Geodesic
Algorithm for summation of divergent continued fractions and some applications
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 4, pp. 558-573 Cet article a éte moissonné depuis la source Math-Net.Ru

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The convergence of continued fractions is defined in a manner other than the conventional definition. A new summation method is used to determine the values of continued fractions and series that diverge in the classical sense. The method is applicable not only to ordinary continued fractions, but also to ones of other classes, for example, to Hessenberg continued fractions. As a result, an original algorithm for finding zeros of $n$th-degree polynomials is constructed. The $r/\varphi$-algorithm proposed is also used to solve infinite systems of linear algebraic equations. 
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G. A. Kirichenko; V. I. Shmoylov. Algorithm for summation of divergent continued fractions and some applications. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 4, pp. 558-573. http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_4_a3/

[1] Lorentzen L., Waadeland H., Continued fraction with application, Amsterdam–London–New-York–Tokyo, 1992

[2] Beiker Dzh., Greivis-Moris P., Approksimatsiya Pade, Mir, M., 1986

[3] Dzhouns U., Tron V., Nepreryvnye drobi. Analiticheskaya teoriya i prilozheniya, Mir, M., 1985 | MR

[4] Shmoilov V. I., Nepreryvnye drobi, v. 2, Raskhodyaschiesya nepreryvnye drobi, Merkator, Lvov, 2004

[5] Kachmar V. S., Rusyn B. P., Shmoilov V. I., “Algoritmy vychisleniya znachenii tsepnykh drobei”, Zh. vychisl. matem. i matem. fiz., 38:9 (1998), 1436–1451 | MR | Zbl

[6] Shmoilov V. I., Nepreryvnye drobi, v. 3, Iz istorii nepreryvnykh drobei, Merkator, Lvov, 2004

[7] Shmoilov V. I., Reshenie algebraicheskikh uravnenii pri pomoschi $r/\varphi$-algoritma, Izd-vo TTI YuFU, Taganrog, 2011

[8] Aitken A., “On Bernoulli's numerical solution of algebraic equations”, Proc. Roy. Soc. Edinburgh. Ser. A, 46 (1925/26), 289–305

[9] Rutiskhauzer G., Algoritm chastnykh i raznostei, IIL, M., 1960

[10] Godunov C. K., Ryabenkii B. C., Raznostnye skhemy, Nauka, M., 1977

[11] Shmoilov V. I., Raskhodyaschiesya sistemy lineinykh algebraicheskikh uravnenii, Izd-vo TTI YuFU, Taganrog, 2010

[12] Shmoilov V. I., Nepreryvnye drobi i $r/\varphi$-algoritm, Izd-vo TTI YuFU, Taganrog, 2012 | Zbl