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。
2. 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](http://maxima.sourceforge.net/) 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: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: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}+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}+29030400\,a^8+38707200\,a^6+29030400\,a^4+11612160\,a^2+1935360}}+\cdots$

### acosh

```maxima: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}+29030400\,a^8-38707200\,a^6+29030400\,a^4-11612160\,a^2+1935360}}+\cdots$

### atanh

```maxima: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: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}+29030400\,a^8-38707200\,a^6+29030400\,a^4-11612160\,a^2+1935360}}+\cdots$

### arccos

```maxima: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}+29030400\,a^8-38707200\,a^6+29030400\,a^4-11612160\,a^2+1935360}}+\cdots$

### arctan

```maxima: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](http://www.finetune.co.jp/~lyuka/technote/fract/sqrt.html).
2. KK62526. 2 の平⽅根. [http://kk62526.server-shared.com/root2/](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://www.staff.science.uu.nl/~zegel101/NUMWISb/gerlach1994.pdf), [http://dx.doi.org/10.1137/1036057](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](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](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](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/](http://maxima.sourceforge.net/).
8. nextliteracy. Maxima ⽇本語ドキュメント. [http://maxima.osdn.jp/maxima.html](http://maxima.osdn.jp/maxima.html).
9. Yasuaki Honda. Maxima on android. [https://sites.google.com/site/maximaonandroid/](https://sites.google.com/site/maximaonandroid/).