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, 57 Shevchenka str., 76018, 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
