高次収束漸化式

この文章について

1. 逆数・平方根・累乗根の高次収束漸化式

• $(1-h)^{-r}$ のマクローリン展開を用いた高次収束漸化式。

1. 自然対数・逆双曲線関数・逆三角関数の高次収束漸化式

• ニュートン法の拡張を用いた高次収束漸化式。

逆数・平方根・累乗根の高次収束漸化式

$(1-h)^{-r}$ のマクローリン展開を用いた高次収束漸化式。 参照:参考資料(1)

$$
\begin{array}{rl}
(1-h)^{-r} \hspace{-1em} \ &= 1+rh+\frac{r(r+1)}{2!}h^2+\frac{r(r+1)(r+2)}{3!}h^3+\frac{r(r+1)(r+2)(r+3)}{4!}h^4+\cdots \\
&= 1+ h \left(r + h \left( \frac{r(r+1)}{2!}+ h \left( \frac{r(r+1)(r+2)}{3!} + h \left( \frac{r(r+1)(r+2)(r+3)}{4!}+\cdots \right) \right) \right) \right) \\
&= 1 + rh \left( 1 + \frac{r+1}{2}h \left( 1 + \frac{r+2}{3}h \left( 1 + \frac{r+3}{4}h \biggl( 1 + \cdots \biggr) \right) \right) \right) \\
\end{array}
$$

$a^{-r}$ のより良い近似値を求めたい場合、

• ${h_n} = 1 - a{x_n}^{1/r}$
• $x_{n+1} = {x_n}(1 + r{h_n} + \frac{r(r+1)}{2!}{h_n}^2 + \frac{r(r+1)(r+2)}{3!}{h_n}^3 + \frac{r(r+1)(r+2)(r+3)}{4!}{h_n}^4 + \cdots ) \simeq {x_n} (1-{h_n})^{-r} = a^{-r}$

とした漸化式によって計算する。$|h_0| \geq 1$ だと収束が保証できないため、初期値 $x_0$ は $|h_0| \ll 1$ となるよう、$a^{-r}$ に十分近い値に定める。

逆数の高次収束漸化式

$$
\begin{array}{rl}
(1-h)^{-1} \hspace{-1em} \ &= 1+h+h^2+h^3+h^4+h^5+h^6+h^7+h^8+\cdots \\
x_0 \hspace{-1em} \ & \simeq 1 / a = a^{-1} \\
{h_n} \hspace{-1em} \ & = 1 - a{x_n} \quad (|{h_n}| \ll 1)
\end{array}
$$

• 2次収束: $x_{n+1} = {x_n} + {x_n}{h_n}$
• 3次収束: $x_{n+1} = {x_n} + {x_n}{h_n} (1 + {h_n})$
• 5次収束: $x_{n+1} = {x_n} + {x_n}{h_n} (1 + {h_n})(1 + {h_n}^2)$
• 7次収束: $x_{n+1} = {x_n} + {x_n}{h_n} (1 + {h_n})(1 + {h_n}^2 + {h_n}^4)$
• 9次収束: $x_{n+1} = {x_n} + {x_n}{h_n} (1 + {h_n})(1 + {h_n}^2)(1 + {h_n}^4)$
• 17次収束: $x_{n+1} = {x_n} + {x_n}{h_n} (1 + {h_n})(1 + {h_n}^2)(1 + {h_n}^4)(1 + {h_n}^8)$
• 33次収束: $x_{n+1} = {x_n} + {x_n}{h_n} (1 + {h_n})(1 + {h_n}^2)(1 + {h_n}^4)(1 + {h_n}^8)(1 + {h_n}^{16})$
• 65次収束: $x_{n+1} = {x_n} + {x_n}{h_n} (1 + {h_n})(1 + {h_n}^2)(1 + {h_n}^4)(1 + {h_n}^8)(1 + {h_n}^{16})(1 + {h_n}^{32})$
• 129次収束: $x_{n+1} = {x_n} + {x_n}{h_n} (1 + {h_n})(1 + {h_n}^2)(1 + {h_n}^4)(1 + {h_n}^8)(1 + {h_n}^{16})(1 + {h_n}^{32})(1 + {h_n}^{64})$
• $k$-次収束: $x_{n+1} = {x_n} \left( \sum_{j=0}^{k-1} {h_n}^j \right)$
• $1/a \simeq x$

逆数平方根の高次収束漸化式

$$
\begin{array}{rl}
(1-h)^{-1/2} \hspace{-1em} \ &= 1+{{h}\over{2}}+{{3,h^2}\over{8}}+{{5,h^3}\over{16}}+{{35,h^4}\over{128}}+{{63,h^5}\over{256}}+{{231,h^6}\over{1024}}+{{429,h^7}\over{2048}}+{{6435,h^8}\over{32768}}+\cdots \\
&= 1 + h \left( \frac{1}{2} + h \left( \frac{3}{8} + h \left( \frac{5}{16} + h \left( \frac{35}{128} + h \biggl( \frac{63}{256} + \cdots \biggr) \right) \right) \right) \right) \\
&= 1 + \frac{1}{2}h \left( 1 + \frac{3}{4}h \left( 1 + \frac{5}{6}h \left( 1 + \frac{7}{8}h \left( 1 + \frac{9}{10}h \biggl( 1 + \cdots \biggr) \right) \right) \right) \right) \\
{x_0} \hspace{-1em} \ & \simeq 1 / \sqrt a = a^{-1/2}\\
{h_n} \hspace{-1em} \ &= 1 - a{{x_n}}^2 \quad (|{h_n}| \ll 1) \\
\end{array}
$$

• 2次収束: $x_{n+1} = {x_n} \left( 1 + \frac{h_n}{2} \right)$
• 3次収束: $x_{n+1} = {x_n} \left( 1 + \frac{1}{2}{h_n} + \frac{3}{8}{h_n}^2 \right)$
• 5次収束: $x_{n+1} = {x_n} \left( 1 + \frac{1}{2}{h_n} + {h_n}^2 \left(\frac{3}{8} + \frac{5}{16}{h_n} + \frac{35}{128}{h_n}^2 \right) \right)$
• 7次収束: $x_{n+1} = {x_n} \left( 1 + \frac{1}{2}{h_n} + {h_n}^2 \left(\frac{3}{8} + \frac{5}{16}{h_n} + {h_n}^2 \left( \frac{35}{128} + \frac{63}{256}{h_n} + \frac{231}{1024}{h_n}^2 \right) \right) \right)$
• 9次収束: $x_{n+1} = {x_n} \left( 1 + \frac{1}{2}{h_n} + {h_n}^2 \left(\frac{3}{8} + \frac{5}{16}{h_n} + {h_n}^2 \left( \frac{35}{128} + \frac{63}{256}{h_n} + {h_n}^2 \left( \frac{231}{1024} + \frac{429}{2048}{h_n} + \frac{6435}{32768}{h_n}^2 \right) \right) \right) \right)$
• $k$-次収束: $x_{n+1} \\= {x_n} \left( 1 + \frac{h_n}{2} \left( 1 + \frac{3{h_n}}{4} \left( 1 + \cdots \left( 1 + \frac{(2k-5){h_n}}{2k-4} \biggl( 1 + \frac{(2k-3){h_n}}{2k-2} \biggr) \right) \cdots \right) \right) \right) \\= {x_n} \left( 1 + \frac{1}{2} {h_n} + \frac{3}{8} {h_n}^2 + \frac{5}{16} {h_n}^3 + \frac{35}{128} {h_n}^4 + \cdots b_{k-1} {h_n}^{k-1} \right) \\= {x_n} \left( \sum_{j=0}^{k-1} b_j {h_n}^j \right) \\ b_j = \frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot \frac{7}{8} \cdots \frac{2j-1}{2j} = \frac{(2j - 1)!!}{(2j)!!} = \frac{ (2j)! }{ 4^j ( j! )^2 }$
• $1/\sqrt{a} \simeq x$
• $\sqrt{a} \simeq ax$

逆数立方根の高次収束漸化式

$$
\begin{array}{rl}
(1-h)^{-1/3} \hspace{-1em} \ &= 1+{{h}\over{3}}+{{2,h^2}\over{9}}+{{14,h^3}\over{81}}+{{35,h^4}\over{243}}+{{91,h^5}\over{729}}+{{728,h^6}\over{6561}}+{{1976,h^7}\over{19683}}+{{5434,h^8}\over{59049}}+\cdots \\
&= 1 + \frac{1}{3}h \left( 1 + \frac{4}{6}h \left( 1 + \frac{7}{9}h \left( 1 + \frac{10}{12}h \left( 1 + \frac{13}{15}h \left( 1 + \frac{16}{18}h \biggl( 1 + \cdots \biggr) \right) \right) \right) \right) \right) \\
x_0 \hspace{-1em} \ & \simeq 1 / \sqrt[3]{a} = a^{-1/3} \\
{h_n} \hspace{-1em} \ &= 1 - a{{x_n}}^3 \quad (|{h_n}| \ll 1) \\
\end{array}
$$

• 2次収束: $x_{n+1} = {x_n} \left( 1 + \frac{1}{3}{h_n} \right)$
• 3次収束: $x_{n+1} = {x_n} \left( 1 + \frac{1}{3}{h_n} + \frac{2}{9} {h_n}^2 \right)$
• 5次収束: $x_{n+1} = {x_n} \left( 1 + \frac{1}{3}{h_n} + {h_n}^2 \left( \frac{2}{9} + \frac{14}{81}{h_n} + \frac{35}{243} {h_n}^2 \right) \right)$
• 7次収束: $x_{n+1} = {x_n} \left( 1 + \frac{1}{3}{h_n} + {h_n}^2 \left( \frac{2}{9} + \frac{14}{81}{h_n} + {h_n}^2 \left( \frac{35}{243} + \frac{91}{729}{h_n} + \frac{728}{6561} {h_n}^2 \right) \right) \right)$
• 9次収束: $x_{n+1} = {x_n} \left( 1 + \frac{1}{3}{h_n} + {h_n}^2 \left( \frac{2}{9} + \frac{14}{81}{h_n} + {h_n}^2 \left( \frac{35}{243} + \frac{91}{729}{h_n} + {h_n}^2 \left( \frac{728}{6561} + \frac{1976}{19683}{h_n} + \frac{5434}{59049} {h_n}^2 \right) \right) \right) \right)$
• $1/\sqrt[3]{a} \simeq x$
• $\sqrt[3]{a} \simeq ax^2$

逆数四乗根の高次収束漸化式

$$
\begin{array}{rl}
(1-h)^{-1/4} \hspace{-1em} \ &= 1+{{h}\over{4}}+{{5,h^2}\over{32}}+{{15,h^3}\over{128}}+{{195,h^4}\over{2048}}+{{663,h^5}\over{8192}}+{{4641,h^6}\over{65536}}+{{16575,h^7}\over{262144}}+{{480675,h^8}\over{8388608}}+\cdots \\
&= 1 + \frac{1}{4}h \left( 1 + \frac{5}{8}h \left( 1 + \frac{9}{12}h \left( 1 + \frac{13}{16}h \left( 1 + \frac{17}{20}h \left( 1 + \frac{21}{24}h \biggl( 1 + \cdots \biggr) \right) \right) \right) \right) \right) \\
x_0 \hspace{-1em} \ & \simeq 1 / \sqrt[4]{a} = a^{-1/4} \\
{h_n} \hspace{-1em} \ &= 1 - a{{x_n}}^4 \quad (|{h_n}| \ll 1)
\end{array}
$$

• 2次収束: $x_{n+1} = {x_n} \left( 1 + \frac{1}{4} {h_n} \right)$
• 3次収束: $x_{n+1} = {x_n} \left( 1 + \frac{1}{4} {h_n} + \frac{5}{32} {h_n}^2 \right)$
• 5次収束: $x_{n+1} = {x_n} \left( 1 + \frac{1}{4} {h_n} + {h_n}^2 \left( \frac{5}{32} + \frac{15}{128} {h_n} + \frac{195}{2048} {h_n}^2 \right) \right)$
• 7次収束: $x_{n+1} = {x_n} \left( 1 + \frac{1}{4} {h_n} + {h_n}^2 \left( \frac{5}{32} + \frac{15}{128} {h_n} + {h_n}^2 \left( \frac{195}{2048} + \frac{663}{8192} {h_n} + \frac{4641}{65536} {h_n}^2 \right) \right) \right)$
• 9次収束: $x_{n+1} = {x_n} \left( 1 + \frac{1}{4} {h_n} + {h_n}^2 \left( \frac{5}{32} + \frac{15}{128} {h_n} + {h_n}^2 \left( \frac{195}{2048} + \frac{663}{8192} {h_n} + {h_n}^2 \left( \frac{4641}{65536} + \frac{16575}{262144} {h_n} + \frac{480675}{8388608} {h_n}^2 \right) \right) \right) \right)$
• $1/\sqrt[4]{a} \simeq x$
• $\sqrt[4]{a} \simeq ax^3$

平方根の高次収束漸化式

$$
\begin{array}{rl}
(1-h)^{1/2} \hspace{-1em} \ &= 1-\frac{h}{2}-\frac{h^2}{8}-\frac{h^3}{16}-\frac{5,h^4}{128}-\frac{7,h^5}{256}-\frac{21,h^6}{1024}-\frac{33,h^7}{2048}-\frac{429,h^8}{32768}-\cdots \\
&= 1 - \frac{1}{2}h \left( 1 + \frac{1}{4}h \left( 1 + \frac{3}{6}h \left( 1 + \frac{5}{8}h \left( 1 + \frac{7}{10}h \left( 1 + \frac{9}{12}h \biggl( 1 + \cdots \biggr) \right) \right) \right) \right) \right) \\
x_0 \hspace{-1em} \ & \simeq \sqrt{a} = a^{1/2} \\
{h_n} \hspace{-1em} \ &= 1 - \frac{a}{{{x_n}}^2} = \frac{{{x_n}}^2 - a}{{{x_n}}^2} \quad (|{h_n}| \ll 1) \\
\end{array}
$$

• 2次収束: $x_{n+1} = {x_n} \left( 1 - \frac{1}{2} {h_n} \right)$
• 3次収束: $x_{n+1} = {x_n} \left( 1 - \frac{1}{2} {h_n} - \frac{1}{8} {h_n}^2 \right)$
• 5次収束: $x_{n+1} = {x_n} \left( 1 - \frac{1}{2} {h_n} - {h_n}^2 \left( \frac{1}{8} + \frac{1}{16} {h_n} + \frac{5}{128} {h_n}^2 \right) \right)$
• 7次収束: $x_{n+1} = {x_n} \left( 1 - \frac{1}{2} {h_n} - {h_n}^2 \left( \frac{1}{8} + \frac{1}{16} {h_n} + {h_n}^2 \left( \frac{5}{128} + \frac{7}{256} {h_n} + \frac{21}{1024} {h_n}^2 \right) \right) \right)$
• 9次収束: $x_{n+1} = {x_n} \left( 1 - \frac{1}{2} {h_n} - {h_n}^2 \left( \frac{1}{8} + \frac{1}{16} {h_n} + {h_n}^2 \left( \frac{5}{128} + \frac{7}{256} {h_n} + {h_n}^2 \left( \frac{21}{1024} + \frac{33}{2048} {h_n} + \frac{429}{32768} {h_n}^2 \right) \right) \right) \right)$
• $\sqrt{a} \simeq x$

ニュートン法の拡張による高次収束漸化式

注: 以下の定理の説明は仮のものです。誤りを含む可能性があります。 参照:参考資料(2)(3)(4)(5)(6)

定理:

十分に微分可能な１変数の実関数 $f$

$$
f : D \subset \mathbb{R} \to \mathbb{R}
$$

が 点 $a$ において、

$$
\begin{array}{l}
f(a)=0, \\
f'(a) \neq 0, \\
f''(a) = 0, \ f'''(a) = 0, \ ... \ , \ f^{(n-1)}(a)=0, \\
f^{(n)}(a) \neq 0.
\end{array}
$$

補足: 例えば、

• $f(a) = 0, \ f'(a) \neq 0, \ f''(a) \neq 0$ であれば $n = 2$
• $f(a) = 0, \ f'(a) \neq 0, \ f''(a) = 0, \ f'''(a) \neq 0$ であれば $n = 3$

とする。

の性質を持つとき、$\forall m \ge n$ において定義される 関数 $Q_m(x)$ を

$$
\begin{array}{l}
Q_n(x)=1, \\
\displaystyle Q_{m+1}(x) = Q_m(x)f'(x) − \frac{1}{m − 1} Q'_m(x)f(x), \quad m \ge n
\end{array}
$$

とすると、次の漸化式

$$
x_{k+1} = x_k − \frac{Q_m(x_k)}{Q_{m+1}(x_k)}f(x_k), \quad k = 0, 1, 2,...
$$

で定義される数列 ${x_k}$ は

$$
|x_{k+1} − a| \le C |x_k − a|^m
$$

を満たす、零でない 定数 $C$ が存在し、 ${x_k}$ は $a$ に少なくとも $m$次収束 する。

以下の例では漸化式の算出に 数式処理システム Maxima を利用しており、 Maxima に投入したコマンドも参考までに併記する。 参照: 参考資料(7)(8)(9)

自然対数の高次収束漸化式

maxima_highorder_convergence_log

[kill(all),reset(),array([Q],N:9),Q[1]:1,G:log(a),H:exp(x),
printf(true,"~%  - $~a$~%  - $~a$~%",tex1(f(x,a)=F:H-a),tex1(g(a)=G)),for m
:2 thru N do([Q[m]:diff(F,x,1)*Q[m-1]-F*diff(Q[m-1],x,1)/(m-1),printf(true,
"- $~a\\\\\\\\=~a\\\\\\\\=~a\\\\\\\\~a$~%",tex1(R[m](x,h)-x),tex1(factor(U:
trigsimp(subst(H-h,a,V:-F*Q[m-1]/Q[m])))),tex1(trigrat(U)),strim(" ",tex1(M
[m](a,E:epsilon)=taylor(subst(G+E,x,V+x),E,0,m+3))))]),printf(true,"~%~%")]$

• $k$-次収束漸化式:
  • $x_{n+1} = R_k({x_n}, {h_n}), \quad h_n = f(x_n, a)$
  • $f(g(a), a) = 0$
  • $M_k(a,\varepsilon) = R_k(g(a)+\varepsilon,f(g(a)+\varepsilon,a))$
  • $\varepsilon \simeq 0 \quad \Longrightarrow \quad M_k(a,\varepsilon) \simeq g(a)$
  • $f\left(x , a\right)=e^{x}-a$
  • $g\left(a\right)=\log a$
• $R_{2}(x,h)-x\\=-h,e^ {- x }\\=-h,e^ {- x }\\M_{2}(a,\varepsilon)=\log a+{{\varepsilon^2}\over{2}}-{{\varepsilon^3}\over{6}}+{{\varepsilon^4}\over{24}}-{{\varepsilon^5}\over{120}}+\cdots$
• $R_{3}(x,h)-x\\=-{{2,h}\over{2,e^{x}-h}}\\=-{{2,h}\over{2,e^{x}-h}}\\M_{3}(a,\varepsilon)=\log a+{{\varepsilon^3}\over{12}}-{{\varepsilon^5}\over{120}}+\cdots$
• $R_{4}(x,h)-x\\=-{{3,h,\left(2,e^{x}-h\right)}\over{6,e^{2,x}-6,h,e^{x}+h^2}}\\=-{{6,h,e^{x}-3,h^2}\over{6,e^{2,x}-6,h,e^{x}+h^2}}\\M_{4}(a,\varepsilon)=\log a+{{\varepsilon^5}\over{180}}-{{\varepsilon^7}\over{1512}}+\cdots$
  • ※ $R_{4}(x,h)$ は 5次収束
• $R_{5}(x,h)-x\\=-{{4,h,\left(6,e^{2,x}-6,h,e^{x}+h^2\right)}\over{\left(2,e^{x}-h\right),\left(12,e^{2,x}-12,h,e^{x}+h^2\right)}}\\=-{{24,h,e^{2,x}-24,h^2,e^{x}+4,h^3}\over{24,e^{3,x}-36,h,e^{2,x}+14,h^2,e^{x}-h^3}}\\M_{5}(a,\varepsilon)=\log a-{{\varepsilon^5}\over{720}}+{{\varepsilon^7}\over{2016}}+\cdots$
• $R_{6}(x,h)-x\\=-{{5,h,\left(2,e^{x}-h\right),\left(12,e^{2,x}-12,h,e^{x}+h^2\right)}\over{120,e^{4,x}-240,h,e^{3,x}+150,h^2,e^{2,x}-30,h^3,e^{x}+h^4}}\\=-{{120,h,e^{3,x}-180,h^2,e^{2,x}+70,h^3,e^{x}-5,h^4}\over{120,e^{4,x}-240,h,e^{3,x}+150,h^2,e^{2,x}-30,h^3,e^{x}+h^4}}\\M_{6}(a,\varepsilon)=\log a-{{\varepsilon^7}\over{5040}}+{{\varepsilon^9}\over{21600}}+\cdots$
  • ※ $R_{6}(x,h)$ は 7次収束
• $R_{7}(x,h)-x\\=-{{6,h,\left(120,e^{4,x}-240,h,e^{3,x}+150,h^2,e^{2,x}-30,h^3,e^{x}+h^4\right)}\over{\left(2,e^{x}-h\right),\left(360,e^{4,x}-720,h,e^{3,x}+420,h^2,e^{2,x}-60,h^3,e^{x}+h^4\right)}}\\=-{{720,h,e^{4,x}-1440,h^2,e^{3,x}+900,h^3,e^{2,x}-180,h^4,e^{x}+6,h^5}\over{720,e^{5,x}-1800,h,e^{4,x}+1560,h^2,e^{3,x}-540,h^3,e^{2,x}+62,h^4,e^{x}-h^5}}\\M_{7}(a,\varepsilon)=\log a+{{\varepsilon^7}\over{30240}}-{{\varepsilon^9}\over{43200}}+\cdots$
• $R_{8}(x,h)-x\\=-{{7,h,\left(2,e^{x}-h\right),\left(360,e^{4,x}-720,h,e^{3,x}+420,h^2,e^{2,x}-60,h^3,e^{x}+h^4\right)}\over{5040,e^{6,x}-15120,h,e^{5,x}+16800,h^2,e^{4,x}-8400,h^3,e^{3,x}+1806,h^4,e^{2,x}-126,h^5,e^{x}+h^6}}\\=-{{5040,h,e^{5,x}-12600,h^2,e^{4,x}+10920,h^3,e^{3,x}-3780,h^4,e^{2,x}+434,h^5,e^{x}-7,h^6}\over{5040,e^{6,x}-15120,h,e^{5,x}+16800,h^2,e^{4,x}-8400,h^3,e^{3,x}+1806,h^4,e^{2,x}-126,h^5,e^{x}+h^6}}\\M_{8}(a,\varepsilon)=\log a+{{\varepsilon^9}\over{151200}}-{{\varepsilon^{11}}\over{399168}}+\cdots$
  • ※ $R_{8}(x,h)$ は 9次収束
• $R_{9}(x,h)-x\\=-{{8,h,\left(5040,e^{6,x}-15120,h,e^{5,x}+16800,h^2,e^{4,x}-8400,h^3,e^{3,x}+1806,h^4,e^{2,x}-126,h^5,e^{x}+h^6\right)}\over{\left(2,e^{x}-h\right),\left(20160,e^{6,x}-60480,h,e^{5,x}+65520,h^2,e^{4,x}-30240,h^3,e^{3,x}+5292,h^4,e^{2,x}-252,h^5,e^{x}+h^6\right)}}\\=-{{40320,h,e^{6,x}-120960,h^2,e^{5,x}+134400,h^3,e^{4,x}-67200,h^4,e^{3,x}+14448,h^5,e^{2,x}-1008,h^6,e^{x}+8,h^7}\over{40320,e^{7,x}-141120,h,e^{6,x}+191520,h^2,e^{5,x}-126000,h^3,e^{4,x}+40824,h^4,e^{3,x}-5796,h^5,e^{2,x}+254,h^6,e^{x}-h^7}}\\M_{9}(a,\varepsilon)=\log a-{{\varepsilon^9}\over{1209600}}+{{\varepsilon^{11}}\over{1064448}}+\cdots$

逆双曲線関数の高次収束漸化式

asinh

maxima_highorder_convergence_asinh

[kill(all),reset(),array([Q],N:9),Q[1]:1,G:asinh(a),H:sinh(x),
printf(true,"~%  - $~a$~%  - $~a$~%",tex1(f(x,a)=F:H-a),tex1(g(a)=G)),for m
:2 thru N do([Q[m]:diff(F,x,1)*Q[m-1]-F*diff(Q[m-1],x,1)/(m-1),printf(true,
"- $~a\\\\\\\\=~a\\\\\\\\=~a\\\\\\\\~a$~%",tex1(R[m](x,h)-x),tex1(factor(U:
trigsimp(subst(H-h,a,V:-F*Q[m-1]/Q[m])))),tex1(trigrat(U)),strim(" ",tex1(M
[m](a,E:epsilon)=taylor(subst(G+E,x,V+x),E,0,m+3))))]),printf(true,"~%~%")]$

• $k$-次収束漸化式: $x_{n+1} = R_k({x_n}, {h_n}), \quad h_n = f(x_n, a)$
  • $f(g(a), a) = 0$
  • $M_k(a,\varepsilon) = R_k(g(a)+\varepsilon,f(g(a)+\varepsilon,a))$
  • $\varepsilon \simeq 0 \quad \Longrightarrow \quad M_k(a,\varepsilon) \simeq g(a)$
  • $f\left(x , a\right)=\sinh x-a$
  • $g\left(a\right)={\rm asinh}; a$
• $R_{2}(x,h)-x\\=-{{h}\over{\cosh x}}\\=-{{2,h,e^{x}}\over{e^{2,x}+1}}\\M_{2}(a,\varepsilon)={\rm asinh}; a+{{\sqrt{a^2+1},a,\varepsilon^2}\over{2,a^2+2}}-{{\left(a^2-2\right),\varepsilon^3}\over{6,a^2+6}}+{{\left(\sqrt{a^2+1},a^3-11,\sqrt{a^2+1},a\right),\varepsilon^4}\over{24,a^4+48,a^2+24}}-{{\left(a^4-43,a^2+16\right),\varepsilon^5}\over{120,a^4+240,a^2+120}}+\cdots$
• $R_{3}(x,h)-x\\={{2,h,\cosh x}\over{h,\sinh x-2,\cosh ^2x}}\\=-{{2,h,e^{3,x}+2,h,e^{x}}\over{e^{4,x}-h,e^{3,x}+2,e^{2,x}+h,e^{x}+1}}\\M_{3}(a,\varepsilon)={\rm asinh}; a+{{\left(a^2-2\right),\varepsilon^3}\over{12,a^2+12}}+{{3,\sqrt{a^2+1},a,\varepsilon^4}\over{8,a^4+16,a^2+8}}-{{\left(a^4+32,a^2-14\right),\varepsilon^5}\over{120,a^4+240,a^2+120}}+{{\left(3,\sqrt{a^2+1},a^3-6,\sqrt{a^2+1},a\right),\varepsilon^6}\over{32,a^6+96,a^4+96,a^2+32}}+\cdots$
• $R_{4}(x,h)-x\\=-{{3,h,\left(h,\sinh x-2,\cosh ^2x\right)}\over{\cosh x,\left(6,h,\sinh x-6,\cosh ^2x-h^2\right)}}\\=-{{6,h,e^{5,x}-6,h^2,e^{4,x}+12,h,e^{3,x}+6,h^2,e^{2,x}+6,h,e^{x}}\over{3,e^{6,x}-6,h,e^{5,x}+\left(2,h^2+9\right),e^{4,x}+\left(2,h^2+9\right),e^{2,x}+6,h,e^{x}+3}}\\M_{4}(a,\varepsilon)={\rm asinh}; a-{{\sqrt{a^2+1},a,\varepsilon^4}\over{8,a^4+16,a^2+8}}+{{\left(a^4+32,a^2-14\right),\varepsilon^5}\over{180,a^4+360,a^2+180}}-{{\left(5,a^3-10,a\right),\varepsilon^6}\over{48,\sqrt{a^2+1},a^4+96,\sqrt{a^2+1},a^2+48,\sqrt{a^2+1}}}-{{\left(2,a^6-120,a^4+699,a^2-124\right),\varepsilon^7}\over{3024,a^6+9072,a^4+9072,a^2+3024}}+\cdots$
• $R_{5}(x,h)-x\\=-{{4,h,\cosh x,\left(6,h,\sinh x-6,\cosh ^2x-h^2\right)}\over{36,h,\cosh ^2x,\sinh x+h^3,\sinh x-24,\cosh ^4x-14,h^2,\cosh ^2x+6,h^2}}\\=-{{6,h,e^{7,x}-12,h^2,e^{6,x}+\left(4,h^3+18,h\right),e^{5,x}+\left(4,h^3+18,h\right),e^{3,x}+12,h^2,e^{2,x}+6,h,e^{x}}\over{3,e^{8,x}-9,h,e^{7,x}+\left(7,h^2+12\right),e^{6,x}+\left(-h^3-9,h\right),e^{5,x}+\left(2,h^2+18\right),e^{4,x}+\left(h^3+9,h\right),e^{3,x}+\left(7,h^2+12\right),e^{2,x}+9,h,e^{x}+3}}\\M_{5}(a,\varepsilon)={\rm asinh}; a-{{\left(a^4+32,a^2-14\right),\varepsilon^5}\over{720,a^4+1440,a^2+720}}+{{\left(5,\sqrt{a^2+1},a^3-10,\sqrt{a^2+1},a\right),\varepsilon^6}\over{96,a^6+288,a^4+288,a^2+96}}+{{\left(2,a^6-120,a^4+699,a^2-124\right),\varepsilon^7}\over{4032,a^6+12096,a^4+12096,a^2+4032}}+{{\left(14,\sqrt{a^2+1},a^5-182,\sqrt{a^2+1},a^3+119,\sqrt{a^2+1},a\right),\varepsilon^8}\over{1152,a^8+4608,a^6+6912,a^4+4608,a^2+1152}}+\cdots$
• $R_{6}(x,h)-x\\=-{{5,h,\left(36,h,\cosh ^2x,\sinh x+h^3,\sinh x-24,\cosh ^4x-14,h^2,\cosh ^2x+6,h^2\right)}\over{\cosh x,\left(240,h,\cosh ^2x,\sinh x+30,h^3,\sinh x-120,\cosh ^4x-150,h^2,\cosh ^2x-h^4+90,h^2\right)}}\\=-{{30,h,e^{9,x}-90,h^2,e^{8,x}+\left(70,h^3+120,h\right),e^{7,x}+\left(-10,h^4-90,h^2\right),e^{6,x}+\left(20,h^3+180,h\right),e^{5,x}+\left(10,h^4+90,h^2\right),e^{4,x}+\left(70,h^3+120,h\right),e^{3,x}+90,h^2,e^{2,x}+30,h,e^{x}}\over{15,e^{10,x}-60,h,e^{9,x}+\left(75,h^2+75\right),e^{8,x}+\left(-30,h^3-120,h\right),e^{7,x}+\left(2,h^4+45,h^2+150\right),e^{6,x}+\left(2,h^4+45,h^2+150\right),e^{4,x}+\left(30,h^3+120,h\right),e^{3,x}+\left(75,h^2+75\right),e^{2,x}+60,h,e^{x}+15}}\\M_{6}(a,\varepsilon)={\rm asinh}; a-{{\left(a^3-2,a\right),\varepsilon^6}\over{96,\sqrt{a^2+1},a^4+192,\sqrt{a^2+1},a^2+96,\sqrt{a^2+1}}}-{{\left(2,a^6-120,a^4+699,a^2-124\right),\varepsilon^7}\over{10080,a^6+30240,a^4+30240,a^2+10080}}-{{\left(14,\sqrt{a^2+1},a^5-182,\sqrt{a^2+1},a^3+119,\sqrt{a^2+1},a\right),\varepsilon^8}\over{1920,a^8+7680,a^6+11520,a^4+7680,a^2+1920}}+{{\left(a^8+64,a^6-1704,a^4+2704,a^2-254\right),\varepsilon^9}\over{21600,a^8+86400,a^6+129600,a^4+86400,a^2+21600}}+\cdots$
• $R_{7}(x,h)-x\\=-{{6,h,\cosh x,\left(240,h,\cosh ^2x,\sinh x+30,h^3,\sinh x-120,\cosh ^4x-150,h^2,\cosh ^2x-h^4+90,h^2\right)}\over{1800,h,\cosh ^4x,\sinh x+540,h^3,\cosh ^2x,\sinh x+h^5,\sinh x-90,h^3,\sinh x-720,\cosh ^6x-1560,h^2,\cosh ^4x-62,h^4,\cosh ^2x+1080,h^2,\cosh ^2x+30,h^4}}\\=-{{90,h,e^{11,x}-360,h^2,e^{10,x}+\left(450,h^3+450,h\right),e^{9,x}+\left(-180,h^4-720,h^2\right),e^{8,x}+\left(12,h^5+270,h^3+900,h\right),e^{7,x}+\left(12,h^5+270,h^3+900,h\right),e^{5,x}+\left(180,h^4+720,h^2\right),e^{4,x}+\left(450,h^3+450,h\right),e^{3,x}+360,h^2,e^{2,x}+90,h,e^{x}}\over{45,e^{12,x}-225,h,e^{11,x}+\left(390,h^2+270\right),e^{10,x}+\left(-270,h^3-675,h\right),e^{9,x}+\left(62,h^4+480,h^2+675\right),e^{8,x}+\left(-2,h^5-90,h^3-450,h\right),e^{7,x}+\left(4,h^4+180,h^2+900\right),e^{6,x}+\left(2,h^5+90,h^3+450,h\right),e^{5,x}+\left(62,h^4+480,h^2+675\right),e^{4,x}+\left(270,h^3+675,h\right),e^{3,x}+\left(390,h^2+270\right),e^{2,x}+225,h,e^{x}+45}}\\M_{7}(a,\varepsilon)={\rm asinh}; a+{{\left(2,a^6-120,a^4+699,a^2-124\right),\varepsilon^7}\over{60480,a^6+181440,a^4+181440,a^2+60480}}+{{\left(14,\sqrt{a^2+1},a^5-182,\sqrt{a^2+1},a^3+119,\sqrt{a^2+1},a\right),\varepsilon^8}\over{5760,a^8+23040,a^6+34560,a^4+23040,a^2+5760}}-{{\left(a^8+64,a^6-1704,a^4+2704,a^2-254\right),\varepsilon^9}\over{43200,a^8+172800,a^6+259200,a^4+172800,a^2+43200}}+{{\left(\sqrt{a^2+1},a^7-60,\sqrt{a^2+1},a^5+192,\sqrt{a^2+1},a^3-62,\sqrt{a^2+1},a\right),\varepsilon^{10}}\over{1920,a^{10}+9600,a^8+19200,a^6+19200,a^4+9600,a^2+1920}}+\cdots$
• $R_{8}(x,h)-x\\=-{{7,h,\left(1800,h,\cosh ^4x,\sinh x+540,h^3,\cosh ^2x,\sinh x+h^5,\sinh x-90,h^3,\sinh x-720,\cosh ^6x-1560,h^2,\cosh ^4x-62,h^4,\cosh ^2x+1080,h^2,\cosh ^2x+30,h^4\right)}\over{\cosh x,\left(15120,h,\cosh ^4x,\sinh x+8400,h^3,\cosh ^2x,\sinh x+126,h^5,\sinh x-2520,h^3,\sinh x-5040,\cosh ^6x-16800,h^2,\cosh ^4x-1806,h^4,\cosh ^2x+12600,h^2,\cosh ^2x-h^6+1260,h^4\right)}}\\=-{{630,h,e^{13,x}-3150,h^2,e^{12,x}+\left(5460,h^3+3780,h\right),e^{11,x}+\left(-3780,h^4-9450,h^2\right),e^{10,x}+\left(868,h^5+6720,h^3+9450,h\right),e^{9,x}+\left(-28,h^6-1260,h^4-6300,h^2\right),e^{8,x}+\left(56,h^5+2520,h^3+12600,h\right),e^{7,x}+\left(28,h^6+1260,h^4+6300,h^2\right),e^{6,x}+\left(868,h^5+6720,h^3+9450,h\right),e^{5,x}+\left(3780,h^4+9450,h^2\right),e^{4,x}+\left(5460,h^3+3780,h\right),e^{3,x}+3150,h^2,e^{2,x}+630,h,e^{x}}\over{315,e^{14,x}-1890,h,e^{13,x}+\left(4200,h^2+2205\right),e^{12,x}+\left(-4200,h^3-7560,h\right),e^{11,x}+\left(1806,h^4+8400,h^2+6615\right),e^{10,x}+\left(-252,h^5-3360,h^3-9450,h\right),e^{9,x}+\left(4,h^6+378,h^4+4200,h^2+11025\right),e^{8,x}+\left(4,h^6+378,h^4+4200,h^2+11025\right),e^{6,x}+\left(252,h^5+3360,h^3+9450,h\right),e^{5,x}+\left(1806,h^4+8400,h^2+6615\right),e^{4,x}+\left(4200,h^3+7560,h\right),e^{3,x}+\left(4200,h^2+2205\right),e^{2,x}+1890,h,e^{x}+315}}\\M_{8}(a,\varepsilon)={\rm asinh}; a-{{\left(2,\sqrt{a^2+1},a^5-26,\sqrt{a^2+1},a^3+17,\sqrt{a^2+1},a\right),\varepsilon^8}\over{5760,a^8+23040,a^6+34560,a^4+23040,a^2+5760}}+{{\left(a^8+64,a^6-1704,a^4+2704,a^2-254\right),\varepsilon^9}\over{151200,a^8+604800,a^6+907200,a^4+604800,a^2+151200}}-{{\left(a^7-60,a^5+192,a^3-62,a\right),\varepsilon^{10}}\over{4480,\sqrt{a^2+1},a^8+17920,\sqrt{a^2+1},a^6+26880,\sqrt{a^2+1},a^4+17920,\sqrt{a^2+1},a^2+4480,\sqrt{a^2+1}}}-{{\left(2,a^{10}-56,a^8+8171,a^6-47896,a^4+35386,a^2-2044\right),\varepsilon^{11}}\over{798336,a^{10}+3991680,a^8+7983360,a^6+7983360,a^4+3991680,a^2+798336}}+\cdots$
• $R_{9}(x,h)-x\\=-{{8,h,\cosh x,\left(15120,h,\cosh ^4x,\sinh x+8400,h^3,\cosh ^2x,\sinh x+126,h^5,\sinh x-2520,h^3,\sinh x-5040,\cosh ^6x-16800,h^2,\cosh ^4x-1806,h^4,\cosh ^2x+12600,h^2,\cosh ^2x-h^6+1260,h^4\right)}\over{141120,h,\cosh ^6x,\sinh x+126000,h^3,\cosh ^4x,\sinh x+5796,h^5,\cosh ^2x,\sinh x-50400,h^3,\cosh ^2x,\sinh x+h^7,\sinh x-1260,h^5,\sinh x-40320,\cosh ^8x-191520,h^2,\cosh ^6x-40824,h^4,\cosh ^4x+151200,h^2,\cosh ^4x-254,h^6,\cosh ^2x+35280,h^4,\cosh ^2x+126,h^6-2520,h^4}}\\=-{{630,h,e^{15,x}-3780,h^2,e^{14,x}+\left(8400,h^3+4410,h\right),e^{13,x}+\left(-8400,h^4-15120,h^2\right),e^{12,x}+\left(3612,h^5+16800,h^3+13230,h\right),e^{11,x}+\left(-504,h^6-6720,h^4-18900,h^2\right),e^{10,x}+\left(8,h^7+756,h^5+8400,h^3+22050,h\right),e^{9,x}+\left(8,h^7+756,h^5+8400,h^3+22050,h\right),e^{7,x}+\left(504,h^6+6720,h^4+18900,h^2\right),e^{6,x}+\left(3612,h^5+16800,h^3+13230,h\right),e^{5,x}+\left(8400,h^4+15120,h^2\right),e^{4,x}+\left(8400,h^3+4410,h\right),e^{3,x}+3780,h^2,e^{2,x}+630,h,e^{x}}\over{315,e^{16,x}-2205,h,e^{15,x}+\left(5985,h^2+2520\right),e^{14,x}+\left(-7875,h^3-11025,h\right),e^{13,x}+\left(5103,h^4+17010,h^2+8820\right),e^{12,x}+\left(-1449,h^5-11025,h^3-19845,h\right),e^{11,x}+\left(127,h^6+2772,h^4+14175,h^2+17640\right),e^{10,x}+\left(-h^7-189,h^5-3150,h^3-11025,h\right),e^{9,x}+\left(2,h^6+378,h^4+6300,h^2+22050\right),e^{8,x}+\left(h^7+189,h^5+3150,h^3+11025,h\right),e^{7,x}+\left(127,h^6+2772,h^4+14175,h^2+17640\right),e^{6,x}+\left(1449,h^5+11025,h^3+19845,h\right),e^{5,x}+\left(5103,h^4+17010,h^2+8820\right),e^{4,x}+\left(7875,h^3+11025,h\right),e^{3,x}+\left(5985,h^2+2520\right),e^{2,x}+2205,h,e^{x}+315}}\\M_{9}(a,\varepsilon)={\rm asinh}; a-{{\left(a^8+64,a^6-1704,a^4+2704,a^2-254\right),\varepsilon^9}\over{1209600,a^8+4838400,a^6+7257600,a^4+4838400,a^2+1209600}}+{{\left(\sqrt{a^2+1},a^7-60,\sqrt{a^2+1},a^5+192,\sqrt{a^2+1},a^3-62,\sqrt{a^2+1},a\right),\varepsilon^{10}}\over{17920,a^{10}+89600,a^8+179200,a^6+179200,a^4+89600,a^2+17920}}+{{\left(2,a^{10}-56,a^8+8171,a^6-47896,a^4+35386,a^2-2044\right),\varepsilon^{11}}\over{2128896,a^{10}[phone],a^8+21288960,a^6+21288960,a^4+10644480,a^2+2128896}}+{{\left(22,\sqrt{a^2+1},a^9-5522,\sqrt{a^2+1},a^7+56067,\sqrt{a^2+1},a^5-79112,\sqrt{a^2+1},a^3+15202,\sqrt{a^2+1},a\right),\varepsilon^{12}}\over{1935360,a^{12}[phone],a^{10}[phone],a^8+38707200,a^6+29030400,a^4+11612160,a^2+1935360}}+\cdots$

acosh

maxima_highorder_convergence_acosh

[kill(all),reset(),array([Q],N:9),Q[1]:1,G:acosh(a),H:cosh(x),
printf(true,"~%  - $~a$~%  - $~a$~%",tex1(f(x,a)=F:H-a),tex1(g(a)=G)),for m
:2 thru N do([Q[m]:diff(F,x,1)*Q[m-1]-F*diff(Q[m-1],x,1)/(m-1),printf(true,
"- $~a\\\\\\\\=~a\\\\\\\\=~a\\\\\\\\~a$~%",tex1(R[m](x,h)-x),tex1(factor(U:
trigsimp(subst(H-h,a,V:-F*Q[m-1]/Q[m])))),tex1(trigrat(U)),strim(" ",tex1(M
[m](a,E:epsilon)=taylor(subst(G+E,x,V+x),E,0,m+3))))]),printf(true,"~%~%")]$

• $k$-次収束漸化式: $x_{n+1} = R_k({x_n}, {h_n}), \quad h_n = f(x_n, a)$
  • $f(g(a), a) = 0$
  • $M_k(a,\varepsilon) = R_k(g(a)+\varepsilon,f(g(a)+\varepsilon,a))$
  • $\varepsilon \simeq 0 \quad \Longrightarrow \quad M_k(a,\varepsilon) \simeq g(a)$
  • $f\left(x , a\right)=\cosh x-a$
  • $g\left(a\right)={\rm acosh}; a$
• $R_{2}(x,h)-x\\=-{{h}\over{\sinh x}}\\=-{{2,h,e^{x}}\over{e^{2,x}-1}}\\M_{2}(a,\varepsilon)={\rm acosh}; a+{{\sqrt{a+1},\sqrt{a-1},a,\varepsilon^2}\over{2,a^2-2}}-{{\left(a^2+2\right),\varepsilon^3}\over{6,a^2-6}}+{{\left(\sqrt{a+1},\sqrt{a-1},a^3+11,\sqrt{a+1},\sqrt{a-1},a\right),\varepsilon^4}\over{24,a^4-48,a^2+24}}-{{\left(a^4+43,a^2+16\right),\varepsilon^5}\over{120,a^4-240,a^2+120}}+\cdots$
• $R_{3}(x,h)-x\\=-{{2,h,\sinh x}\over{2,\sinh ^2x-h,\cosh x}}\\=-{{2,h,e^{3,x}-2,h,e^{x}}\over{e^{4,x}-h,e^{3,x}-2,e^{2,x}-h,e^{x}+1}}\\M_{3}(a,\varepsilon)={\rm acosh}; a+{{\left(a^2+2\right),\varepsilon^3}\over{12,a^2-12}}-{{3,\sqrt{a+1},\sqrt{a-1},a,\varepsilon^4}\over{8,a^4-16,a^2+8}}-{{\left(a^4-32,a^2-14\right),\varepsilon^5}\over{120,a^4-240,a^2+120}}-{{\left(3,\sqrt{a+1},\sqrt{a-1},a^3+6,\sqrt{a+1},\sqrt{a-1},a\right),\varepsilon^6}\over{32,a^6-96,a^4+96,a^2-32}}+\cdots$
• $R_{4}(x,h)-x\\=-{{3,h,\left(2,\sinh ^2x-h,\cosh x\right)}\over{\left(6,\cosh ^2x-6,h,\cosh x+h^2-6\right),\sinh x}}\\=-{{6,h,e^{5,x}-6,h^2,e^{4,x}-12,h,e^{3,x}-6,h^2,e^{2,x}+6,h,e^{x}}\over{3,e^{6,x}-6,h,e^{5,x}+\left(2,h^2-9\right),e^{4,x}+\left(9-2,h^2\right),e^{2,x}+6,h,e^{x}-3}}\\M_{4}(a,\varepsilon)={\rm acosh}; a+{{\sqrt{a+1},\sqrt{a-1},a,\varepsilon^4}\over{8,a^4-16,a^2+8}}+{{\left(a^4-32,a^2-14\right),\varepsilon^5}\over{180,a^4-360,a^2+180}}+{{\left(5,a^3+10,a\right),\varepsilon^6}\over{48,\sqrt{a+1},\sqrt{a-1},a^4-96,\sqrt{a+1},\sqrt{a-1},a^2+48,\sqrt{a+1},\sqrt{a-1}}}-{{\left(2,a^6+120,a^4+699,a^2+124\right),\varepsilon^7}\over{3024,a^6-9072,a^4+9072,a^2-3024}}+\cdots$
• $R_{5}(x,h)-x\\=-{{4,h,\sinh x,\left(6,\sinh ^2x-6,h,\cosh x+h^2\right)}\over{24,\sinh ^4x-36,h,\cosh x,\sinh ^2x+14,h^2,\sinh ^2x-h^3,\cosh x+6,h^2}}\\=-{{6,h,e^{7,x}-12,h^2,e^{6,x}+\left(4,h^3-18,h\right),e^{5,x}+\left(18,h-4,h^3\right),e^{3,x}+12,h^2,e^{2,x}-6,h,e^{x}}\over{3,e^{8,x}-9,h,e^{7,x}+\left(7,h^2-12\right),e^{6,x}+\left(9,h-h^3\right),e^{5,x}+\left(18-2,h^2\right),e^{4,x}+\left(9,h-h^3\right),e^{3,x}+\left(7,h^2-12\right),e^{2,x}-9,h,e^{x}+3}}\\M_{5}(a,\varepsilon)={\rm acosh}; a-{{\left(a^4-32,a^2-14\right),\varepsilon^5}\over{720,a^4-1440,a^2+720}}-{{\left(5,\sqrt{a+1},\sqrt{a-1},a^3+10,\sqrt{a+1},\sqrt{a-1},a\right),\varepsilon^6}\over{96,a^6-288,a^4+288,a^2-96}}+{{\left(2,a^6+120,a^4+699,a^2+124\right),\varepsilon^7}\over{4032,a^6-12096,a^4+12096,a^2-4032}}-{{\left(14,\sqrt{a+1},\sqrt{a-1},a^5+182,\sqrt{a+1},\sqrt{a-1},a^3+119,\sqrt{a+1},\sqrt{a-1},a\right),\varepsilon^8}\over{1152,a^8-4608,a^6+6912,a^4-4608,a^2+1152}}+\cdots$
• $R_{6}(x,h)-x\\=-{{5,h,\left(24,\sinh ^4x-36,h,\cosh x,\sinh ^2x+14,h^2,\sinh ^2x-h^3,\cosh x+6,h^2\right)}\over{\sinh x,\left(120,\sinh ^4x-240,h,\cosh x,\sinh ^2x+150,h^2,\sinh ^2x-30,h^3,\cosh x+h^4+90,h^2\right)}}\\=-{{30,h,e^{9,x}-90,h^2,e^{8,x}+\left(70,h^3-120,h\right),e^{7,x}+\left(90,h^2-10,h^4\right),e^{6,x}+\left(180,h-20,h^3\right),e^{5,x}+\left(90,h^2-10,h^4\right),e^{4,x}+\left(70,h^3-120,h\right),e^{3,x}-90,h^2,e^{2,x}+30,h,e^{x}}\over{15,e^{10,x}-60,h,e^{9,x}+\left(75,h^2-75\right),e^{8,x}+\left(120,h-30,h^3\right),e^{7,x}+\left(2,h^4-45,h^2+150\right),e^{6,x}+\left(-2,h^4+45,h^2-150\right),e^{4,x}+\left(30,h^3-120,h\right),e^{3,x}+\left(75-75,h^2\right),e^{2,x}+60,h,e^{x}-15}}\\M_{6}(a,\varepsilon)={\rm acosh}; a+{{\left(a^3+2,a\right),\varepsilon^6}\over{96,\sqrt{a+1},\sqrt{a-1},a^4-192,\sqrt{a+1},\sqrt{a-1},a^2+96,\sqrt{a+1},\sqrt{a-1}}}-{{\left(2,a^6+120,a^4+699,a^2+124\right),\varepsilon^7}\over{10080,a^6-30240,a^4+30240,a^2-10080}}+{{\left(14,\sqrt{a+1},\sqrt{a-1},a^5+182,\sqrt{a+1},\sqrt{a-1},a^3+119,\sqrt{a+1},\sqrt{a-1},a\right),\varepsilon^8}\over{1920,a^8-7680,a^6+11520,a^4-7680,a^2+1920}}+{{\left(a^8-64,a^6-1704,a^4-2704,a^2-254\right),\varepsilon^9}\over{21600,a^8-86400,a^6+129600,a^4-86400,a^2+21600}}+\cdots$
• $R_{7}(x,h)-x\\=-{{6,h,\sinh x,\left(120,\sinh ^4x-240,h,\cosh x,\sinh ^2x+150,h^2,\sinh ^2x-30,h^3,\cosh x+h^4+90,h^2\right)}\over{720,\sinh ^6x-1800,h,\cosh x,\sinh ^4x+1560,h^2,\sinh ^4x-540,h^3,\cosh x,\sinh ^2x+62,h^4,\sinh ^2x+1080,h^2,\sinh ^2x-h^5,\cosh x-90,h^3,\cosh x+30,h^4}}\\=-{{90,h,e^{11,x}-360,h^2,e^{10,x}+\left(450,h^3-450,h\right),e^{9,x}+\left(720,h^2-180,h^4\right),e^{8,x}+\left(12,h^5-270,h^3+900,h\right),e^{7,x}+\left(-12,h^5+270,h^3-900,h\right),e^{5,x}+\left(180,h^4-720,h^2\right),e^{4,x}+\left(450,h-450,h^3\right),e^{3,x}+360,h^2,e^{2,x}-90,h,e^{x}}\over{45,e^{12,x}-225,h,e^{11,x}+\left(390,h^2-270\right),e^{10,x}+\left(675,h-270,h^3\right),e^{9,x}+\left(62,h^4-480,h^2+675\right),e^{8,x}+\left(-2,h^5+90,h^3-450,h\right),e^{7,x}+\left(-4,h^4+180,h^2-900\right),e^{6,x}+\left(-2,h^5+90,h^3-450,h\right),e^{5,x}+\left(62,h^4-480,h^2+675\right),e^{4,x}+\left(675,h-270,h^3\right),e^{3,x}+\left(390,h^2-270\right),e^{2,x}-225,h,e^{x}+45}}\\M_{7}(a,\varepsilon)={\rm acosh}; a+{{\left(2,a^6+120,a^4+699,a^2+124\right),\varepsilon^7}\over{60480,a^6-181440,a^4+181440,a^2-60480}}-{{\left(14,\sqrt{a+1},\sqrt{a-1},a^5+182,\sqrt{a+1},\sqrt{a-1},a^3+119,\sqrt{a+1},\sqrt{a-1},a\right),\varepsilon^8}\over{5760,a^8-23040,a^6+34560,a^4-23040,a^2+5760}}-{{\left(a^8-64,a^6-1704,a^4-2704,a^2-254\right),\varepsilon^9}\over{43200,a^8-172800,a^6+259200,a^4-172800,a^2+43200}}-{{\left(\sqrt{a+1},\sqrt{a-1},a^7+60,\sqrt{a+1},\sqrt{a-1},a^5+192,\sqrt{a+1},\sqrt{a-1},a^3+62,\sqrt{a+1},\sqrt{a-1},a\right),\varepsilon^{10}}\over{1920,a^{10}-9600,a^8+19200,a^6-19200,a^4+9600,a^2-1920}}+\cdots$
• $R_{8}(x,h)-x\\=-{{7,h,\left(720,\sinh ^6x-1800,h,\cosh x,\sinh ^4x+1560,h^2,\sinh ^4x-540,h^3,\cosh x,\sinh ^2x+62,h^4,\sinh ^2x+1080,h^2,\sinh ^2x-h^5,\cosh x-90,h^3,\cosh x+30,h^4\right)}\over{\sinh x,\left(5040,\sinh ^6x-15120,h,\cosh x,\sinh ^4x+16800,h^2,\sinh ^4x-8400,h^3,\cosh x,\sinh ^2x+1806,h^4,\sinh ^2x+12600,h^2,\sinh ^2x-126,h^5,\cosh x-2520,h^3,\cosh x+h^6+1260,h^4\right)}}\\=-{{630,h,e^{13,x}-3150,h^2,e^{12,x}+\left(5460,h^3-3780,h\right),e^{11,x}+\left(9450,h^2-3780,h^4\right),e^{10,x}+\left(868,h^5-6720,h^3+9450,h\right),e^{9,x}+\left(-28,h^6+1260,h^4-6300,h^2\right),e^{8,x}+\left(-56,h^5+2520,h^3-12600,h\right),e^{7,x}+\left(-28,h^6+1260,h^4-6300,h^2\right),e^{6,x}+\left(868,h^5-6720,h^3+9450,h\right),e^{5,x}+\left(9450,h^2-3780,h^4\right),e^{4,x}+\left(5460,h^3-3780,h\right),e^{3,x}-3150,h^2,e^{2,x}+630,h,e^{x}}\over{315,e^{14,x}-1890,h,e^{13,x}+\left(4200,h^2-2205\right),e^{12,x}+\left(7560,h-4200,h^3\right),e^{11,x}+\left(1806,h^4-8400,h^2+6615\right),e^{10,x}+\left(-252,h^5+3360,h^3-9450,h\right),e^{9,x}+\left(4,h^6-378,h^4+4200,h^2-11025\right),e^{8,x}+\left(-4,h^6+378,h^4-4200,h^2+11025\right),e^{6,x}+\left(252,h^5-3360,h^3+9450,h\right),e^{5,x}+\left(-1806,h^4+8400,h^2-6615\right),e^{4,x}+\left(4200,h^3-7560,h\right),e^{3,x}+\left(2205-4200,h^2\right),e^{2,x}+1890,h,e^{x}-315}}\\M_{8}(a,\varepsilon)={\rm acosh}; a+{{\left(2,\sqrt{a+1},\sqrt{a-1},a^5+26,\sqrt{a+1},\sqrt{a-1},a^3+17,\sqrt{a+1},\sqrt{a-1},a\right),\varepsilon^8}\over{5760,a^8-23040,a^6+34560,a^4-23040,a^2+5760}}+{{\left(a^8-64,a^6-1704,a^4-2704,a^2-254\right),\varepsilon^9}\over{151200,a^8-604800,a^6+907200,a^4-604800,a^2+151200}}+{{\left(a^7+60,a^5+192,a^3+62,a\right),\varepsilon^{10}}\over{4480,\sqrt{a+1},\sqrt{a-1},a^8-17920,\sqrt{a+1},\sqrt{a-1},a^6+26880,\sqrt{a+1},\sqrt{a-1},a^4-17920,\sqrt{a+1},\sqrt{a-1},a^2+4480,\sqrt{a+1},\sqrt{a-1}}}-{{\left(2,a^{10}+56,a^8+8171,a^6+47896,a^4+35386,a^2+2044\right),\varepsilon^{11}}\over{798336,a^{10}-3991680,a^8+7983360,a^6-7983360,a^4+3991680,a^2-798336}}+\cdots$
• $R_{9}(x,h)-x\\=-{{8,h,\sinh x,\left(5040,\sinh ^6x-15120,h,\cosh x,\sinh ^4x+16800,h^2,\sinh ^4x-8400,h^3,\cosh x,\sinh ^2x+1806,h^4,\sinh ^2x+12600,h^2,\sinh ^2x-126,h^5,\cosh x-2520,h^3,\cosh x+h^6+1260,h^4\right)}\over{40320,\sinh ^8x-141120,h,\cosh x,\sinh ^6x+191520,h^2,\sinh ^6x-126000,h^3,\cosh x,\sinh ^4x+40824,h^4,\sinh ^4x+151200,h^2,\sinh ^4x-5796,h^5,\cosh x,\sinh ^2x-50400,h^3,\cosh x,\sinh ^2x+254,h^6,\sinh ^2x+35280,h^4,\sinh ^2x-h^7,\cosh x-1260,h^5,\cosh x+126,h^6+2520,h^4}}\\=-{{630,h,e^{15,x}-3780,h^2,e^{14,x}+\left(8400,h^3-4410,h\right),e^{13,x}+\left(15120,h^2-8400,h^4\right),e^{12,x}+\left(3612,h^5-16800,h^3+13230,h\right),e^{11,x}+\left(-504,h^6+6720,h^4-18900,h^2\right),e^{10,x}+\left(8,h^7-756,h^5+8400,h^3-22050,h\right),e^{9,x}+\left(-8,h^7+756,h^5-8400,h^3+22050,h\right),e^{7,x}+\left(504,h^6-6720,h^4+18900,h^2\right),e^{6,x}+\left(-3612,h^5+16800,h^3-13230,h\right),e^{5,x}+\left(8400,h^4-15120,h^2\right),e^{4,x}+\left(4410,h-8400,h^3\right),e^{3,x}+3780,h^2,e^{2,x}-630,h,e^{x}}\over{315,e^{16,x}-2205,h,e^{15,x}+\left(5985,h^2-2520\right),e^{14,x}+\left(11025,h-7875,h^3\right),e^{13,x}+\left(5103,h^4-17010,h^2+8820\right),e^{12,x}+\left(-1449,h^5+11025,h^3-19845,h\right),e^{11,x}+\left(127,h^6-2772,h^4+14175,h^2-17640\right),e^{10,x}+\left(-h^7+189,h^5-3150,h^3+11025,h\right),e^{9,x}+\left(-2,h^6+378,h^4-6300,h^2+22050\right),e^{8,x}+\left(-h^7+189,h^5-3150,h^3+11025,h\right),e^{7,x}+\left(127,h^6-2772,h^4+14175,h^2-17640\right),e^{6,x}+\left(-1449,h^5+11025,h^3-19845,h\right),e^{5,x}+\left(5103,h^4-17010,h^2+8820\right),e^{4,x}+\left(11025,h-7875,h^3\right),e^{3,x}+\left(5985,h^2-2520\right),e^{2,x}-2205,h,e^{x}+315}}\\M_{9}(a,\varepsilon)={\rm acosh}; a-{{\left(a^8-64,a^6-1704,a^4-2704,a^2-254\right),\varepsilon^9}\over{1209600,a^8-4838400,a^6+7257600,a^4-4838400,a^2+1209600}}-{{\left(\sqrt{a+1},\sqrt{a-1},a^7+60,\sqrt{a+1},\sqrt{a-1},a^5+192,\sqrt{a+1},\sqrt{a-1},a^3+62,\sqrt{a+1},\sqrt{a-1},a\right),\varepsilon^{10}}\over{17920,a^{10}-89600,a^8+179200,a^6-179200,a^4+89600,a^2-17920}}+{{\left(2,a^{10}+56,a^8+8171,a^6+47896,a^4+35386,a^2+2044\right),\varepsilon^{11}}\over{2128896,a^{10}-10644480,a^8+21288960,a^6-21288960,a^4+10644480,a^2-2128896}}-{{\left(22,\sqrt{a+1},\sqrt{a-1},a^9+5522,\sqrt{a+1},\sqrt{a-1},a^7+56067,\sqrt{a+1},\sqrt{a-1},a^5+79112,\sqrt{a+1},\sqrt{a-1},a^3+15202,\sqrt{a+1},\sqrt{a-1},a\right),\varepsilon^{12}}\over{1935360,a^{12}-11612160,a^{10}[phone],a^8-38707200,a^6+29030400,a^4-11612160,a^2+1935360}}+\cdots$

atanh

maxima_highorder_convergence_atanh

[kill(all),reset(),array([Q],N:9),Q[1]:1,G:atanh(a),H:tanh(x),
printf(true,"~%  - $~a$~%  - $~a$~%",tex1(f(x,a)=F:H-a),tex1(g(a)=G)),for m
:2 thru N do([Q[m]:diff(F,x,1)*Q[m-1]-F*diff(Q[m-1],x,1)/(m-1),printf(true,
"- $~a\\\\\\\\=~a\\\\\\\\=~a\\\\\\\\~a$~%",tex1(R[m](x,h)-x),tex1(factor(U:
trigsimp(subst(H-h,a,V:-F*Q[m-1]/Q[m])))),tex1(trigrat(U)),strim(" ",tex1(M
[m](a,E:epsilon)=taylor(subst(G+E,x,V+x),E,0,m+3))))]),printf(true,"~%~%")]$

• $k$-次収束漸化式: $x_{n+1} = R_k({x_n}, {h_n}), \quad h_n = f(x_n, a)$
  • $f(g(a), a) = 0$
  • $M_k(a,\varepsilon) = R_k(g(a)+\varepsilon,f(g(a)+\varepsilon,a))$
  • $\varepsilon \simeq 0 \quad \Longrightarrow \quad M_k(a,\varepsilon) \simeq g(a)$
  • $f\left(x , a\right)=\tanh x-a$
  • $g\left(a\right)={\rm atanh}; a$
• $R_{2}(x,h)-x\\=-h,\cosh ^2x\\=-{{e^ {- 2,x },\left(h,e^{4,x}+2,h,e^{2,x}+h\right)}\over{4}}\\M_{2}(a,\varepsilon)={\rm atanh}; a-a,\varepsilon^2-{{2,\varepsilon^3}\over{3}}-{{a,\varepsilon^4}\over{3}}-{{2,\varepsilon^5}\over{15}}+\cdots$
• $R_{3}(x,h)-x\\=-{{h,\cosh ^2x}\over{h,\cosh x,\sinh x+1}}\\=-{{h,e^{4,x}+2,h,e^{2,x}+h}\over{h,e^{4,x}+4,e^{2,x}-h}}\\M_{3}(a,\varepsilon)={\rm atanh}; a+{{\varepsilon^3}\over{3}}-{{2,\varepsilon^5}\over{15}}+\cdots$
• $R_{4}(x,h)-x\\=-{{3,h,\cosh ^2x,\left(h,\cosh x,\sinh x+1\right)}\over{6,h,\cosh x,\sinh x+2,h^2,\cosh ^4x-3,h^2,\cosh ^2x+3}}\\=-{{3,h^2,e^{8,x}+\left(6,h^2+12,h\right),e^{6,x}+24,h,e^{4,x}+\left(12,h-6,h^2\right),e^{2,x}-3,h^2}\over{2,h^2,e^{8,x}+\left(24,h-4,h^2\right),e^{6,x}+\left(48-12,h^2\right),e^{4,x}+\left(-4,h^2-24,h\right),e^{2,x}+2,h^2}}\\M_{4}(a,\varepsilon)={\rm atanh}; a+{{4,\varepsilon^5}\over{45}}-{{8,\varepsilon^7}\over{189}}+\cdots$
  • ※ $R_{4}(x,h)$ は 5次収束
• $R_{5}(x,h)-x\\=-{{h,\cosh ^2x,\left(6,h,\cosh x,\sinh x+2,h^2,\cosh ^4x-3,h^2,\cosh ^2x+3\right)}\over{h^3,\cosh ^5x,\sinh x-3,h^3,\cosh ^3x,\sinh x+9,h,\cosh x,\sinh x+7,h^2,\cosh ^4x-9,h^2,\cosh ^2x+3}}\\=-{{2,h^3,e^{12,x}+24,h^2,e^{10,x}+\left(-18,h^3+48,h^2+48,h\right),e^{8,x}+\left(96,h-32,h^3\right),e^{6,x}+\left(-18,h^3-48,h^2+48,h\right),e^{4,x}-24,h^2,e^{2,x}+2,h^3}\over{h^3,e^{12,x}+\left(28,h^2-8,h^3\right),e^{10,x}+\left(-19,h^3-32,h^2+144,h\right),e^{8,x}+\left(192-120,h^2\right),e^{6,x}+\left(19,h^3-32,h^2-144,h\right),e^{4,x}+\left(8,h^3+28,h^2\right),e^{2,x}-h^3}}\\M_{5}(a,\varepsilon)={\rm atanh}; a-{{\varepsilon^5}\over{45}}+{{2,\varepsilon^7}\over{63}}+\cdots$
• $R_{6}(x,h)-x\\=-{{5,h,\cosh ^2x,\left(h^3,\cosh ^5x,\sinh x-3,h^3,\cosh ^3x,\sinh x+9,h,\cosh x,\sinh x+7,h^2,\cosh ^4x-9,h^2,\cosh ^2x+3\right)}\over{30,h^3,\cosh ^5x,\sinh x-60,h^3,\cosh ^3x,\sinh x+60,h,\cosh x,\sinh x+2,h^4,\cosh ^8x-15,h^4,\cosh ^6x+15,h^4,\cosh ^4x+75,h^2,\cosh ^4x-90,h^2,\cosh ^2x+15}}\\=-{{5,h^4,e^{16,x}+\left(140,h^3-30,h^4\right),e^{14,x}+\left(-170,h^4+120,h^3+720,h^2\right),e^{12,x}+\left(-230,h^4-780,h^3+1440,h^2+960,h\right),e^{10,x}+\left(1920,h-1520,h^3\right),e^{8,x}+\left(230,h^4-780,h^3-1440,h^2+960,h\right),e^{6,x}+\left(170,h^4+120,h^3-720,h^2\right),e^{4,x}+\left(30,h^4+140,h^3\right),e^{2,x}-5,h^4}\over{2,h^4,e^{16,x}+\left(120,h^3-44,h^4\right),e^{14,x}+\left(-64,h^4-480,h^3+1200,h^2\right),e^{12,x}+\left(172,h^4-1320,h^3-960,h^2+3840,h\right),e^{10,x}+\left(380,h^4-4320,h^2+3840\right),e^{8,x}+\left(172,h^4+1320,h^3-960,h^2-3840,h\right),e^{6,x}+\left(-64,h^4+480,h^3+1200,h^2\right),e^{4,x}+\left(-44,h^4-120,h^3\right),e^{2,x}+2,h^4}}\\M_{6}(a,\varepsilon)={\rm atanh}; a-{{4,\varepsilon^7}\over{315}}+{{8,\varepsilon^9}\over{675}}+\cdots$
  • ※ $R_{6}(x,h)$ は 7次収束
• $R_{7}(x,h)-x\\=-{{3,h,\cosh ^2x,\left(30,h^3,\cosh ^5x,\sinh x-60,h^3,\cosh ^3x,\sinh x+60,h,\cosh x,\sinh x+2,h^4,\cosh ^8x-15,h^4,\cosh ^6x+15,h^4,\cosh ^4x+75,h^2,\cosh ^4x-90,h^2,\cosh ^2x+15\right)}\over{2,h^5,\cosh ^9x,\sinh x-30,h^5,\cosh ^7x,\sinh x+45,h^5,\cosh ^5x,\sinh x+270,h^3,\cosh ^5x,\sinh x-450,h^3,\cosh ^3x,\sinh x+225,h,\cosh x,\sinh x+62,h^4,\cosh ^8x-270,h^4,\cosh ^6x+225,h^4,\cosh ^4x+390,h^2,\cosh ^4x-450,h^2,\cosh ^2x+45}}\\=-{{3,h^5,e^{20,x}+\left(180,h^4-60,h^5\right),e^{18,x}+\left(-225,h^5-360,h^4+1800,h^3\right),e^{16,x}+\left(-3240,h^4+2160,h^3+5760,h^2\right),e^{14,x}+\left(990,h^5-4680,h^4-7560,h^3+11520,h^2+5760,h\right),e^{12,x}+\left(1656,h^5-15840,h^3+11520,h\right),e^{10,x}+\left(990,h^5+4680,h^4-7560,h^3-11520,h^2+5760,h\right),e^{8,x}+\left(3240,h^4+2160,h^3-5760,h^2\right),e^{6,x}+\left(-225,h^5+360,h^4+1800,h^3\right),e^{4,x}+\left(-60,h^5-180,h^4\right),e^{2,x}+3,h^5}\over{h^5,e^{20,x}+\left(124,h^4-52,h^5\right),e^{18,x}+\left(27,h^5-1168,h^4+2160,h^3\right),e^{16,x}+\left(648,h^5-2288,h^4-5760,h^3+12480,h^2\right),e^{14,x}+\left(1002,h^5+3344,h^4-18000,h^3-7680,h^2+28800,h\right),e^{12,x}+\left(8680,h^4-40320,h^2+23040\right),e^{10,x}+\left(-1002,h^5+3344,h^4+18000,h^3-7680,h^2-28800,h\right),e^{8,x}+\left(-648,h^5-2288,h^4+5760,h^3+12480,h^2\right),e^{6,x}+\left(-27,h^5-1168,h^4-2160,h^3\right),e^{4,x}+\left(52,h^5+124,h^4\right),e^{2,x}-h^5}}\\M_{7}(a,\varepsilon)={\rm atanh}; a+{{2,\varepsilon^7}\over{945}}-{{4,\varepsilon^9}\over{675}}+\cdots$
• $R_{8}(x,h)-x\\=-{{7,h,\cosh ^2x,\left(2,h^5,\cosh ^9x,\sinh x-30,h^5,\cosh ^7x,\sinh x+45,h^5,\cosh ^5x,\sinh x+270,h^3,\cosh ^5x,\sinh x-450,h^3,\cosh ^3x,\sinh x+225,h,\cosh x,\sinh x+62,h^4,\cosh ^8x-270,h^4,\cosh ^6x+225,h^4,\cosh ^4x+390,h^2,\cosh ^4x-450,h^2,\cosh ^2x+45\right)}\over{252,h^5,\cosh ^9x,\sinh x-1680,h^5,\cosh ^7x,\sinh x+1890,h^5,\cosh ^5x,\sinh x+4200,h^3,\cosh ^5x,\sinh x-6300,h^3,\cosh ^3x,\sinh x+1890,h,\cosh x,\sinh x+4,h^6,\cosh ^{12}x-126,h^6,\cosh ^{10}x+420,h^6,\cosh ^8x+1806,h^4,\cosh ^8x-315,h^6,\cosh ^6x-6300,h^4,\cosh ^6x+4725,h^4,\cosh ^4x+4200,h^2,\cosh ^4x-4725,h^2,\cosh ^2x+315}}\\=-{{7,h^6,e^{24,x}+\left(868,h^5-350,h^6\right),e^{22,x}+\left(-532,h^6-6440,h^5+15120,h^4\right),e^{20,x}+\left(4550,h^6-31500,h^5-10080,h^4+87360,h^3\right),e^{18,x}+\left(16275,h^6-16800,h^5-191520,h^4+120960,h^3+201600,h^2\right),e^{16,x}+\left(18564,h^6+91560,h^5-292320,h^4-302400,h^3+403200,h^2+161280,h\right),e^{14,x}+\left(168336,h^5-672000,h^3+322560,h\right),e^{12,x}+\left(-18564,h^6+91560,h^5+292320,h^4-302400,h^3-403200,h^2+161280,h\right),e^{10,x}+\left(-16275,h^6-16800,h^5+191520,h^4+120960,h^3-201600,h^2\right),e^{8,x}+\left(-4550,h^6-31500,h^5+10080,h^4+87360,h^3\right),e^{6,x}+\left(532,h^6-6440,h^5-15120,h^4\right),e^{4,x}+\left(350,h^6+868,h^5\right),e^{2,x}-7,h^6}\over{2,h^6,e^{24,x}+\left(504,h^5-228,h^6\right),e^{22,x}+\left(972,h^6-9408,h^5+14448,h^4\right),e^{20,x}+\left(5900,h^6-6552,h^5-86016,h^4+134400,h^3\right),e^{18,x}+\left(4350,h^6+77952,h^5-200256,h^4-268800,h^3+537600,h^2\right),e^{16,x}+\left(-14376,h^6+135408,h^5+204288,h^4-940800,h^3-268800,h^2+967680,h\right),e^{14,x}+\left(-28056,h^6+608160,h^4-1612800,h^2+645120\right),e^{12,x}+\left(-14376,h^6-135408,h^5+204288,h^4+940800,h^3-268800,h^2-967680,h\right),e^{10,x}+\left(4350,h^6-77952,h^5-200256,h^4+268800,h^3+537600,h^2\right),e^{8,x}+\left(5900,h^6+6552,h^5-86016,h^4-134400,h^3\right),e^{6,x}+\left(972,h^6+9408,h^5+14448,h^4\right),e^{4,x}+\left(-228,h^6-504,h^5\right),e^{2,x}+2,h^6}}\\M_{8}(a,\varepsilon)={\rm atanh}; a+{{8,\varepsilon^9}\over{4725}}-{{16,\varepsilon^{11}}\over{6237}}+\cdots$
  • ※ $R_{8}(x,h)$ は 9次収束
• $R_{9}(x,h)-x\\=-{{h,\cosh ^2x,\left(252,h^5,\cosh ^9x,\sinh x-1680,h^5,\cosh ^7x,\sinh x+1890,h^5,\cosh ^5x,\sinh x+4200,h^3,\cosh ^5x,\sinh x-6300,h^3,\cosh ^3x,\sinh x+1890,h,\cosh x,\sinh x+4,h^6,\cosh ^{12}x-126,h^6,\cosh ^{10}x+420,h^6,\cosh ^8x+1806,h^4,\cosh ^8x-315,h^6,\cosh ^6x-6300,h^4,\cosh ^6x+4725,h^4,\cosh ^4x+4200,h^2,\cosh ^4x-4725,h^2,\cosh ^2x+315\right)}\over{h^7,\cosh ^{13}x,\sinh x-63,h^7,\cosh ^{11}x,\sinh x+315,h^7,\cosh ^9x,\sinh x+1449,h^5,\cosh ^9x,\sinh x-315,h^7,\cosh ^7x,\sinh x-6930,h^5,\cosh ^7x,\sinh x+6615,h^5,\cosh ^5x,\sinh x+7875,h^3,\cosh ^5x,\sinh x-11025,h^3,\cosh ^3x,\sinh x+2205,h,\cosh x,\sinh x+127,h^6,\cosh ^{12}x-1449,h^6,\cosh ^{10}x+3465,h^6,\cosh ^8x+5103,h^4,\cosh ^8x-2205,h^6,\cosh ^6x-15750,h^4,\cosh ^6x+11025,h^4,\cosh ^4x+5985,h^2,\cosh ^4x-6615,h^2,\cosh ^2x+315}}\\=-{{4,h^7,e^{28,x}+\left(1008,h^6-448,h^7\right),e^{26,x}+\left(1036,h^7-16800,h^6+28896,h^5\right),e^{24,x}+\left(15232,h^7-49728,h^6-114240,h^5+268800,h^4\right),e^{22,x}+\left(34244,h^7+110880,h^6-715680,h^5+1075200,h^3\right),e^{20,x}+\left(448,h^7+569520,h^6-564480,h^5-2688000,h^4+1612800,h^3+1935360,h^2\right),e^{18,x}+\left(-104916,h^7+697536,h^6+1632960,h^5-4300800,h^4-3225600,h^3+3870720,h^2+1290240,h\right),e^{16,x}+\left(-169728,h^7+3249792,h^5-7526400,h^3+2580480,h\right),e^{14,x}+\left(-104916,h^7-697536,h^6+1632960,h^5+4300800,h^4-3225600,h^3-3870720,h^2+1290240,h\right),e^{12,x}+\left(448,h^7-569520,h^6-564480,h^5+2688000,h^4+1612800,h^3-1935360,h^2\right),e^{10,x}+\left(34244,h^7-110880,h^6-715680,h^5+1075200,h^3\right),e^{8,x}+\left(15232,h^7+49728,h^6-114240,h^5-268800,h^4\right),e^{6,x}+\left(1036,h^7+16800,h^6+28896,h^5\right),e^{4,x}+\left(-448,h^7-1008,h^6\right),e^{2,x}+4,h^7}\over{h^7,e^{28,x}+\left(508,h^6-240,h^7\right),e^{26,x}+\left(2585,h^7-17088,h^6+23184,h^5\right),e^{24,x}+\left(9280,h^7+23448,h^6-258048,h^5+326592,h^4\right),e^{22,x}+\left(-12171,h^7+278080,h^6-341712,h^5-1419264,h^4+2016000,h^3\right),e^{20,x}+\left(-81328,h^7+291780,h^6+1677312,h^5-3757824,h^4-3225600,h^3+6128640,h^2\right),e^{18,x}+\left(-103395,h^7-514944,h^6+3231648,h^5+2967552,h^4-12499200,h^3-2580480,h^2+9031680,h\right),e^{16,x}+\left(-1139376,h^6+9959040,h^4-17418240,h^2+5160960\right),e^{14,x}+\left(103395,h^7-514944,h^6-3231648,h^5+2967552,h^4+12499200,h^3-2580480,h^2-9031680,h\right),e^{12,x}+\left(81328,h^7+291780,h^6-1677312,h^5-3757824,h^4+3225600,h^3+6128640,h^2\right),e^{10,x}+\left(12171,h^7+278080,h^6+341712,h^5-1419264,h^4-2016000,h^3\right),e^{8,x}+\left(-9280,h^7+23448,h^6+258048,h^5+326592,h^4\right),e^{6,x}+\left(-2585,h^7-17088,h^6-23184,h^5\right),e^{4,x}+\left(240,h^7+508,h^6\right),e^{2,x}-h^7}}\\M_{9}(a,\varepsilon)={\rm atanh}; a-{{\varepsilon^9}\over{4725}}+{{2,\varepsilon^{11}}\over{2079}}+\cdots$

逆三角関数の高次収束漸化式

arcsin

maxima_highorder_convergence_asin

[kill(all),reset(),array([Q],N:9),Q[1]:1,G:asin(a),H:sin(x),
printf(true,"~%  - $~a$~%  - $~a$~%",tex1(f(x,a)=F:H-a),tex1(g(a)=G)),for m
:2 thru N do([Q[m]:diff(F,x,1)*Q[m-1]-F*diff(Q[m-1],x,1)/(m-1),printf(true,
"- $~a\\\\\\\\=~a\\\\\\\\=~a\\\\\\\\~a$~%",tex1(R[m](x,h)-x),tex1(factor(U:
trigsimp(subst(H-h,a,V:-F*Q[m-1]/Q[m])))),tex1(trigrat(U)),strim(" ",tex1(M
[m](a,E:epsilon)=taylor(subst(G+E,x,V+x),E,0,m+3))))]),printf(true,"~%~%")]$

• $k$-次収束漸化式: $x_{n+1} = R_k({x_n}, {h_n}), \quad h_n = f(x_n, a)$
  • $f(g(a), a) = 0$
  • $M_k(a,\varepsilon) = R_k(g(a)+\varepsilon,f(g(a)+\varepsilon,a))$
  • $\varepsilon \simeq 0 \quad \Longrightarrow \quad M_k(a,\varepsilon) \simeq g(a)$
  • $f\left(x , a\right)=\sin x-a$
  • $g\left(a\right)=\arcsin a$
• $R_{2}(x,h)-x\\=-{{h}\over{\cos x}}\\=-{{h}\over{\cos x}}\\M_{2}(a,\varepsilon)=\arcsin a+{{\sqrt{-a^2+1},a,\varepsilon^2}\over{2,a^2-2}}+{{\left(a^2+2\right),\varepsilon^3}\over{6,a^2-6}}-{{\left(\sqrt{-a^2+1},a^3+11,\sqrt{-a^2+1},a\right),\varepsilon^4}\over{24,a^4-48,a^2+24}}-{{\left(a^4+43,a^2+16\right),\varepsilon^5}\over{120,a^4-240,a^2+120}}+\cdots$
• $R_{3}(x,h)-x\\=-{{2,h,\cos x}\over{h,\sin x+2,\cos ^2x}}\\=-{{2,h,\cos x}\over{\cos \left(2,x\right)+h,\sin x+1}}\\M_{3}(a,\varepsilon)=\arcsin a-{{\left(a^2+2\right),\varepsilon^3}\over{12,a^2-12}}+{{3,\sqrt{-a^2+1},a,\varepsilon^4}\over{8,a^4-16,a^2+8}}-{{\left(a^4-32,a^2-14\right),\varepsilon^5}\over{120,a^4-240,a^2+120}}-{{\left(3,\sqrt{-a^2+1},a^3+6,\sqrt{-a^2+1},a\right),\varepsilon^6}\over{32,a^6-96,a^4+96,a^2-32}}+\cdots$
• $R_{4}(x,h)-x\\=-{{3,h,\left(h,\sin x+2,\cos ^2x\right)}\over{\cos x,\left(6,h,\sin x+6,\cos ^2x-h^2\right)}}\\=-{{6,h,\cos \left(2,x\right)+6,h^2,\sin x+6,h}\over{3,\cos \left(3,x\right)+6,h,\sin \left(2,x\right)+\left(9-2,h^2\right),\cos x}}\\M_{4}(a,\varepsilon)=\arcsin a-{{\sqrt{-a^2+1},a,\varepsilon^4}\over{8,a^4-16,a^2+8}}+{{\left(a^4-32,a^2-14\right),\varepsilon^5}\over{180,a^4-360,a^2+180}}-{{\left(5,a^3+10,a\right),\varepsilon^6}\over{48,\sqrt{-a^2+1},a^4-96,\sqrt{-a^2+1},a^2+48,\sqrt{-a^2+1}}}+{{\left(2,a^6+120,a^4+699,a^2+124\right),\varepsilon^7}\over{3024,a^6-9072,a^4+9072,a^2-3024}}+\cdots$
• $R_{5}(x,h)-x\\=-{{4,h,\cos x,\left(6,h,\sin x+6,\cos ^2x-h^2\right)}\over{36,h,\cos ^2x,\sin x-h^3,\sin x+24,\cos ^4x-14,h^2,\cos ^2x+6,h^2}}\\=-{{6,h,\cos \left(3,x\right)+12,h^2,\sin \left(2,x\right)+\left(18,h-4,h^3\right),\cos x}\over{3,\cos \left(4,x\right)+9,h,\sin \left(3,x\right)+\left(12-7,h^2\right),\cos \left(2,x\right)+\left(9,h-h^3\right),\sin x-h^2+9}}\\M_{5}(a,\varepsilon)=\arcsin a-{{\left(a^4-32,a^2-14\right),\varepsilon^5}\over{720,a^4-1440,a^2+720}}-{{\left(5,\sqrt{-a^2+1},a^3+10,\sqrt{-a^2+1},a\right),\varepsilon^6}\over{96,a^6-288,a^4+288,a^2-96}}-{{\left(2,a^6+120,a^4+699,a^2+124\right),\varepsilon^7}\over{4032,a^6-12096,a^4+12096,a^2-4032}}+{{\left(14,\sqrt{-a^2+1},a^5+182,\sqrt{-a^2+1},a^3+119,\sqrt{-a^2+1},a\right),\varepsilon^8}\over{1152,a^8-4608,a^6+6912,a^4-4608,a^2+1152}}+\cdots$
• $R_{6}(x,h)-x\\=-{{5,h,\left(36,h,\cos ^2x,\sin x-h^3,\sin x+24,\cos ^4x-14,h^2,\cos ^2x+6,h^2\right)}\over{\cos x,\left(240,h,\cos ^2x,\sin x-30,h^3,\sin x+120,\cos ^4x-150,h^2,\cos ^2x+h^4+90,h^2\right)}}\\=-{{30,h,\cos \left(4,x\right)+90,h^2,\sin \left(3,x\right)+\left(120,h-70,h^3\right),\cos \left(2,x\right)+\left(90,h^2-10,h^4\right),\sin x-10,h^3+90,h}\over{15,\cos \left(5,x\right)+60,h,\sin \left(4,x\right)+\left(75-75,h^2\right),\cos \left(3,x\right)+\left(120,h-30,h^3\right),\sin \left(2,x\right)+\left(2,h^4-45,h^2+150\right),\cos x}}\\M_{6}(a,\varepsilon)=\arcsin a-{{\left(a^3+2,a\right),\varepsilon^6}\over{96,\sqrt{-a^2+1},a^4-192,\sqrt{-a^2+1},a^2+96,\sqrt{-a^2+1}}}+{{\left(2,a^6+120,a^4+699,a^2+124\right),\varepsilon^7}\over{10080,a^6-30240,a^4+30240,a^2-10080}}-{{\left(14,\sqrt{-a^2+1},a^5+182,\sqrt{-a^2+1},a^3+119,\sqrt{-a^2+1},a\right),\varepsilon^8}\over{1920,a^8-7680,a^6+11520,a^4-7680,a^2+1920}}+{{\left(a^8-64,a^6-1704,a^4-2704,a^2-254\right),\varepsilon^9}\over{21600,a^8-86400,a^6+129600,a^4-86400,a^2+21600}}+\cdots$
• $R_{7}(x,h)-x\\=-{{6,h,\cos x,\left(240,h,\cos ^2x,\sin x-30,h^3,\sin x+120,\cos ^4x-150,h^2,\cos ^2x+h^4+90,h^2\right)}\over{1800,h,\cos ^4x,\sin x-540,h^3,\cos ^2x,\sin x+h^5,\sin x+90,h^3,\sin x+720,\cos ^6x-1560,h^2,\cos ^4x+62,h^4,\cos ^2x+1080,h^2,\cos ^2x-30,h^4}}\\=-{{90,h,\cos \left(5,x\right)+360,h^2,\sin \left(4,x\right)+\left(450,h-450,h^3\right),\cos \left(3,x\right)+\left(720,h^2-180,h^4\right),\sin \left(2,x\right)+\left(12,h^5-270,h^3+900,h\right),\cos x}\over{45,\cos \left(6,x\right)+225,h,\sin \left(5,x\right)+\left(270-390,h^2\right),\cos \left(4,x\right)+\left(675,h-270,h^3\right),\sin \left(3,x\right)+\left(62,h^4-480,h^2+675\right),\cos \left(2,x\right)+\left(2,h^5-90,h^3+450,h\right),\sin x+2,h^4-90,h^2+450}}\\M_{7}(a,\varepsilon)=\arcsin a-{{\left(2,a^6+120,a^4+699,a^2+124\right),\varepsilon^7}\over{60480,a^6-181440,a^4+181440,a^2-60480}}+{{\left(14,\sqrt{-a^2+1},a^5+182,\sqrt{-a^2+1},a^3+119,\sqrt{-a^2+1},a\right),\varepsilon^8}\over{5760,a^8-23040,a^6+34560,a^4-23040,a^2+5760}}-{{\left(a^8-64,a^6-1704,a^4-2704,a^2-254\right),\varepsilon^9}\over{43200,a^8-172800,a^6+259200,a^4-172800,a^2+43200}}-{{\left(\sqrt{-a^2+1},a^7+60,\sqrt{-a^2+1},a^5+192,\sqrt{-a^2+1},a^3+62,\sqrt{-a^2+1},a\right),\varepsilon^{10}}\over{1920,a^{10}-9600,a^8+19200,a^6-19200,a^4+9600,a^2-1920}}+\cdots$
• $R_{8}(x,h)-x\\=-{{7,h,\left(1800,h,\cos ^4x,\sin x-540,h^3,\cos ^2x,\sin x+h^5,\sin x+90,h^3,\sin x+720,\cos ^6x-1560,h^2,\cos ^4x+62,h^4,\cos ^2x+1080,h^2,\cos ^2x-30,h^4\right)}\over{\cos x,\left(15120,h,\cos ^4x,\sin x-8400,h^3,\cos ^2x,\sin x+126,h^5,\sin x+2520,h^3,\sin x+5040,\cos ^6x-16800,h^2,\cos ^4x+1806,h^4,\cos ^2x+12600,h^2,\cos ^2x-h^6-1260,h^4\right)}}\\=-{{630,h,\cos \left(6,x\right)+3150,h^2,\sin \left(5,x\right)+\left(3780,h-5460,h^3\right),\cos \left(4,x\right)+\left(9450,h^2-3780,h^4\right),\sin \left(3,x\right)+\left(868,h^5-6720,h^3+9450,h\right),\cos \left(2,x\right)+\left(28,h^6-1260,h^4+6300,h^2\right),\sin x+28,h^5-1260,h^3+6300,h}\over{315,\cos \left(7,x\right)+1890,h,\sin \left(6,x\right)+\left(2205-4200,h^2\right),\cos \left(5,x\right)+\left(7560,h-4200,h^3\right),\sin \left(4,x\right)+\left(1806,h^4-8400,h^2+6615\right),\cos \left(3,x\right)+\left(252,h^5-3360,h^3+9450,h\right),\sin \left(2,x\right)+\left(-4,h^6+378,h^4-4200,h^2+11025\right),\cos x}}\\M_{8}(a,\varepsilon)=\arcsin a-{{\left(2,\sqrt{-a^2+1},a^5+26,\sqrt{-a^2+1},a^3+17,\sqrt{-a^2+1},a\right),\varepsilon^8}\over{5760,a^8-23040,a^6+34560,a^4-23040,a^2+5760}}+{{\left(a^8-64,a^6-1704,a^4-2704,a^2-254\right),\varepsilon^9}\over{151200,a^8-604800,a^6+907200,a^4-604800,a^2+151200}}-{{\left(a^7+60,a^5+192,a^3+62,a\right),\varepsilon^{10}}\over{4480,\sqrt{-a^2+1},a^8-17920,\sqrt{-a^2+1},a^6+26880,\sqrt{-a^2+1},a^4-17920,\sqrt{-a^2+1},a^2+4480,\sqrt{-a^2+1}}}+{{\left(2,a^{10}+56,a^8+8171,a^6+47896,a^4+35386,a^2+2044\right),\varepsilon^{11}}\over{798336,a^{10}-3991680,a^8+7983360,a^6-7983360,a^4+3991680,a^2-798336}}+\cdots$
• $R_{9}(x,h)-x\\=-{{8,h,\cos x,\left(15120,h,\cos ^4x,\sin x-8400,h^3,\cos ^2x,\sin x+126,h^5,\sin x+2520,h^3,\sin x+5040,\cos ^6x-16800,h^2,\cos ^4x+1806,h^4,\cos ^2x+12600,h^2,\cos ^2x-h^6-1260,h^4\right)}\over{141120,h,\cos ^6x,\sin x-126000,h^3,\cos ^4x,\sin x+5796,h^5,\cos ^2x,\sin x+50400,h^3,\cos ^2x,\sin x-h^7,\sin x-1260,h^5,\sin x+40320,\cos ^8x-191520,h^2,\cos ^6x+40824,h^4,\cos ^4x+151200,h^2,\cos ^4x-254,h^6,\cos ^2x-35280,h^4,\cos ^2x+126,h^6+2520,h^4}}\\=-{{630,h,\cos \left(7,x\right)+3780,h^2,\sin \left(6,x\right)+\left(4410,h-8400,h^3\right),\cos \left(5,x\right)+\left(15120,h^2-8400,h^4\right),\sin \left(4,x\right)+\left(3612,h^5-16800,h^3+13230,h\right),\cos \left(3,x\right)+\left(504,h^6-6720,h^4+18900,h^2\right),\sin \left(2,x\right)+\left(-8,h^7+756,h^5-8400,h^3+22050,h\right),\cos x}\over{315,\cos \left(8,x\right)+2205,h,\sin \left(7,x\right)+\left(2520-5985,h^2\right),\cos \left(6,x\right)+\left(11025,h-7875,h^3\right),\sin \left(5,x\right)+\left(5103,h^4-17010,h^2+8820\right),\cos \left(4,x\right)+\left(1449,h^5-11025,h^3+19845,h\right),\sin \left(3,x\right)+\left(-127,h^6+2772,h^4-14175,h^2+17640\right),\cos \left(2,x\right)+\left(-h^7+189,h^5-3150,h^3+11025,h\right),\sin x-h^6+189,h^4-3150,h^2+11025}}\\M_{9}(a,\varepsilon)=\arcsin a-{{\left(a^8-64,a^6-1704,a^4-2704,a^2-254\right),\varepsilon^9}\over{1209600,a^8-4838400,a^6+7257600,a^4-4838400,a^2+1209600}}-{{\left(\sqrt{-a^2+1},a^7+60,\sqrt{-a^2+1},a^5+192,\sqrt{-a^2+1},a^3+62,\sqrt{-a^2+1},a\right),\varepsilon^{10}}\over{17920,a^{10}-89600,a^8+179200,a^6-179200,a^4+89600,a^2-17920}}-{{\left(2,a^{10}+56,a^8+8171,a^6+47896,a^4+35386,a^2+2044\right),\varepsilon^{11}}\over{2128896,a^{10}-10644480,a^8+21288960,a^6-21288960,a^4+10644480,a^2-2128896}}+{{\left(22,\sqrt{-a^2+1},a^9+5522,\sqrt{-a^2+1},a^7+56067,\sqrt{-a^2+1},a^5+79112,\sqrt{-a^2+1},a^3+15202,\sqrt{-a^2+1},a\right),\varepsilon^{12}}\over{1935360,a^{12}-11612160,a^{10}[phone],a^8-38707200,a^6+29030400,a^4-11612160,a^2+1935360}}+\cdots$

arccos

maxima_highorder_convergence_acos

[kill(all),reset(),array([Q],N:9),Q[1]:1,G:acos(a),H:cos(x),
printf(true,"~%  - $~a$~%  - $~a$~%",tex1(f(x,a)=F:H-a),tex1(g(a)=G)),for m
:2 thru N do([Q[m]:diff(F,x,1)*Q[m-1]-F*diff(Q[m-1],x,1)/(m-1),printf(true,
"- $~a\\\\\\\\=~a\\\\\\\\=~a\\\\\\\\~a$~%",tex1(R[m](x,h)-x),tex1(factor(U:
trigsimp(subst(H-h,a,V:-F*Q[m-1]/Q[m])))),tex1(trigrat(U)),strim(" ",tex1(M
[m](a,E:epsilon)=taylor(subst(G+E,x,V+x),E,0,m+3))))]),printf(true,"~%~%")]$

• $k$-次収束漸化式: $x_{n+1} = R_k({x_n}, {h_n}), \quad h_n = f(x_n, a)$
  • $f(g(a), a) = 0$
  • $M_k(a,\varepsilon) = R_k(g(a)+\varepsilon,f(g(a)+\varepsilon,a))$
  • $\varepsilon \simeq 0 \quad \Longrightarrow \quad M_k(a,\varepsilon) \simeq g(a)$
  • $f\left(x , a\right)=\cos x-a$
  • $g\left(a\right)=\arccos a$
• $R_{2}(x,h)-x\\={{h}\over{\sin x}}\\={{h}\over{\sin x}}\\M_{2}(a,\varepsilon)=\arccos a-{{\sqrt{-a^2+1},a,\varepsilon^2}\over{2,a^2-2}}+{{\left(a^2+2\right),\varepsilon^3}\over{6,a^2-6}}+{{\left(\sqrt{-a^2+1},a^3+11,\sqrt{-a^2+1},a\right),\varepsilon^4}\over{24,a^4-48,a^2+24}}-{{\left(a^4+43,a^2+16\right),\varepsilon^5}\over{120,a^4-240,a^2+120}}+\cdots$
• $R_{3}(x,h)-x\\={{2,h,\sin x}\over{2,\sin ^2x+h,\cos x}}\\=-{{2,h,\sin x}\over{\cos \left(2,x\right)-h,\cos x-1}}\\M_{3}(a,\varepsilon)=\arccos a-{{\left(a^2+2\right),\varepsilon^3}\over{12,a^2-12}}-{{3,\sqrt{-a^2+1},a,\varepsilon^4}\over{8,a^4-16,a^2+8}}-{{\left(a^4-32,a^2-14\right),\varepsilon^5}\over{120,a^4-240,a^2+120}}+{{\left(3,\sqrt{-a^2+1},a^3+6,\sqrt{-a^2+1},a\right),\varepsilon^6}\over{32,a^6-96,a^4+96,a^2-32}}+\cdots$
• $R_{4}(x,h)-x\\=-{{3,h,\left(2,\sin ^2x+h,\cos x\right)}\over{\left(6,\cos ^2x-6,h,\cos x+h^2-6\right),\sin x}}\\={{6,h,\cos \left(2,x\right)-6,h^2,\cos x-6,h}\over{3,\sin \left(3,x\right)-6,h,\sin \left(2,x\right)+\left(2,h^2-9\right),\sin x}}\\M_{4}(a,\varepsilon)=\arccos a+{{\sqrt{-a^2+1},a,\varepsilon^4}\over{8,a^4-16,a^2+8}}+{{\left(a^4-32,a^2-14\right),\varepsilon^5}\over{180,a^4-360,a^2+180}}+{{\left(5,a^3+10,a\right),\varepsilon^6}\over{48,\sqrt{-a^2+1},a^4-96,\sqrt{-a^2+1},a^2+48,\sqrt{-a^2+1}}}+{{\left(2,a^6+120,a^4+699,a^2+124\right),\varepsilon^7}\over{3024,a^6-9072,a^4+9072,a^2-3024}}+\cdots$
• $R_{5}(x,h)-x\\={{4,h,\sin x,\left(6,\sin ^2x+6,h,\cos x-h^2\right)}\over{24,\sin ^4x+36,h,\cos x,\sin ^2x-14,h^2,\sin ^2x-h^3,\cos x+6,h^2}}\\=-{{6,h,\sin \left(3,x\right)-12,h^2,\sin \left(2,x\right)+\left(4,h^3-18,h\right),\sin x}\over{3,\cos \left(4,x\right)-9,h,\cos \left(3,x\right)+\left(7,h^2-12\right),\cos \left(2,x\right)+\left(9,h-h^3\right),\cos x-h^2+9}}\\M_{5}(a,\varepsilon)=\arccos a-{{\left(a^4-32,a^2-14\right),\varepsilon^5}\over{720,a^4-1440,a^2+720}}+{{\left(5,\sqrt{-a^2+1},a^3+10,\sqrt{-a^2+1},a\right),\varepsilon^6}\over{96,a^6-288,a^4+288,a^2-96}}-{{\left(2,a^6+120,a^4+699,a^2+124\right),\varepsilon^7}\over{4032,a^6-12096,a^4+12096,a^2-4032}}-{{\left(14,\sqrt{-a^2+1},a^5+182,\sqrt{-a^2+1},a^3+119,\sqrt{-a^2+1},a\right),\varepsilon^8}\over{1152,a^8-4608,a^6+6912,a^4-4608,a^2+1152}}+\cdots$
• $R_{6}(x,h)-x\\={{5,h,\left(24,\sin ^4x+36,h,\cos x,\sin ^2x-14,h^2,\sin ^2x-h^3,\cos x+6,h^2\right)}\over{\sin x,\left(120,\sin ^4x+240,h,\cos x,\sin ^2x-150,h^2,\sin ^2x-30,h^3,\cos x+h^4+90,h^2\right)}}\\={{30,h,\cos \left(4,x\right)-90,h^2,\cos \left(3,x\right)+\left(70,h^3-120,h\right),\cos \left(2,x\right)+\left(90,h^2-10,h^4\right),\cos x-10,h^3+90,h}\over{15,\sin \left(5,x\right)-60,h,\sin \left(4,x\right)+\left(75,h^2-75\right),\sin \left(3,x\right)+\left(120,h-30,h^3\right),\sin \left(2,x\right)+\left(2,h^4-45,h^2+150\right),\sin x}}\\M_{6}(a,\varepsilon)=\arccos a+{{\left(a^3+2,a\right),\varepsilon^6}\over{96,\sqrt{-a^2+1},a^4-192,\sqrt{-a^2+1},a^2+96,\sqrt{-a^2+1}}}+{{\left(2,a^6+120,a^4+699,a^2+124\right),\varepsilon^7}\over{10080,a^6-30240,a^4+30240,a^2-10080}}+{{\left(14,\sqrt{-a^2+1},a^5+182,\sqrt{-a^2+1},a^3+119,\sqrt{-a^2+1},a\right),\varepsilon^8}\over{1920,a^8-7680,a^6+11520,a^4-7680,a^2+1920}}+{{\left(a^8-64,a^6-1704,a^4-2704,a^2-254\right),\varepsilon^9}\over{21600,a^8-86400,a^6+129600,a^4-86400,a^2+21600}}+\cdots$
• $R_{7}(x,h)-x\\={{6,h,\sin x,\left(120,\sin ^4x+240,h,\cos x,\sin ^2x-150,h^2,\sin ^2x-30,h^3,\cos x+h^4+90,h^2\right)}\over{720,\sin ^6x+1800,h,\cos x,\sin ^4x-1560,h^2,\sin ^4x-540,h^3,\cos x,\sin ^2x+62,h^4,\sin ^2x+1080,h^2,\sin ^2x+h^5,\cos x+90,h^3,\cos x-30,h^4}}\\=-{{90,h,\sin \left(5,x\right)-360,h^2,\sin \left(4,x\right)+\left(450,h^3-450,h\right),\sin \left(3,x\right)+\left(720,h^2-180,h^4\right),\sin \left(2,x\right)+\left(12,h^5-270,h^3+900,h\right),\sin x}\over{45,\cos \left(6,x\right)-225,h,\cos \left(5,x\right)+\left(390,h^2-270\right),\cos \left(4,x\right)+\left(675,h-270,h^3\right),\cos \left(3,x\right)+\left(62,h^4-480,h^2+675\right),\cos \left(2,x\right)+\left(-2,h^5+90,h^3-450,h\right),\cos x-2,h^4+90,h^2-450}}\\M_{7}(a,\varepsilon)=\arccos a-{{\left(2,a^6+120,a^4+699,a^2+124\right),\varepsilon^7}\over{60480,a^6-181440,a^4+181440,a^2-60480}}-{{\left(14,\sqrt{-a^2+1},a^5+182,\sqrt{-a^2+1},a^3+119,\sqrt{-a^2+1},a\right),\varepsilon^8}\over{5760,a^8-23040,a^6+34560,a^4-23040,a^2+5760}}-{{\left(a^8-64,a^6-1704,a^4-2704,a^2-254\right),\varepsilon^9}\over{43200,a^8-172800,a^6+259200,a^4-172800,a^2+43200}}+{{\left(\sqrt{-a^2+1},a^7+60,\sqrt{-a^2+1},a^5+192,\sqrt{-a^2+1},a^3+62,\sqrt{-a^2+1},a\right),\varepsilon^{10}}\over{1920,a^{10}-9600,a^8+19200,a^6-19200,a^4+9600,a^2-1920}}+\cdots$
• $R_{8}(x,h)-x\\={{7,h,\left(720,\sin ^6x+1800,h,\cos x,\sin ^4x-1560,h^2,\sin ^4x-540,h^3,\cos x,\sin ^2x+62,h^4,\sin ^2x+1080,h^2,\sin ^2x+h^5,\cos x+90,h^3,\cos x-30,h^4\right)}\over{\sin x,\left(5040,\sin ^6x+15120,h,\cos x,\sin ^4x-16800,h^2,\sin ^4x-8400,h^3,\cos x,\sin ^2x+1806,h^4,\sin ^2x+12600,h^2,\sin ^2x+126,h^5,\cos x+2520,h^3,\cos x-h^6-1260,h^4\right)}}\\={{630,h,\cos \left(6,x\right)-3150,h^2,\cos \left(5,x\right)+\left(5460,h^3-3780,h\right),\cos \left(4,x\right)+\left(9450,h^2-3780,h^4\right),\cos \left(3,x\right)+\left(868,h^5-6720,h^3+9450,h\right),\cos \left(2,x\right)+\left(-28,h^6+1260,h^4-6300,h^2\right),\cos x-28,h^5+1260,h^3-6300,h}\over{315,\sin \left(7,x\right)-1890,h,\sin \left(6,x\right)+\left(4200,h^2-2205\right),\sin \left(5,x\right)+\left(7560,h-4200,h^3\right),\sin \left(4,x\right)+\left(1806,h^4-8400,h^2+6615\right),\sin \left(3,x\right)+\left(-252,h^5+3360,h^3-9450,h\right),\sin \left(2,x\right)+\left(4,h^6-378,h^4+4200,h^2-11025\right),\sin x}}\\M_{8}(a,\varepsilon)=\arccos a+{{\left(2,\sqrt{-a^2+1},a^5+26,\sqrt{-a^2+1},a^3+17,\sqrt{-a^2+1},a\right),\varepsilon^8}\over{5760,a^8-23040,a^6+34560,a^4-23040,a^2+5760}}+{{\left(a^8-64,a^6-1704,a^4-2704,a^2-254\right),\varepsilon^9}\over{151200,a^8-604800,a^6+907200,a^4-604800,a^2+151200}}+{{\left(a^7+60,a^5+192,a^3+62,a\right),\varepsilon^{10}}\over{4480,\sqrt{-a^2+1},a^8-17920,\sqrt{-a^2+1},a^6+26880,\sqrt{-a^2+1},a^4-17920,\sqrt{-a^2+1},a^2+4480,\sqrt{-a^2+1}}}+{{\left(2,a^{10}+56,a^8+8171,a^6+47896,a^4+35386,a^2+2044\right),\varepsilon^{11}}\over{798336,a^{10}-3991680,a^8+7983360,a^6-7983360,a^4+3991680,a^2-798336}}+\cdots$
• $R_{9}(x,h)-x\\={{8,h,\sin x,\left(5040,\sin ^6x+15120,h,\cos x,\sin ^4x-16800,h^2,\sin ^4x-8400,h^3,\cos x,\sin ^2x+1806,h^4,\sin ^2x+12600,h^2,\sin ^2x+126,h^5,\cos x+2520,h^3,\cos x-h^6-1260,h^4\right)}\over{40320,\sin ^8x+141120,h,\cos x,\sin ^6x-191520,h^2,\sin ^6x-126000,h^3,\cos x,\sin ^4x+40824,h^4,\sin ^4x+151200,h^2,\sin ^4x+5796,h^5,\cos x,\sin ^2x+50400,h^3,\cos x,\sin ^2x-254,h^6,\sin ^2x-35280,h^4,\sin ^2x-h^7,\cos x-1260,h^5,\cos x+126,h^6+2520,h^4}}\\=-{{630,h,\sin \left(7,x\right)-3780,h^2,\sin \left(6,x\right)+\left(8400,h^3-4410,h\right),\sin \left(5,x\right)+\left(15120,h^2-8400,h^4\right),\sin \left(4,x\right)+\left(3612,h^5-16800,h^3+13230,h\right),\sin \left(3,x\right)+\left(-504,h^6+6720,h^4-18900,h^2\right),\sin \left(2,x\right)+\left(8,h^7-756,h^5+8400,h^3-22050,h\right),\sin x}\over{315,\cos \left(8,x\right)-2205,h,\cos \left(7,x\right)+\left(5985,h^2-2520\right),\cos \left(6,x\right)+\left(11025,h-7875,h^3\right),\cos \left(5,x\right)+\left(5103,h^4-17010,h^2+8820\right),\cos \left(4,x\right)+\left(-1449,h^5+11025,h^3-19845,h\right),\cos \left(3,x\right)+\left(127,h^6-2772,h^4+14175,h^2-17640\right),\cos \left(2,x\right)+\left(-h^7+189,h^5-3150,h^3+11025,h\right),\cos x-h^6+189,h^4-3150,h^2+11025}}\\M_{9}(a,\varepsilon)=\arccos a-{{\left(a^8-64,a^6-1704,a^4-2704,a^2-254\right),\varepsilon^9}\over{1209600,a^8-4838400,a^6+7257600,a^4-4838400,a^2+1209600}}+{{\left(\sqrt{-a^2+1},a^7+60,\sqrt{-a^2+1},a^5+192,\sqrt{-a^2+1},a^3+62,\sqrt{-a^2+1},a\right),\varepsilon^{10}}\over{17920,a^{10}-89600,a^8+179200,a^6-179200,a^4+89600,a^2-17920}}-{{\left(2,a^{10}+56,a^8+8171,a^6+47896,a^4+35386,a^2+2044\right),\varepsilon^{11}}\over{2128896,a^{10}-10644480,a^8+21288960,a^6-21288960,a^4+10644480,a^2-2128896}}-{{\left(22,\sqrt{-a^2+1},a^9+5522,\sqrt{-a^2+1},a^7+56067,\sqrt{-a^2+1},a^5+79112,\sqrt{-a^2+1},a^3+15202,\sqrt{-a^2+1},a\right),\varepsilon^{12}}\over{1935360,a^{12}-11612160,a^{10}[phone],a^8-38707200,a^6+29030400,a^4-11612160,a^2+1935360}}+\cdots$

arctan

maxima_highorder_convergence_atan

[kill(all),reset(),array([Q],N:9),Q[1]:1,G:atan(a),H:tan(x),
printf(true,"~%  - $~a$~%  - $~a$~%",tex1(f(x,a)=F:H-a),tex1(g(a)=G)),for m
:2 thru N do([Q[m]:diff(F,x,1)*Q[m-1]-F*diff(Q[m-1],x,1)/(m-1),printf(true,
"- $~a\\\\\\\\=~a\\\\\\\\=~a\\\\\\\\~a$~%",tex1(R[m](x,h)-x),tex1(factor(U:
trigsimp(subst(H-h,a,V:-F*Q[m-1]/Q[m])))),tex1(trigrat(U)),strim(" ",tex1(M
[m](a,E:epsilon)=taylor(subst(G+E,x,V+x),E,0,m+3))))]),printf(true,"~%~%")]$

• $k$-次収束漸化式: $x_{n+1} = R_k({x_n}, {h_n}), \quad h_n = f(x_n, a)$
  • $f(g(a), a) = 0$
  • $M_k(a,\varepsilon) = R_k(g(a)+\varepsilon,f(g(a)+\varepsilon,a))$
  • $\varepsilon \simeq 0 \quad \Longrightarrow \quad M_k(a,\varepsilon) \simeq g(a)$
  • $f\left(x , a\right)=\tan x-a$
  • $g\left(a\right)=\arctan a$
• $R_{2}(x,h)-x\\=-h,\cos ^2x\\=-{{h,\cos \left(2,x\right)+h}\over{2}}\\M_{2}(a,\varepsilon)=\arctan a+a,\varepsilon^2+{{2,\varepsilon^3}\over{3}}-{{a,\varepsilon^4}\over{3}}-{{2,\varepsilon^5}\over{15}}+\cdots$
• $R_{3}(x,h)-x\\={{h,\cos ^2x}\over{h,\cos x,\sin x-1}}\\={{h,\cos \left(2,x\right)+h}\over{h,\sin \left(2,x\right)-2}}\\M_{3}(a,\varepsilon)=\arctan a-{{\varepsilon^3}\over{3}}-{{2,\varepsilon^5}\over{15}}+\cdots$
• $R_{4}(x,h)-x\\={{3,h,\cos ^2x,\left(h,\cos x,\sin x-1\right)}\over{2,h^2,\cos ^2x,\sin ^2x-6,h,\cos x,\sin x+h^2,\cos ^2x+3}}\\=-{{3,h^2,\sin \left(4,x\right)+6,h^2,\sin \left(2,x\right)-12,h,\cos \left(2,x\right)-12,h}\over{2,h^2,\cos \left(4,x\right)+24,h,\sin \left(2,x\right)-4,h^2,\cos \left(2,x\right)-6,h^2-24}}\\M_{4}(a,\varepsilon)=\arctan a+{{4,\varepsilon^5}\over{45}}+{{8,\varepsilon^7}\over{189}}+\cdots$
  • ※ $R_{4}(x,h)$ は 5次収束
• $R_{5}(x,h)-x\\=-{{h,\cos ^2x,\left(2,h^2,\cos ^2x,\sin ^2x-6,h,\cos x,\sin x+h^2,\cos ^2x+3\right)}\over{h^3,\cos ^5x,\sin x-3,h^3,\cos ^3x,\sin x-9,h,\cos x,\sin x-7,h^2,\cos ^4x+9,h^2,\cos ^2x+3}}\\={{2,h^3,\cos \left(6,x\right)+24,h^2,\sin \left(4,x\right)+48,h^2,\sin \left(2,x\right)+\left(-18,h^3-48,h\right),\cos \left(2,x\right)-16,h^3-48,h}\over{h^3,\sin \left(6,x\right)-8,h^3,\sin \left(4,x\right)-28,h^2,\cos \left(4,x\right)+\left(-19,h^3-144,h\right),\sin \left(2,x\right)+32,h^2,\cos \left(2,x\right)+60,h^2+96}}\\M_{5}(a,\varepsilon)=\arctan a-{{\varepsilon^5}\over{45}}-{{2,\varepsilon^7}\over{63}}+\cdots$
• $R_{6}(x,h)-x\\=-{{5,h,\cos ^2x,\left(h^3,\cos ^5x,\sin x-3,h^3,\cos ^3x,\sin x-9,h,\cos x,\sin x-7,h^2,\cos ^4x+9,h^2,\cos ^2x+3\right)}\over{30,h^3,\cos ^5x,\sin x-60,h^3,\cos ^3x,\sin x-60,h,\cos x,\sin x+2,h^4,\cos ^8x-15,h^4,\cos ^6x+15,h^4,\cos ^4x-75,h^2,\cos ^4x+90,h^2,\cos ^2x+15}}\\=-{{5,h^4,\sin \left(8,x\right)-30,h^4,\sin \left(6,x\right)-140,h^3,\cos \left(6,x\right)+\left(-170,h^4-720,h^2\right),\sin \left(4,x\right)-120,h^3,\cos \left(4,x\right)+\left(-230,h^4-1440,h^2\right),\sin \left(2,x\right)+\left(780,h^3+960,h\right),\cos \left(2,x\right)+760,h^3+960,h}\over{2,h^4,\cos \left(8,x\right)+120,h^3,\sin \left(6,x\right)-44,h^4,\cos \left(6,x\right)-480,h^3,\sin \left(4,x\right)+\left(-64,h^4-1200,h^2\right),\cos \left(4,x\right)+\left(-1320,h^3-3840,h\right),\sin \left(2,x\right)+\left(172,h^4+960,h^2\right),\cos \left(2,x\right)+190,h^4+2160,h^2+1920}}\\M_{6}(a,\varepsilon)=\arctan a+{{4,\varepsilon^7}\over{315}}+{{8,\varepsilon^9}\over{675}}+\cdots$
  • ※ $R_{6}(x,h)$ は 7次収束
• $R_{7}(x,h)-x\\={{3,h,\cos ^2x,\left(30,h^3,\cos ^5x,\sin x-60,h^3,\cos ^3x,\sin x-60,h,\cos x,\sin x+2,h^4,\cos ^8x-15,h^4,\cos ^6x+15,h^4,\cos ^4x-75,h^2,\cos ^4x+90,h^2,\cos ^2x+15\right)}\over{2,h^5,\cos ^9x,\sin x-30,h^5,\cos ^7x,\sin x+45,h^5,\cos ^5x,\sin x-270,h^3,\cos ^5x,\sin x+450,h^3,\cos ^3x,\sin x+225,h,\cos x,\sin x-62,h^4,\cos ^8x+270,h^4,\cos ^6x-225,h^4,\cos ^4x+390,h^2,\cos ^4x-450,h^2,\cos ^2x-45}}\\={{3,h^5,\cos \left(10,x\right)+180,h^4,\sin \left(8,x\right)-60,h^5,\cos \left(8,x\right)-360,h^4,\sin \left(6,x\right)+\left(-225,h^5-1800,h^3\right),\cos \left(6,x\right)+\left(-3240,h^4-5760,h^2\right),\sin \left(4,x\right)-2160,h^3,\cos \left(4,x\right)+\left(-4680,h^4-11520,h^2\right),\sin \left(2,x\right)+\left(990,h^5+7560,h^3+5760,h\right),\cos \left(2,x\right)+828,h^5+7920,h^3+5760,h}\over{h^5,\sin \left(10,x\right)-52,h^5,\sin \left(8,x\right)-124,h^4,\cos \left(8,x\right)+\left(27,h^5-2160,h^3\right),\sin \left(6,x\right)+1168,h^4,\cos \left(6,x\right)+\left(648,h^5+5760,h^3\right),\sin \left(4,x\right)+\left(2288,h^4+12480,h^2\right),\cos \left(4,x\right)+\left(1002,h^5+18000,h^3+28800,h\right),\sin \left(2,x\right)+\left(-3344,h^4-7680,h^2\right),\cos \left(2,x\right)-4340,h^4-20160,h^2-11520}}\\M_{7}(a,\varepsilon)=\arctan a-{{2,\varepsilon^7}\over{945}}-{{4,\varepsilon^9}\over{675}}+\cdots$
• $R_{8}(x,h)-x\\=-{{7,h,\cos ^2x,\left(2,h^5,\cos ^9x,\sin x-30,h^5,\cos ^7x,\sin x+45,h^5,\cos ^5x,\sin x-270,h^3,\cos ^5x,\sin x+450,h^3,\cos ^3x,\sin x+225,h,\cos x,\sin x-62,h^4,\cos ^8x+270,h^4,\cos ^6x-225,h^4,\cos ^4x+390,h^2,\cos ^4x-450,h^2,\cos ^2x-45\right)}\over{252,h^5,\cos ^9x,\sin x-1680,h^5,\cos ^7x,\sin x+1890,h^5,\cos ^5x,\sin x-4200,h^3,\cos ^5x,\sin x+6300,h^3,\cos ^3x,\sin x+1890,h,\cos x,\sin x+4,h^6,\cos ^{12}x-126,h^6,\cos ^{10}x+420,h^6,\cos ^8x-1806,h^4,\cos ^8x-315,h^6,\cos ^6x+6300,h^4,\cos ^6x-4725,h^4,\cos ^4x+4200,h^2,\cos ^4x-4725,h^2,\cos ^2x-315}}\\=-{{7,h^6,\sin \left(12,x\right)-350,h^6,\sin \left(10,x\right)-868,h^5,\cos \left(10,x\right)+\left(-532,h^6-15120,h^4\right),\sin \left(8,x\right)+6440,h^5,\cos \left(8,x\right)+\left(4550,h^6+10080,h^4\right),\sin \left(6,x\right)+\left(31500,h^5+87360,h^3\right),\cos \left(6,x\right)+\left(16275,h^6+191520,h^4+201600,h^2\right),\sin \left(4,x\right)+\left(16800,h^5+120960,h^3\right),\cos \left(4,x\right)+\left(18564,h^6+292320,h^4+403200,h^2\right),\sin \left(2,x\right)+\left(-91560,h^5-302400,h^3-161280,h\right),\cos \left(2,x\right)-84168,h^5-336000,h^3-161280,h}\over{2,h^6,\cos \left(12,x\right)+504,h^5,\sin \left(10,x\right)-228,h^6,\cos \left(10,x\right)-9408,h^5,\sin \left(8,x\right)+\left(972,h^6-14448,h^4\right),\cos \left(8,x\right)+\left(-6552,h^5-134400,h^3\right),\sin \left(6,x\right)+\left(5900,h^6+86016,h^4\right),\cos \left(6,x\right)+\left(77952,h^5+268800,h^3\right),\sin \left(4,x\right)+\left(4350,h^6+200256,h^4+537600,h^2\right),\cos \left(4,x\right)+\left(135408,h^5+940800,h^3+967680,h\right),\sin \left(2,x\right)+\left(-14376,h^6-204288,h^4-268800,h^2\right),\cos \left(2,x\right)-14028,h^6-304080,h^4-806400,h^2-322560}}\\M_{8}(a,\varepsilon)=\arctan a+{{8,\varepsilon^9}\over{4725}}+{{16,\varepsilon^{11}}\over{6237}}+\cdots$
  • ※ $R_{8}(x,h)$ は 9次収束
• $R_{9}(x,h)-x\\={{h,\cos ^2x,\left(252,h^5,\cos ^9x,\sin x-1680,h^5,\cos ^7x,\sin x+1890,h^5,\cos ^5x,\sin x-4200,h^3,\cos ^5x,\sin x+6300,h^3,\cos ^3x,\sin x+1890,h,\cos x,\sin x+4,h^6,\cos ^{12}x-126,h^6,\cos ^{10}x+420,h^6,\cos ^8x-1806,h^4,\cos ^8x-315,h^6,\cos ^6x+6300,h^4,\cos ^6x-4725,h^4,\cos ^4x+4200,h^2,\cos ^4x-4725,h^2,\cos ^2x-315\right)}\over{h^7,\cos ^{13}x,\sin x-63,h^7,\cos ^{11}x,\sin x+315,h^7,\cos ^9x,\sin x-1449,h^5,\cos ^9x,\sin x-315,h^7,\cos ^7x,\sin x+6930,h^5,\cos ^7x,\sin x-6615,h^5,\cos ^5x,\sin x+7875,h^3,\cos ^5x,\sin x-11025,h^3,\cos ^3x,\sin x-2205,h,\cos x,\sin x-127,h^6,\cos ^{12}x+1449,h^6,\cos ^{10}x-3465,h^6,\cos ^8x+5103,h^4,\cos ^8x+2205,h^6,\cos ^6x-15750,h^4,\cos ^6x+11025,h^4,\cos ^4x-5985,h^2,\cos ^4x+6615,h^2,\cos ^2x+315}}\\={{4,h^7,\cos \left(14,x\right)+1008,h^6,\sin \left(12,x\right)-448,h^7,\cos \left(12,x\right)-16800,h^6,\sin \left(10,x\right)+\left(1036,h^7-28896,h^5\right),\cos \left(10,x\right)+\left(-49728,h^6-268800,h^4\right),\sin \left(8,x\right)+\left(15232,h^7+114240,h^5\right),\cos \left(8,x\right)+110880,h^6,\sin \left(6,x\right)+\left(34244,h^7+715680,h^5+1075200,h^3\right),\cos \left(6,x\right)+\left(569520,h^6+2688000,h^4+1935360,h^2\right),\sin \left(4,x\right)+\left(448,h^7+564480,h^5+1612800,h^3\right),\cos \left(4,x\right)+\left(697536,h^6+4300800,h^4+3870720,h^2\right),\sin \left(2,x\right)+\left(-104916,h^7-1632960,h^5-3225600,h^3-1290240,h\right),\cos \left(2,x\right)-84864,h^7-1624896,h^5-3763200,h^3-1290240,h}\over{h^7,\sin \left(14,x\right)-240,h^7,\sin \left(12,x\right)-508,h^6,\cos \left(12,x\right)+\left(2585,h^7-23184,h^5\right),\sin \left(10,x\right)+17088,h^6,\cos \left(10,x\right)+\left(9280,h^7+258048,h^5\right),\sin \left(8,x\right)+\left(326592,h^4-23448,h^6\right),\cos \left(8,x\right)+\left(-12171,h^7+341712,h^5+2016000,h^3\right),\sin \left(6,x\right)+\left(-278080,h^6-1419264,h^4\right),\cos \left(6,x\right)+\left(-81328,h^7-1677312,h^5-3225600,h^3\right),\sin \left(4,x\right)+\left(-291780,h^6-3757824,h^4-6128640,h^2\right),\cos \left(4,x\right)+\left(-103395,h^7-3231648,h^5-12499200,h^3-9031680,h\right),\sin \left(2,x\right)+\left(514944,h^6+2967552,h^4+2580480,h^2\right),\cos \left(2,x\right)+569688,h^6+4979520,h^4+8709120,h^2+2580480}}\\M_{9}(a,\varepsilon)=\arctan a-{{\varepsilon^9}\over{4725}}-{{2,\varepsilon^{11}}\over{2079}}+\cdots$

参考資料

1. Lyuka. 逆数と平⽅根を求める⾼次収束アルゴリズム. http://www.finetune.co.jp/~lyuka/technote/fract/sqrt.html.
2. KK62526. 2 の平⽅根. http://kk62526.server-shared.com/root2/.
3. Jürgen Gerlach. Accelerated convergence in newton’s method. Siam Review, 36(2):272–276, 1994. http://www.staff.science.uu.nl/~zegel101/NUMWISb/gerlach1994.pdf, http://dx.doi.org/10.1137/1036057.
4. William F Ford and James A Pennline. Accelerated convergence in newton’s method. SIAM review, 38(4): 658–659, 1996. http://dx.doi.org/10.1137/S0036144594292972.
5. Latasa, Manuel. (2001). Kepler equation and accelerated Newton method.. Pre-publicaciones del Seminario Matemático " García de Galdeano ", Nº. 16, 200116 pags.. https://www.researchgate.net/publication/28131582_Kepler_equation_and_accelerated_Newton_method.
6. ⻑⽥直樹. お話：数値解析第 1 回収束の種類. http://www.cis.twcu.ac.jp/~osada/rikei/rikei2008-5.pdf.
7. Macsyma group at Project MAC and volunteer contributors. Maxima, a computer algebra system. http://maxima.sourceforge.net/.
8. nextliteracy. Maxima ⽇本語ドキュメント. http://maxima.osdn.jp/maxima.html.
9. Yasuaki Honda. Maxima on android. https://sites.google.com/site/maximaonandroid/.