on the extended Fibonacci sequence (C,,). The use of Shanks’ transformation will be carried out in Section 3 and the rates of convergence of the various sequences obtained will be compared. In Section 4 we shall study the iterates of some methods for computing the dominant zero of x2 - x - a. There are two methods for proving the results. The first one consists ill using C,, = u,+,/u,, and various relations satisfied by the u,,. These relations generalize similar relations for the Fibonacci numbers. l’his is the method followed by Phillips. It is also possible to use (1) and (3). The second way for proving the results is to use directly (2) and (4). It will be our method. Due to a linearity property of Aitken’s and Shanks’ transformations all the results stated also hold for continued fractions of the form

al al h+~i_+v+

l

...

where hEc.

C. Brezinski, A. Lmharki

3

/ Extended Fihonacci sequences

2, Aitken’s method Let us apply repeatedly Aitken’s A* process to the sequence ( C, ), that is we set x0

(n) =

l,...

n=o,

cn9

.

we construct the sequence (xp$)

Then, for k=O, l,...,

(n+l),

xp2l =

xk

(n)

Xk

(n) _ (P;+*)

_

xk

zxy+

)

by

*

1) +

Xy)

n=o, l,...

’

.

xk

It must be noticed that our notations slightly differ from that of Phillips. It is just a matter of conveniency for our purpose. Theorem 1. Vk,n >, 0

x:.)l)= c(n+k_t1~~1.

Proof. If a # - $, then for 1 < i < n c,,+ic,_i - c,2 = x;Y”-i+i(l

- r)(l - yl)*(l - !.2,r+3)/D(1 - rn+‘),

c,,i - 2c,, + c,,_i = xlF+i(l with D = (1 - yn-i+i)(l

- r)(l - v’)*(l + Y”+l)/D,

- y”+‘)(l - y”+‘+‘). Thus

cv+iCn_i- C,’

l- r2n+3 =c

Cn,i - 2Cn + C,r_i = x1 1 - yZfr+*

2,r+1-

If a = - $ we have cn+-icn-i

-

c~;’

=

i2(2n + 3)/4( r’1+ l)*[ (n + l)* - i'] ,

Cn+i- 2C,, + Cn_i = 2i2/2( n + l)[ (n -I-l)* - i'] . Thus Cn+jcn__i- c,” 2n+3 Cn,i - 2Cn + C,, = 2(2n + 2)

=c

In both cases, taking i = 1, we get xi”)= #-== L

(CZ,t+,C*n+3

-

G*+Mc2,,+7

2rr+l*

C2,,+3, -

Vn. We have 2CZ,r+5

+

C2t?+3!.

Replacing n by 2n + 5 and setting i = 2 we obtain _.$) = C,,,, ll. It is easy to see that Xpj= c N(k.tr) where N(O,n) = n and N( k,n) = 2N( k - 1, n + 1) + 1. Thus N(k,n)

= 2”n + Mk

with M, = OaFld M,=2M,_,+2”+1

whosesolutionis

M,=(k+1)2k-1.

u

This proof follows the lines of Phillips’ proof. Another one can also be easily obtained by induction but it is longer,

C. Brezinski, A. Lembarki / Extended Fibonacci sequences

4

3. Shanks’ transformation Instead of Aitken’s process let us now try to transform transformation or, equivalently, by Wynn’s c-algorithm. Shanks’ transformation is defined by [6]

the sequence (Cm) by Shanks’

k,n =O, L...

ek(C,,)= Hk+,(Cn)/Hk(A2Cjn), where . . . Yn+l Y, Ytl+1 Ytl+* .‘* : HJyJ =.’ : . . Yn+k-1 Y,t+k ‘*

Yn+li-1

Yn+k . . .

.

Ytl+*k-2

l

The e,( C,,) can be recursively computed by the c-algorithm of Wynn [7] which is as follows: &y=o,

cb”‘=

c,,

=o,

n

l,...;

c:-“+‘, = c In_:l)+[cy+l)-p]-‘, It is related to Shanks’ transformation $‘=

k, n =0,

e,(C,),

k,n=O,l,....

by

l,....

When k = 1, $‘I = e,( S,) = $I. If the c-algorithm is applied to our sequence (C,,) we get the Theorem

2. vk,n > 0

cg)= Ctk+,j,1+k(k+2j.

Proof. Vn >, 0, rr) = C, and rp) = xi”) = C2,,+3 which is true by Theorem 1. The property is assumed to be true up to the index k and Vn > 0. To prove that it is stil 1 true for k + 1 we shall make use of Wynn’s cross rule [l] which states that (n+l) -E:“k)]-‘+[f:;-l)_L~)]-l= [ c2k

[Eg;:)_&q-l

+[E$p2LE:“kq-’

with $‘)= C, and E?; = 00. All the four brackets of this relaiior have the form (C, - CM)-‘, where M = (k + 1)n + k( k + 2). If a Z - $ WE have _ CM)_’

(c,

(1- fy’)(l

=x-1

- r”+l) - y)(rN-M -

r”+*(l

1)

l

The cross rule reduces to 1

_

rM+lrk+l #+I

_

1

M+lr-(k+l)

+1-r

+k+l)_

1

I_ rM+lyn+k+l =

p+k+l

_

1

which is obviously true. If a=-- i, the cross rule becomes - M+k+2+M-k --= k-d k-+1

which is evident.

0

- M+n+k+2+M-n-k n+k+l n+k+l

+1-r

,Gf+iy--rrr+k+ ).-(n+k+l)

_

1) 1

C. Brezinski, A. Lembarki / Extended Fibonacci sequences

5

We shall now compare the rates of convergence of the various sequences obtained by the c-algorithm and the repeated application of Aitken’s process. To speak of convergence we have, of course, to assume that (C,, ) converges, that is a z - $ + c where c is a real nonpositive number. It must be noticed that the computations of xj$ and EYE)both need the same terms Cn9**.9C*+2k of the initial sequence. We have Zfa#

Theorem3.

- $ +c, cc0

(1)

lim (xi”+‘)n-00

(2)

xp2, - xr = o(x:“+2)-x,) and

x,)/(xy)-

lim (xf:,

x1) = Y2”; (n + 00)

- x1)/( x:~+~) - xl)’ =

F2”“/xl(l

-

r);

n-+oc,

(3)

xp;,

-

x1 = o(xp-

x,)’

(k

-+

Do),

,limJxi’;‘, --_,

vq>2,

and

(4)

xk

jimm(x::)l -~~)/(x:)I+‘)-x~)~=x-~(l --_, (n+Lx = o(xyXl) (k + 00)

-x,)/(x:.)1)-xJ=

-Y)-‘;

1

and

)irnm (xy+‘)‘4

QqH.

x,)/(x:.)‘)-

(5)

lim (~(;~+‘)-x~)/(E~)-x,)=y~+~; n-00

(6)

E:“k)+Z - xr = and

lim

(n+2)

O(‘2k

( Eg)+2

_

-x1) x1)/(

x1)’ = 00;

b-4 c$+‘)

_

x,)(k+2)‘(k+“z

y-(k+‘)[

x,(1

tt + 00

(7)

&#)+2

-

x,

=

and

Qq>1,

o(c:“k)-x,)

(It-+

00)

lim (~:nk)+~-x~)/(~~‘-x,)~=

00;

k-+oo

(8)

(n+lLx

‘2k

and

1

= o( E(Znkf)-

Qq> 1,

Xl)

(k-4

lim (~:nk+‘)-x,)/(f:;.‘-X~)~=

00;

k-+oo

(9

(xp)_

lim I1 *

and

00)

xr)-x~=o(E~)-x,)

lim (xi”)-- x,)/
n-r00

and

(11)

x,)/(cyL)_

J-)~“/(~+‘)=[x,(1

_

r)]1-2”/(k+1)

00

Qq> 1.

xp)-x1

(k-urn))) - x1)2” = r-li2” [x,(1 - #-2L =o(Cn+2k-x1)Y

lim (@-x1)/(C~+2k-xr)k+1=~-k(k+1’[xr(1

(k-,

w); -r)]-”

n+w

and

Qq>1,

Zf a = - 4 we have:

~:“k)-x,=o(c~+~~-x~)~

(k+oo).

-

r)]

-l’(k+l);

00

C. Bre:inski, A. Lembarki / Extended Fibonacci sequences

(12) (13) (14) (1% (16) (17) (18) (1% (20)

lim (x:.)l+‘)-x,)/(xr)-x1)=1; I1- x lim (~:.n+)~ - x1)/(xr+2)x1) = f; II- cc lim ( x.F~~- x,)/(x:“’ - x1) = $ ; k-cx:

lim (xp+*)-xl)/(xp)-xX.l)=l; k+X

lim (d/i+*)-x&+$)-x1)=1; I1 -

x

lim (c~)+Z-xJ/(~~~+~)-xJ=(k+l)/(k+2): t1- x lim (E!$)+~-x,)/(47-x1)=1; /i-+X

lim (E:‘;+ ‘) - s, )/( 6:‘;) - x1 ) = 1:

h-+cc

lim (s:.“‘-_~1)/(~~~)-_~1)=(k+1)2-~ t1 +

x

and

(21)

vq>

1

xy-x,

=O(E:l;)-Xl)Y

lim (x:.)l)- x,)/(C,~+~~ I1 -

(k-,

-x,)

co);

= 2~”

x

md

vq

>,

lim (.xf’)-

1 ,

x~)/‘(C,~+~~ - x1)y = 0;

k+X

(22)

lim (c~~~-xl)/(C,l,,,-x,)=l/(k+l) t1 --,

x

arzd

jimx ($)-

x,)/(C,,+~~

--_,

- xJ2

= 2”.

Proof. Since C,, - x1 = x1( 1 - +.“+‘/(l

- r”+‘) when a f - i + c with c < 0 and C,l - .I., = l/2(11 + 1) when u = - i the proofs are all obvious. q

Let us now comment on these results. The numbers (xi”)) double entry tables as follows: .@I c:“’

and (c:‘;.)) are usually displayed in

X0

.(l’

X0

(2’

x0

.(3) x0

. . .

_y(o’

(‘1

1

x(l) 1 Xl

. . .

(2)

EO (0)

(2’

X2

(0’

c2

(‘1

Eo

c2

c(4O)

(1) X2

(0) X3

(3) co

(2) E2

E’6’

(0, C6

. . .

.. . . . .

. . .

. . .

. . .

.. . . . .

Aitken’s array

Shanks’ array

Thus the lower index indicates a column and the upper index a descending diagonal. When a z - a + c with c < 0, for the iterated application of Aitken’s process, each column is linearly convergent (result (l)), each column converges faster than the preceding one and the acceleration has degree two (result (2)). Result (3) is very important since it shows that a linearly convergent sequence can be transformed into a diagonal converging super-quadratically. Each diagonal converges faster than the preceding one (result (4)). For Shanks’ transformation each column is linearly convergent (result (5)), each column converges faster than the preceding one and the acceleration has degree (k + 2)/( k + 1) (result (6)). The linearly converging sequence

C. Brezinski, A. Lenzbarki

/ Estertded Fibonacci sequemvs

7

( S, ) is tran,f ormcd into a diagonal converging super-linearly (result (7)). Each diagonal converges faster than the preceding one (result (8)). Result (9) shows that the columns and the diagonals of Aitken’s array converge faster than the corresponding ones in Shanks’ array. Results (10) and (11) evaluate the degree of acceleration with respect to the initial sequence (C,,). Now when a= - $, for the iterated application of Aitken’s process, each column converges logarithmically (result (12)) and each column does not converge faster than the preceding one (result (13)). R.exult (14) shows that the logarithmic sequence (C,,) can be transformed into a diagonal corverging linearly, but there is no acceleration from one diagonal to the other (result (15)). For Shanks’ transformation each column is logarithmic (result (16)) and each column does not converge faster than the preceding one (result (17)). Each diagonal is also logarithmic (result (18)) and there is no acceleration from one diagonal to the other (result (19)). Result (20) shows that the columns of Aitken’s array do not converge faster than that of Shanks’ array but that the diagonal do. Results (21) and (22) evaluate the degree of acceleration with respect to the initial sequence. The results of Theorem 3 provide an illustration of the notions concerning the speed of convergence of a sequence introduced in [2]. 4. Fixed point methods When a f - d + c, c < 0 the sequence (C,) of the previous sections converges to the dominant zero of x2-x - a. In this section we shall study if some classical fixed point methods yield subsequences of the sequence (C.) thus generalizing Phillips and much older results. We begin with Newton’s method: y0 given,

yn+r =(,$+a)/(3j:,-

J),

n=O,

l,....

It is easy to check in both cases (a f - i + c with c < 0 and a = - 4) that if k exists such that YV= C, then Y,~+,= C2k+l. This is a generalization of a result proved by Serret in 1847 [5] relating Newton’s method for the computation of the square root and its continued fraction expansion

Thus, if y0 = C, = 1, then by induction it is easy to see that Y, = C2”-r thus generalizing a result obtained in 1878 by Moret-Blanc [3] for the square root. It can be checked that (y,,) has order two if ti f - ;‘4+ c with c < 0 and has only order one when a = - i e zero x1 =x2= i. that iswhen x2-xLet us now turn to the secant method: -I), ~~=1,2,... Y,,+ 1 =(_vnyn_, +a)/(y,l+y,,-l YQ’Yl given, If p and k exist such that y,_ 1 = Cp and y,, = C,, then in both cases, l’n+1=

Ck+p+l’

Thus if y0 = yr = Co = 1 then y, = C& k,=k,=l,

.

k,+,=k.,+k,,_,,

where k,, are the Fibonacci numbers given by r1=1,2

,...

.

If a f $ + c with c c 0 it is easy to check that (.y,,) has order j( 1 + 6) if a= -i.

and that the order is one

C. Brezinski, A. Lembarki / Extended Fibonacci sequences

8

Let US now use

Steffensen’s method v1 = 1 +a&

vo=ylP

x,+1 = vo -(v,

-

vo)2/(v2

v2= 1 +a/v,, -

20,

+

vo)*

If k exists such that y, = C, then, by Theorem y. = Co = 1 then Y, =

1, y,,+, = C2k+3. Thus, in both cases, if

c3(2”-1)

andtheorderistwoifaf-_++withc
qk+p+l)2P-l’

Thus if y. = C, = 1 then y,, = CN,,where N,, = [(p + 1)2” - 1](2P” - 1)/(2” - 1). Ifa#-i + c with c’< 0, ( y,, ) has order 2 P while it has order one only when a = - $ . Finally we can also use the E-algorithm as follows vo =Y,P

v1 = 1 +a&,

....

v2P= 1 +a/v2p_,.

Then, applying the c-algorithm to v,, . . . , vaP we set y,, , = c$-j. If k exists such that y, = C, then, from Theorem 2, we have in both cases y,,+ , = CcP+ ,jk +P(P+ 2j. Thus if y. = Co = 1, y,, = CN,, with Iv,=(P+2)[(P+1)“-1]. Ifa#-: The

+ c with c’< 0, ( y,, ) has order p + 1 and it has order one if a = - i. repeated application of Aitken’s process is more powerful than Shanks’ transformation.

References de la convergence en analyse numCrique. Lecture Notes in Math. 584 (Springer. Heidelberg, 1977). PI C. Brezinski. Vitesse de convergence d’une suite, Relt. Rnwnuine Muth. Pures Appl. 30 (1985) 403-417. PI A. Moret-blanc. Question 1111, Ivow. Ann. Muth. I2 (2) (1878) 477-480. PI G.M. Phillips. Aitken sequences and Fibonacci numbers, Amer. Muth. Month& 91 (1984) 354-357. 151J.A. Serret, Sur le developpement en fraction continue de la racine carree d’un nombre entier, J. Math. Pures. Appl.

PI C. Brezinski, Acc&rdtion

M

I,1 (1847) 518-520. D. Shanks. Non linear transformations

of divergent and slowly convergent sequences. J. Math. Pl~_v.s.34 (1955) 1-42. (71 P. Wynn. On a device for computing the e,,,(&) transformation, MTAC’ /O (1956) 9l--96.