ISSN 2075-9827
http://www.journals.pu.if.ua/index.php/cmp
Carpathian Math. Publ. 2014, 6 (1), 32–43
Карпатськi матем. публ. 2014, Т.6, №1, С.32–43
doi:10.15330/cmp.6.1.32-43
Z ATORSKY R.A., S EMENCHUK A.V.
CALCULATION ALGORITHM OF RATIONAL ESTIMATIONS OF RECURRENCE
PERIODICAL FOURTH ORDER FRACTION
Recurrence fourth order fractions are studied. Connection with algebraic fourth order equations
is established. Calculation algorithms of rational contractions of such fractions are built.
Key words and phrases: periodical recurrence fraction, triangular matrix, parapermanent, paradeterminant, rational approximation.
Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine
E-mail: [email protected] (Zatorsky R.A.), [email protected] (Semenchuk A.V.)
I NTRODUCTION
Continued fractions are generalized by quite a few Ukrainian [12, 13] and foreign mathematicians [1]–[11], [14].
The important conditions for generalization of continued fractions are following:
– construction of an easy-to-use algebraic object, the form of which would be similar to the
form of continued fractions, would make it possible to naturally introduce the notion of their
order and to single out the class of periodic objects generalizing periodic continued fractions;
– the algorithm for calculating the value of rational contractions of mathematical objects is
to be simple in realization and efficient;
– by analogy with periodic chain fractions, random periodic algebraic objects of higher
orders are to be of the forms of some algebraic irrationalities of higher orders.
In [15] it is suggested new generalization of continued fractions — recurrence fractions,
which satisfy the above-mentioned conditions. In addition, the connection between singly periodic recurrence fractions of order n and algebraic equations of order n has been established.
Recurrence fractions of order three have been studied in [16].
This article focuses on recurrence fractions of order four, proves their connection with corresponding algebraic equations of order four and determines algorithms for constructing rational approximations of order four.
УДК 511.14
2010 Mathematics Subject Classification: 11J70.
c Zatorsky R.A., Semenchuk A.V., 2014
C ALCULATION ALGORITHM OF RATIONAL ESTIMATIONS OF
1
P ERIODIC
RECURRENCE FRACTION OF
RECURRENCE FRACTIONS OF ORDER
A recurrence fraction of order four takes the form

q1
 p2
q2
 q2
 r3
p3
 p
q3
q3
 3
p4
r4
 s4
q4
 r4
p4
q4

p5
s5
r5
 0
q5
r5
p5
q5

 ..
.
 .
. . . . . . . . . . . . ..

p
 0
0 . . . rsmm prmm qmm qm

..
.
.
. . . . . . . . . . . . . . . . . . ..
4-TH
ORDER
33
4








 .







(1)
∞
Its rational contractions






Pn

=

Qn





q1
p2
q2
r3
p3
s4
r4
0
..
.
0
q2
p3
q3
r4
p4
s5
r5
q3
p4
q4
r5
p5
q4
p5
q5
q5
... ... ... ...
0 . . . rsnn prnn
..
.
pn
qn
qn












n
satisfy the recurrence equations
Pn = qn Pn−1 + pn Pn−2 + rn Pn−3 + sn Pn−4,
n = 1, 2, 3, . . . ,
Q n = q n Q n−1 + pn Q n−2 + r n Q n−3 + s n Q n−4 ,
n = 2, 3, 4, . . .
(2)
with the initial conditions
P0 = 1,
Pi<0 = 0,
Q1 = 1,
Qi<1 = 0.
Definition. The recurrence fraction (1) of order 4, the elements of which satisfy the conditions
prk+m = pm , qrk+m = qm , rrk+m = rm , srk+m = sm ,
m = 1, 2, . . . , k, r = 0, 1, 2, . . .
(3)
is a periodic recurrence fraction of order 4 with the period k.
We shall determine the connections between periodic recurrence fractions of order four
and real positive roots of quartic equations.
1. Consider a singly periodic recurrence fraction of order four. Let us decompose the parapermanent of the numerator of the rational contraction
34
Z ATORSKY R.A., S EMENCHUK A.V.










Pn
=

Qn









q
p
q
r
p
s
r
q
p
q
r
p
s
r
0
..
.
0
0
0
0
q
p
q
r
p
q
p
q
q
... ... ... ...
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
...
... q
p
... q
. . . pr
. . . rs
q
p
q
r
p
q
p
q
q
by the elements of the first column. We get the equality
Pn
= q+
Qn
p
Pn −1
Q n −1
+
r
Pn −1
Q n −2
+
s
Pn −1
Q n −3
= q+
p
Pn −1
Q n −1
+
r
Pn −1
Q n −1
·
Pn −2
Q n −2
+



















n+1
s
Pn −1
Q n −1
·
Pn −2
Q n −2
·
Pn −3
Q n −3
.
(4)
Let us take the limit
Pn
Pn
= lim
= x,
n→∞ Pn−1
n→ ∞ Q n
p
then the equality (4) is written as x = q + x + xr2 + xs3 , or x4 = qx3 + px2 + rx + s. One of the
roots of this equation is
s p
2β
α + 2y + − 3α + 2y + √
α+2y
q
,
x= +
4
2
lim
where
5
y = − α+
6
s
3
Q
− +
2
r
Q2
P3
+
−
4
27
P
3·
r
3
− Q2 +
q
,
Q2
4
+
P3
27
3
α = − q2 − p,
8
3q4
pq2 qr
α2
α3
αγ β2
q3 qp
−
− r, γ = −
−
− − s, P = − − γ, Q = −
−
− .
8
2
256
16
4
12
108
3
8
It is easy to establish
that
if
q
=
4,
p
=
−
6,
r
=
4,
the
singly
periodic
fraction
will
represent
√
the irrationality 4 1 + s.
β=−
Example 1. If
q = 4, p = −6, r = 4, s = 2,
then the recurrence fraction is written as

4
 −3
4
 2
 2
− 32 4
 −3
 1
 2
− 23 − 32 4

1
 0
− 23 − 23 4
2

1
 0
− 23 − 32 4
0

2
..
.
.
. . . . . . . . . . . . . . . ..






 .





∞
C ALCULATION ALGORITHM OF RATIONAL ESTIMATIONS OF
RECURRENCE FRACTION OF
4-TH
ORDER
35
The relevant algebraic equation of order four is of the form √
x4 = 4x3 − 6x2 + 4x + 2. The
rational approximations to the maximum modulo root x = 1 + 4 3 ≈ 2, 31607401295249246 of
this equation can be found with the help of the linear recurrence relations of order four
Pn = 4Pn−1 − 6Pn−2 + 4Pn−3 + 2Pn−4 ,
P0 = 1,
while x = lim PPmm−1 .
m→∞
Here are the first 35 rational approximations to this root:
u1 = 4,
u8 = 1164,
u15 = 404736,
u22 = 144235520,
u29 = 51545829376,
u2 = 10,
u9 = 2704,
u16 = 937104,
u23 = 334031360,
u30 = 119382376448,
u3 = 20,
u10 = 6136,
u17 = 2165568,
u24 = 773463744,
u31 = 276492099584,
u4 = 38, u11 = 13936, u18 = 5006752, u25 = 1791122688, u32 = 640367841536,
u5 = 80, u12 = 32072, u19 = 11591488, u26 = 4148304768, u33 = 1483139933184,
u6 = 192, u13 = 74624, u20 = 26861920, u27 = 9608400640, u34 = 3435085834752,
u7 = 480, u14 = 174080, u21 = 62256896, u28 = 22255192192, u35 = 7955959305216,
while
u35
u34
=
7955959305216
3435085834752
≈ 2.3160874.
2. Consider a doubly periodic recurrence fraction of order four


q1

 p2
q2

 q2

 r1
p1

 p
q1
q1

 s1
p2
r2

 2
q
2
 ,
 r2
p2
q2


p1
s1
r1

 0
q
1
r1
p1
q1


p
s
r
2
2
2

 0
0
q
2
r
p
q


2
2
2
..
...
... ... ... ... ...
.
∞
where qi , pi , ri , si are some rational positive numbers.
Let us decompose the parapermanent of the numerator of the rational contraction by the
elements of the first column
[ q1 ] n
q [q ]
+ p2 [ q 1 ] n − 2 + r 1 [ q 2 ] n − 3 + s 2 [ q 1 ] n − 4
= 1 2 n−1
[ q2 ] n−1
[ q2 ] n−1
(5)
r1
s1
p
+
.
= q1 + [ q ] 2 + [ q ]
[ q2 ] n −1 [ q1 ] n −2 [ q2 ] n −3
2 n −1
2 n −1
· [[qq1 ]]n−2
· [q ] · [q ]
[q ]
[q ]
[q ]
1 n −2
1 n −2
2 n −3
1 n −2
2 n −3
1 n −4
In this equality, the parapermanent of order i with the upper element q j , j = 1, 2 is denoted
[q ]
by [q j ]i . Likewise, we decompose the numerator of the fraction [q2 ]n−1 by the elements of the
1 n −2
first column
p
[ q2 ] n−1
r2
s1
= q2 + [ q ] 1 + [ q ]
.
(6)
+ [q ]
[ q2 ] n −3
[ q2 ] n −3 [ q1 ] n −4
1 n −2
1 n −2
1 n −2
[ q1 ] n−2
·
·
·
Let us take the limits
[ q2 ] n −3
[ q2 ] n −3
[ q1 ] n −4
[ q2 ] n −3
[ q1 ] n −4
[ q2 ] n −5
[ q1 ] m
[ q2 ] m
= x, lim
= y.
m → ∞ [ q2 ] m −1
m → ∞ [ q1 ] m − 1
Passing n to infinity in the equalities (5), (6), we get simultaneous equations

 x = q1 + p2 + r1 + s22 ,
y
xy
xy
y = q2 + p1 + r2 + s21 ,
lim
x
xy
x y
36
Z ATORSKY R.A., S EMENCHUK A.V.
from which we find that
p
( q 2 x + p1 )2 + 4(r 2 x + s 1 )
,
2x
and x is the positive root of the equation of order four
y=
q 2 x + p1 +
(q2 p2 r2 + r22 − q22 s2 ) x4 = (q1 q2 p2 r2 + p22 r2 + 2q1 r22 + 2q2 p1 s2 + 2r2 s2 − p1 p2 r2 − q2 r1 r2
− 2r2 s1 − q2 p2 s1 − q1 q22 s2 − q2 p2 s2 ) x3 + (q1 p1 p2 r2 + q1 q2 r1 r2 + 2p2 r1 r2 + p22 s1 + q1 q2 p2 s1
+ 4q1 r2 s1 + p21 s2 + 2s1 s2 − q21 r22 − p1 r1 r2 − q2 r1 s1 − p1 p2 s1 − s21 − s22 − 2q1 q2 p1 s2
(7)
2
− 2q1 r2 s2 − p1 p2 s2 − q2 r1 s2 ) x + (q1 p1 p2 s1 + q1 q2 r1 s1 + 2p2 r1 s1 + r12 r2 + q1 p1 r1 r2
+ 2q1 s21 − 2q21 r2 s1 − p1 r1 s1 − q1 p21 s2 − 2q1 s1 s2 − p1 r1 s2 ) x + s1 (q21 s1 − q1 p1 r1 − r12 ).
Thus, the following theorem is proved.
Theorem 1. If qi , pi , ri , si are some rational positive numbers and there are limits
[ q1 ] m
lim
m → ∞ [ q2 ] m −1
= x,
[ q2 ] m
lim
m → ∞ [ q1 ] m − 1
= y,
then x is the positive root of the equation (7) of order four.
Example 2. If q1 = 3, p1 = 3, r1 = 3, s1 = 3, q2 = 2, p2 = 2, r2 = 2, s2 = 2, then the recurrence
fraction is written as


3
 2

2
 2

 3

3
3
 3

3
 2

2
2
 2

2
2
2


3
3
3
 0

3
3
3
3


2
2
2
 0

0
2


2
2
2
..
...
... ... ... ... ...
.
and the rational contractions, which approximate the maximum modulo real root


v
r
u
1 1
29
27

u 87
x = +  − + 2y + t − 2y + q
 ≈ 3, 978743113,
2 2
4
4
−29
+ 2y
4
where
1
y=
145 +
24
of the fourth order equation
q
3
√
−63197 + 972 7226 − p
3
1415
√
−63197 + 972 7226
!
,
4x4 = 8x3 + 23x2 + 27x + 27,
are equal to
3
8
36
96
429
= 3, δ2 = = 4, δ3 =
= 4, δ4 =
= 4, δ5 =
≈ 3, 9722,
1
2
9
24
108
1138
5097
13520
δ6 =
≈ 3, 9790, δ7 =
≈ 3, 97892, δ8 =
≈ 3, 97881,
286
1281
3398
160614
719349
60552
≈ 3, 978711, δ10 =
≈ 3, 9787455, δ11 =
≈ 3, 9787442,
δ9 =
15219
40368
180798
1908070
8545755
22667576
=
≈ 3, 97874328, δ13 =
≈ 3, 97874296, δ14 =
≈ 3, 97874313,
479566
2147853
5697170
269287302
101522250
≈ 3, 978743118, δ16 =
≈ 3, 9787431129.
δ15 =
25516161
67681500
δ1 =
δ12
C ALCULATION ALGORITHM OF RATIONAL
2
A LGORITHM
ESTIMATIONS OF RECURRENCE FRACTION OF
4-TH
ORDER
37
FOR CALCULATING RATIONAL CONTRACTIONS OF PERIODIC RECURRENCE
FRACTIONS OF ORDER FOUR
Let us construct a new algorithm for calculating rational contractions of periodic recurrence
fractions of order four.
Let k be the period of a recurrence fraction, and n — the order of the parapermanent of its
rational contraction, while n = sk, s = 1, 2, 3, . . .
Then the following theorem is true.
Theorem 2. The rational contraction
Pn
Qn
δn =
of the periodic recurrence fraction (1) of order four, with the period k > 2, the elements of
which satisfy the conditions (3), is equal to the value of the expression
q 0 + p1 ·
s −1
Bsk
−1
Assk
+ r2 ·
s −1
Csk
−2
Assk
+ s2 ·
s −1
Dsk
−3
Assk
,
(8)
s −1
s −1
s −1
where Assk , Bsk
−1, Csk −2 and Dsk −3 are defined by the recurrence equalities
Assk=s3 ϕk−1 Dks−(s2−1)−3 +(s2 ϕk−2 + r2 ϕk−1 )Cks−(s2−1)−2 +(s1 ϕk−3 + r1 ϕk−2 + p1 ϕk−1 ) Bks−(s2−1)−1 + ϕk Ask−(1s−1) ,
(9)
s−1
s−2
s−2
s−1
s−2
Bsk
−1=s3 ψk−2 Dk ( s−1)−3 +(s2 ψk−3 + r2 ψk−2 )Ck ( s−1)−2 +(s1 ψk−4 + r1 ψk−3 + p1 ψk−2 ) Bk ( s−1)−1 + ψk−1 Ak ( s−1) ,
(10)
s−1
s−2
s−2
s−1
s−2
Csk
−2=s3 τk−3 Dk ( s−1)−3 +(s2 τk−4 + r2 τk−3 )Ck ( s−1)−2 +(s1 τk−5 + r1 τk−4 + p1 τk−3 ) Bk ( s−1)−1 + τk−2 Ak ( s−1) ,
(11)
s−1
s−2
s−2
s−1
s−2
Dsk
−3=s3 ξ k−4 Dk ( s−1)−3 +(s2 ξ k−5 + r2 ξ k−4 )Ck ( s−1)−2 +(s1 ξ k−6 + r1 ξ k−5 + p1 ξ k−4 ) Bk ( s−1)−1 + ξ k−3 Ak ( s−1) ,
(12)
where










ϕk = 










q1
p2
q2
r3
p3
s4
r4
0
..
.
0
0
0
0
q2
p3
q3
r4
p4
s5
r5
q3
p4
q4
r5
p5
q4
p5
q5
q5
... ... ... ...
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
...
. . . q k −3
p
. . . q k −2 q k −2
k −2
r
p
. . . pk−1 q k−1 qk−1
k −1
k −1
pk
rk
qk
. . . rsk
p
q
k
k
k









 ,









k
(13)
38
Z ATORSKY R.A., S EMENCHUK A.V.

ψk − 1









=










τk−2









=










ξ k −3









=










q2
p3
q3
r4
p4
s5
r5
0
..
.
0
0
0
0
q3
p4
q4
r5
p5
s6
r6
q4
p5
q5
r6
p6
q5
p6
q6
q6
... ... ... ...
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
...
. . . q k −3
p
. . . q k −2 q k −2
k −2
r
p
. . . pk−1 q k−1 qk−1
k −1
k −1
pk
rk
qk
. . . rsk
p
q
k
k
k
0
..
.
0
0
0
0
q4
p5
q5
r6
p6
s7
r7
q5
p6
q6
r7
p7
q6
p7
q7
q7
... ... ... ...
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
...
. . . q k −3
p
. . . q k −2 q k −2
k −2
p
r
. . . pk−1 q k−1 qk−1
k −1
k −1
pk
rk
. . . rsk
qk
p
q
k
k
k
0
..
.
0
0
0
0
q5
p6
q6
r7
p7
s8
r8
q6
p7
q7
r8
p8
(14)
,
(15)
,
(16)
k −1



















k −2

q4
p5
q5
r6
p6
s7
r7
,

q3
p4
q4
r5
p5
s6
r6



















q7
p8
q8
q8
... ... ... ...
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
...
. . . q k −3
p
. . . q k −2 q k −2
k −2
p
r
. . . pk−1 q k−1 qk−1
k −1
k −1
pk
rk
. . . rsk
qk
p
q
k
k
and if k = 2, 3, 4, we assume that
k



















k −3
ξ <0 = τ<0 = ψ<0 = ϕ<0 = 0,
ϕ0 = ψ0 = τ0 = ξ 0 = 1.
Proof. If n = sk, then the numerator and the dominator of the n-th rational contraction of the
periodic recurrence fraction (1) of order four, with the period of k > 2, the elements of which
satisfy the conditions (3), are respectively in the form
C ALCULATION ALGORITHM OF RATIONAL ESTIMATIONS OF


















Psk = 
















Qsk
RECURRENCE FRACTION OF
4-TH
ORDER

q0
p1
q1
r2
p2
s3
r3
q1
p2
q2
r3
p3
s4
r4
q2
p3
q3
r4
p4
39
q3
p
4
0
q4 q4
..
.
. . . . . . . . . . . . . ..
0 0 0 0 0 . . . q k −3
p
0 0 0 0 0 . . . q k −2 q k −2
k −2
p
r
0 0 0 0 0 . . . pk−1 q k−1 qk−1
k −1
k −1
pk
rk
0 0 0 0 0 . . . rsk
qk
pk
qk
k
p1
s1
r1
0 0 0 0 0 ... 0
r1
p1
q1 q1
..
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
0 0 0 0 0 ... 0
0
0
0 0 . . . q k −3
p
0 0 0 0 0 ... 0
0
0
0 0 . . . q k −2 q k −2
k −2
r
p
0 0 0 0 0 ... 0
0
0
0 0 . . . pk−1 q k−1 qk−1
k −1
k −1
pk
rk
qk
0 0 0 0 0 ... 0
0
0
0 0 . . . rsk
pk
qk
k

q1
 p2 q 2
 q2
 r 3 p3
 p3 q 3 q 3
 s r
 4 4 p4 q 4
 r 4 p4 q 4

p
r
s
 0 r55 p55 q55 q5

 ..
.
 . . . . . . . . . . . . . ..

 0 0 0 0 0 . . . q k −3

 0 0 0 0 0 . . . pk −2 q k −2

q k −2
=
p
r
 0 0 0 0 0 . . . pkk−−11 qkk−−11 qk−1

pk
rk
 0 0 0 0 0 . . . sk
qk
rk
pk
qk


p1
s1
r1
 0 0 0 0 0 ... 0
r1
p1
q1 q1
 .
.
 .
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

 0 0 0 0 0 ... 0
0
0
0 0 . . . q k −3

p
 0 0 0 0 0 ... 0
0
0
0 0 . . . q k −2 q k −2

k −2
r
p

0
0
0 0 . . . pk−1 q k−1 qk−1
 0 0 0 0 0 ... 0
k −1
k −1
p

















,
















(17)


















.
















rk
k
0 0 0 0 0 ... 0
0
0
0 0 . . . rsk
qk
pk
qk
k
Let us denote the parapermanent, formed from the parapermanent (17) as a result of deleting the first column, by Assk , the parapermanent, formed as a result of deleting the first two
s −1
columns, — by Bsk
−1, the parapermanent, formed as a result of deleting the first three columns,
s −1
— by Csk−2, and the parapermanent, formed as a result of deleting the first four columns, —
s −1
by Dsk
−3 (in the four cases, the superscript denotes the number of complete periods containing
these parapermanents).
Let us decompose the parapermanent (17) by the elements of the first column and get the
equality
40
Z ATORSKY R.A., S EMENCHUK A.V.
s −1
s −1
s −1
Psk = q0 Assk + p1 Bsk
−1 + r2 Csk −2 + s3 Dsk −3.
(18)
Let us decompose the parapermanent Assk by the elements of the inscribed rectangular table
T (k + 1), then we get the recurrence (9). In the same way, let us decompose the parapermas −1
s −1
s −1
nents Bsk
−1, Csk −2 , and Dsk −3 by the elements of the tables T (k), T (k − 1), i T (k − 2). At that
we get the recurrences (10), (11), (12).
As Qsk = Assk , considering (18), we conclude that the rational contraction δn = QPnn of the
periodic recurrence fraction is equal to
s −1
s −1
s −1
s −1
s −1
s −1
Csk
Dsk
q0 Assk + p1 Bsk
Bsk
Psk
−2
−3
−1 + r2 Csk −2 + s3 Dsk −3
−1
=
=
q
+
p
+
r
+
s
.
0
2
3
1
Qsk
Assk
Assk
Assk
Assk
Example 3. Let us have a periodic recurrence fraction of order four with the period, where
q1 = 1, p2 = 1, r3 = 1, s4 = 1, q2 = 1, p3 = 1, r4 = 1, s5 = 1, q3 = 2, p4 = 2, r5 = 2, s1 = 2,
q4 = 1, p5 = 1, r1 = 1, s2 = 1, q5 = 2, p1 = 2, r2 = 2, s3 = 2.
This periodic recurrence fraction approximates to the maximum modulo real root
r

v
u
109
1 1
11
u 33
 ≈ 1, 969558741906025,
x = +  − + 2y + t − 2y + q 36
4 2
8
8
−11
+ 2y
8
where
1
y=
2
55 1
+
24 9
the equation of order four
q
3
√
−2007 + 144 622 − p
3
23
√
−2007 + 144 622
9x4 − 9x3 − 9x2 − 8x − 16 = 0.
!
,
(19)
Let us find the rational contractions (20) of the relevant recurrence fraction first with the
help of the recurrences (2), and then by the algorithm of the theorem 2.


1
 1

1


 1

1
2


2


1
 1

2
1
2


1
1
 0

2
2
2
2


1
 0

0
2
2
1


2
Pn
1
 0

0 0
2 1 1
(20)
=

2


Qn
1
0 0 0 2 1
2
 0

2


1

 0
2 1
0 0 0 0 1
2


1
1
 0

2
2
0 0 0 0 0
2
2


1
 0

2
1
0 0 0 0 0 0 2


2
 ..

.
.

 .
.
... ... ... ... ... ... ... ... ...
0
0 0 0 0 0 0 0 0 0 0 ... 2
By means of the recurrences (2) we shall have:
C ALCULATION ALGORITHM OF
δ1 =
2
= 2,
1
δ2 =
RATIONAL ESTIMATIONS OF RECURRENCE FRACTION OF
6
= 2,
3
δ3 =
12
= 2,
6
δ4 =
35
≈ 1.9444,
18
δ5 =
4-TH
ORDER
41
69
= 1.97142,
35
134
396
768
2269
≈ 1.97059, δ7 =
≈ 1.970149, δ8 =
≈ 1.9692308, δ9 =
≈ 1.969618,
68
201
390
1152
8670
25614
4469
≈ 1.9695901, δ11 =
≈ 1.96955929, δ12 =
≈ 1.96955017,
δ10 =
2269
4402
13005
146807
289145
49692
≈ 1.96956005, δ14 =
≈ 1.96955915, δ15 =
≈ 1.96955867,
δ13 =
25230
74538
146807
1657236
3215088
560950
≈ 1.969558653, δ17 =
≈ 1.969558784, δ18 =
≈ 1.969558745,
δ16 =
284810
841425
1632390
9498457
18707769
δ19 =
≈ 1.969558739, δ20 =
≈ 1.9695587399,
4822632
9498457
107223678
36293638
≈ 1.9695587426, δ22 =
≈ 1.969558741948,
δ21 =
18427294
54440457
614553083
208017180
≈ 1.96955874185, δ24 =
≈ 1.96955874189,
δ23 =
105616134
312025770
2348209518
1210398397
≈ 1.969558741926, δ26 =
≈ 1.9695587419051,
δ25 =
614553083
1192251578
6937404876
13458775392
δ27 =
≈ 1.9695587419044, δ28 =
≈ 1.969558741906103,
3522314277
6833396286
78313147789
39761773093
≈ 1.96955874190628, δ30 =
≈ 1.96955874190598.
δ29 =
20188163088
39761773093
δ6 =
Let us do similar calculations with the help of the algorithm of the theorem 2.
We shall calculate ξ −1 , ξ 0 , ξ 1 , ξ 2 , τ0 , τ1 , τ2, τ3 , ψ1 , ψ2, ψ3, ψ4 , ϕ2 , ϕ3, ϕ4 , ϕ5 from the equalities
(13), (14), (15), (16):




1


1
 1 2

1
 1

 2

2




 = 6,
ϕ5 =  12 2 1
 = 35, ϕ4 =  21
 = 18, ϕ3 =  21 2
 1

2
1


1
1
2
 2 2 2 2

1
2 2 1
2 12 2
1
2
0 2 2 2 1




2
2
 2 1

1


 = 12,
= 3, ψ4 = 
ϕ2 = 1
 = 24, ψ3 =  2 1
1
2
2
2


1
2
2
2 2 2
2 12 2 1


1
1
2
1
 = 6, τ2 = 1
= 3, τ1 = 1, τ0 = 1,
= 4, ψ1 = 2, τ3 =  2 2
ψ2 =
2
2 1
1
2
2 2 1
2
ξ2 =
= 4, ξ 1 = 2, ξ 0 = 1, ξ −1 = 0.
2 1
Consequently, the recurrences (9), (10), (11), (12) will be written as:
s
s −2
s −2
s −2
s −1
A5s
=18D5s
−8 + 30C5s −7 + 33B5s −6 + 35A5s −5 ,
s −1
s −2
s −2
s −2
s −1
B5s
−1 =12D5s −8 + 20C5s −7 + 22B5s −6 + 24A5s −5 ,
s −1
s −2
s −2
s −2
s −1
C5s
−2 =3D5s −8 + 5C5s −7 + 6B5s −6 + 6A5s −5 ,
s −1
s −2
s −2
s −2
s −1
D5s
−3 =2D5s −8 + 4C5s −7 + 4B5s −6 + 4A5s −5 .
42
Z ATORSKY R.A., SEMENCHUK A.V.
The s-th approximation to the value of the given recurrence fraction, by the algorithm of the
theorem 2 is of the form
B s −1
C s −1
D s −1
γs = 1 + 5ss−1 + 5ss−2 + 5ss−3 .
A5s
A5s
A5s
0
0
0
1
Since, D2 = ξ 2 = 4, C3 = τ3 = 6, B4 = ψ4 = 24, A5 = ϕ5 = 35, then
γ1 =
4469
289145
69
= 1.97143, γ2 =
≈ 1.9695901, γ3 =
≈ 1.969558672,
35
2269
146807
γ4 =
1210398397
18707769
≈ 1.9695587399, γ5 =
≈ 1.969558741926,
9498457
614553083
78313147789
≈ 1.96955874190598.
γ6 =
39761773093
Thus, from this example it is clear that the s-th approximation γs , found by means of the algorithm of Theorem 2 coincides with the (5s)-th approximation δ5s , found by the algorithm (2).
3
CONCLUSIONS
Therefore, recurrence fractions of order four are natural generalization of chain fractions.
Periodic recurrence fractions of order four are connected with corresponding algebraic equations of order four and show irrationalities of order four, while Theorem 2 provides an effective
algorithm for constructing rational approximations to these irrationalities.
REFERENCES
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[2] Bruno A.D. Algorithm of the generalizationued continued fraction. Preprint No. 10, Keldysh Inst. Appl. Math.,
Russian Acad. Sci., Moscow, 2004. (in Russian)
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[4] Hermite C. Extraits de lettres de M. Ch. Hermite à M. Jacobi sur différents objects de la théorie des nombres. J. Reine
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[5] Jacobi C.G.J. Allgemeine Theorie der Kettenbruchanlichen Algorithmen, in welchen jede Zahl aus drei vorhergehenden
gebildet wird. J. Reine Angew. Math. 1868, 69, 29–64.
[6] Klein F. Über eine geometrische Auffassung der gewöhnlichen Kettenbruchentwicklung. Nachr. Ges. Wiss. Gottingen Math.-Phys. Klasse 1895, 357–359.
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auf die Theorie der Zahlen. S. B. Preuss. Akad. Wiss. 1842, 93–95.
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41–60.
[9] Perron O. Grundlagen für eine Theorie des Jacobischen Ketten-bruchalgorithmus. Math. Ann. 1907, 64, 1–76.
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C ALCULATION ALGORITHM OF RATIONAL ESTIMATIONS OF
RECURRENCE FRACTION OF
4-TH
ORDER
43
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Received 07.02.2014
Заторський Р.А., Семенчук А.В. Алгоритм обчислення рацiональних наближень перiодичного дробу 4-го порядку // Карпатськi матем. публ. — 2014. — Т.6, №1. — C. 32–43.
Вивчаються рекурентнi дроби четвертого порядку, встановлюються їх зв’язки з алгебраїчними рiвняннями четвертого порядку i будуються алгоритми обчислення рацiональних наближень.
Ключовi слова i фрази: перiодичний дрiб, трикутна матриця, параперманент, парадетермiнант, рацiональне наближення.
Заторский Р.А., Семенчук А.В. Алгоритм вычисления рациональных приближений периодической
рекуррентной дроби 4-го порядка // Карпатские матем. публ. — 2014. — Т.6, №1. — C. 32–43.
Изучаются рекуррентные дроби четвертого порядка, устанавливаются их связи с алгебраическими уравнениями четвертого порядка и строятся алгоритмы вычисления их рациональных приближений.
Ключевые слова и фразы: периодическая рекуррентная дробь, треугольная матрица, параперманент, парадетерминант, рациональное приближение.