Department of Economics, Niigata Sangyo University, Niigata, Japan

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1 Appled Matheatcs, 0, 5, Publshed Onlne March 0 n ScRes. On Relatons between the General Recurrence Forula o the Etenson o Murase-Newton s Method (the Etenson o Tsuchura * -Horguch s Method and Horner s Method Shunj Horguch Departent o Econocs, Ngata Sangyo Unversty, Ngata, Japan Eal: shor@econ.nsu.ac.jp Receved Noveber 0; revsed Deceber 0; accepted January 0 Copyrght 0 by author and Scentc Research Publshng Inc. Ths wor s lcensed under the Creatve Coons Attrbuton Internatonal Lcense (CC BY. Abstract In 67, Yoshasu Murase ade a cubc equaton to obtan the thcness o a hearth. He ntroduced two nds o recurrence orulas o square and the deoraton (Re. []. We nd that the three orulas lead to the etenson o Newton-Raphson s ethod and Horner s ethod at the sae te. Ths shows orgnalty o Japanese natve atheatcs (Wasan n the Edo era ( Suzu (Re. [] estates Murase to be a rare atheatcan n not only the hstory o Wasan but also the hstory o atheatcs n the world. Secton ntroduces Murase s three solutons o the cubc equaton o the hearth. Secton eplans the Horner s ethod. We gve the generalzaton o three orulas and the relaton between these orulas and Horner s ethod. Secton gves dentons o Murase-Newton s ethod (Tsuchura-Horguch s ethod, general recurrence orula o Murase-Newton s ethod (Tsuchura-Horguch s ethod, and general recurrence orula o the etenson o Murase-Newton s ethod (the etenson o Tsuchura-Horguch s ethod concernng n-degree polynoal equaton. Secton s contents o the ttle o ths paper. Keywords Recurrence Forula; Newton-Raphson s Method (Newton s Method; Etensons o Murase-Newton s Method; Horner s Method * Tsuchura s Taotsu Tsuchura, the proessor eertus o Tohou Unversty. How to cte ths paper: Horguch, S. (0 On Relatons between the General Recurrence Forula o the Etenson o Murase-Newton s Method (the Etenson o Tsuchura*-Horguch s Method and Horner s Method. Appled Matheatcs, 5,

2 . Introducton All the reerences are wrtten n Japanese. We wrote ths paper ro two nds o recurrence orulas o the square and the deoraton o a cubc equaton wrtten n Re. [], and a hnt o Tsuchura. Thereore, t s enough or readers to now these three orulas. But t s very dcult even or Japanese people to read the Murase s boo wrtten n the Japanese ancent wrtng. Thereore, the readers do not need to read the boo. Furtherore, the readers do not need to nd Japanese reerences. Fro now on, we eplan the Murase s three orulas as ntroducton. The readers can now the orgn o ths paper. Murase ade the cubc equaton or the net proble n 67. There s a rectangular sold (base s a square. We put t together our and ae the hearth as Fgure. Fgure. Hearth. We cla one sde o length o the square that one sde s, and a volue becoes 9 o the hearth. Let one sde o length o the square be then the net cubc equaton s obtaned. that s ( ( (. Ths has three solutons o real nuber, 6 ± 5. Murase derved two ollowng recurrence orulas and deored equaton ro (.. The rst ethod 8 + 0,,, (. + Usng on an abacus, Murase calculates to 0 0 (ntal value,.85,.97,.996, and decdes a soluton wth. The second ethod 8 + ( 0,,, (. here he calculates to 0 0,.85,.976,.9989, , and decdes a soluton wth. An epresson (. has better precson than that (., and convergence becoes ast. The thrd ethod was nonrecurrng n spte o a short sentence or any years. However, Yasuo Fuj (Taaazu Se Matheatcs Research Insttute o Yoach Unversty succeeds n decodng n May, 009. It s the net equaton. The thrd ethod 8 (.5 The studes o three orulas o Murase progress by the thrd ethod (already decoded.. Horner s Method Let n n a + a + + a+ a (. n n+ 778

3 be a n-th degree polynoal where a,, a n + are real nubers. The Horner s ethod s an algorth to calculate ( α. Dvdng orula. Thereore, orula. α becoes R. Net, dvdng Here, R s the derental coecent ( α Furtherore, dvdng n n ( α ( n n by α, we obtan the net b + b + + b + b + R (. b + b + + b + b by α, we obtan the net n n n { } n n ( α ( α( n n c + c + + c + c + R + R (.. n n c + c + + c + c by α, we obtan the net orula. n n n n ( α ( α {( α( n n } d + d + + d + d + R + R + R (. Coparng the coecents o orula (. and (., (. and (., (. and (. respectvely, we obtan the net calculatng orula o b, R, c, R, d, R. b a b a + b α,,, n + c b c b + c α (,,, n d c ( + ( d c d α,,, n ( α ( α n+ n n n (.5 (.6 (.7 b R, c R, d R (.8 Slarly, we can contnue calculatng. The ndcaton o calculatng orula by synthetc dvson s net Table. Table. Synthetc dvson or an epresson (. (Re. []. a a a. a n a n a n+ + b α b α. b n a b n a b n α b b b. b n b n R (α + c α c α. c n a c n a c c c. c n R '(α + d α d α. d n a d d d. R Eaple.. I we apply Horner s ethod to an epresson (. n case o soluton, then t s calculated n Table. Table. Synthetc dvson or an epresson (

4 We obtan the net theore ro the Murase s three orulas and Table o Horner s ethod. Theore.. ( We epand the rst, second, thrd ethod o Murase, and obtan the net recurrence orulas where s a real nuber. + ( 8 (, o denonator o orula (., (. respectvely, and o orula (.5 change 0. Thereore, these changes correspond to the second lne o the Table o the calculaton o Horner s ethod n Eaple. (Re. [].. Epansons Recurrence Forula o Murase-Newton In 009, we ound the etenson o Newton-Raphson s ethod ro the Murase s three orulas and a hnt o Taotsu Tsuchura, and called t the Murase-Newton s ethod or the Tsuchura-Horguch s ethod. We obtaned the etenson o Newton-Raphson s ethod as ollows. q Let t g t such as where q s a real nuber that s not 0. We dene the uncton ( q (.9 g t : t (., we have the net denton. Denton.. For equaton ( 0, we call the net recurrence orula the Murase-Newton s ethod or the Tsuchura-Horguch s ethod (009 where q s a real nuber that s not 0. Applyng the Newton-Raphson s ethod to g( t and epress t agan n ( + q q q R (. q q q 0, here, q, then the Forula (. becoes Newton-Raphson s ethod. Furtherore, we call the net orula general recurrence orula o the Murase-Newton s ethod or general recurrence orula o the Tsuchura-Horguch s ethod. Here q and λ are real nubers that are not 0. ( (,,, q q r + λ q λ 0 q λ R (. The Forula (. swtches n varous recurrence orula by q, λ, r,. In partcular, λ qr, q,, then (. becoes Tsuchura-Horguch s ethod (Re. [5]. a,, n+ be a real nuber. The j-th ter o polynoal o n-th degree Let n j+ s a ( j,, n+ j n n n a + a + a + + a + a+ a (. n n n+. The j-th ter o -th derved uncton ( (,, o n n j( ( n j+ ( n j( n j ( n j( aj ( j,,, n( s ths. (.5 n j ( We replace the coecent ( n j+ ( n j( n j ( n j( aj o wth a orula ncludng a real varable or constant, and denote such ( n (. We can assocate the lne o the ndcaton o calculatng orula o the Horner s ethod, and t becoes substtute a certan real nuber n. For an understandng o the notaton (, see ( wth ( ( when we n the net eaple and Forulas (.-(.5 n Secton. Eaple.. In Murase s orula ( + 8, let ( : 8. I we tae n /, then ( becoes (. Furtherore we tae n /, /, 5/ and n, then ( becoes 0,,, respectvely. These correspond to the second lne o the Table o the calculaton o Horner s ethod n Eaple.. 780

5 We ae net recurrence orula. Ths s equal to Forula (.9 n Secton. Denton.. The orula + 8 ( ( ( q q r + λ λ 0, q 0, q, λ R (.7 s called general recurrence orula o the etenson o Murase-Newton s ethod or general recurrence orula o the etenson o Tsuchura-Horguch s ethod concernng o n-th degree polynoal equaton.. On Relatons between General Recurrence Forula o the Etenson o Tsuchura-Horguch s Method (the Etenson o Murase-Newton s Method and Horner s Method We easly eplan by the net th-degree equaton. Here a,, a6 are real nubers. 5.. Horner s Method or an Epresson (. + a + a + a + a+ a ( Let α be a real nuber. I we apply the Horner s ethod to polynoal Table. Table. Synthetc dvson or an epresson (.., then we obtan the calculaton n a a a a 5 a 6 a a α a α ( ( ( α ( a + α α a + ( a + α α α a + a + ( a + α α α α a + a + a + ( a + α α α α α a + ( a + ( a + α α α a5 + a + ( a + ( a + α α α α + + ( + ( + ( α a ( a α α ( α ( a + α α a + ( a + α α α a + a + ( a + α α α α ( + a + ( a + α α a + a + ( a + α α α a5 + a + a + ( a + 5α α α α α ( a + α α a + ( a + 6α α α + a + ( a + 6α α a + a + ( 6a + 0α α α α ( a + α α a α a + a + 0α α + α a + 5α a a a a a α α α α α.. In the Case o General Fth-Degree Equaton (. ( ( corresponds to the Theore.. There ests,,, so that t equals to +. Furtherore, α and,, +, then 6 -th lne! o Table o Horner s ethod, respectvely. Proo. We should dene the orulas, as ollows.,, ( 78

6 ( : (( ( + + ( a ( a a + a (. 5 ( :! ( ( a a + a (. ( :! ( a a (. ( :! + a (.5 Fro Theore., we obtan the net theore. Theore.. There ests the ratonal recurrence orula + obtaned ro Forula (. so that the denonator equals to (.. Slarly, there ests the ratonal recurrence orula +, +, + so that the denonator equals to (! o (., (! o (., (! o (.5, respectvely. ( Proo. We should choose the orulas +, +, +, + as ollows. + + a + + a + a a a + + a + a + a ( ( ( ( ( (( ( ( ( ( + ( + a + ( a a a ( + + ( + a + a + a ( + + ( a a a a ( + + a + a a a a a 5 6 Furtherore, we obtan the net theore by a sple calculaton. Theore.. The recurrence Forula (.6 obtaned ro Forula (. s equal to general recurrence Forula (.0 o the etenson o Tsuchura-Horguch s ethod o o Forula (.. ( a + + a + a + a5 + a + a + a + a + a + a (( ( ( ( (.6 (.7 (.8 (.9 (.0 Slarly recurrence Forulas (.7-(.9 s equal to general recurrence Forulas (.-(. o the etenson o Tsuchura-Horguch s ethod o,, o Forula (., respectvely. ( + a + a + a + a + a ! ( + + ( + a + a + a ( ( + a + a + a + a + a ! + + a + a ( a (. (. + a + a + a + a + a -! (. 78

7 .. In the Case o Specal Fth-Degree Equatons (. o Murase s Type a + a6 0,,,5 (. We transor the th-degree Equatons (., and obtan the net our recurrence orulas. ( + a a (,,,5 (.5 Because t s a sple atter, we gve only theores wthout proo n the ollowng. Theore.. I α, and,,,, 5, then the denonator a o the recurrence Forula (.5 corresponds to the second lne o the calculaton o Horner s ethod. Slarly, α, ( and,, 6, 0, ( and,, 0, and ( 5 and, 5, then the denonator o recurrence Forula (.5 corresponds to the thrd, orth, and th lne o the calculaton o Horner s ethod, respectvely. Theore.5. The recurrence Forula (.5 (,,,5 s equal to the net general recurrence orula 6 o (.6 o the etenson o Tsuchura-Horguch s ethod o ( a + a + a + a 6 6,,, ( ( ( + ( 6 a Corollary.6. I, then (. becoes the net orula. 5 In ths case, Forula (. becoes the net orula. 6 0 (.6 + a + a (. + a + a + a + a - - ( ( ( + a Forula (.7 s equal to (.6. In the case o,, 5, a slar thng holds, respectvely. Acnowledgeents Dr. Taotsu Tsuchura gave a hnt to e. I a deeply grateul to h. Reerences [] Murase, Y. (67 Sanpouutsudana. Nshda, T., Ed., Kensesha Co., Ltd., Toyo. (n Japanese [] Suzu, T. (00 Wasan no Sertsu. Kousesha Kouseau Co., Ltd., Toyo. (n Japanese [] Nagasaa, H. (980 Coputer and Nuercal Calculatons. Asaura Publshng Co., Ltd., Toyo. (n Japanese [] Horguch, S., Kaneo, T. and Fuj, Y. (0 On Relaton between the Yoshasu Murase s Three Solutons o a Cubc Equaton o Hearth and Horner s Method. The Bulletn o Wasan Insttute,, -8. (n Japanese [5] Horguch, S. (0 General Recurrence Forula Obtaned ro the Murase Yoshasu s Recurrence Forulas and Newton s Method. RIMS, 79, -. (n Japanese 78