前提など
プロローグ
どうも、安田です。
私は現在数列$Y$と名付けた難解な漸化式を持つ数列について研究をしているのですが、解法を探している過程で数列の近似方法を発見しました。
本記事ではその近似方法、微分近似と積分近似について書き留めます。
数列$Y$で微分近似と積分近似を使った記事
定義しておくこと
本記事では以下を数列$a$と定義しておきます。
\begin{equation}
\begin{cases}
a_{n+1} = a_n + f(a_n) \\
a_1 = x
\end{cases}
\end{equation}
プロローグ
どうも、安田です。
私は現在数列$Y$と名付けた難解な漸化式を持つ数列について研究をしているのですが、解法を探している過程で数列の近似方法を発見しました。
本記事ではその近似方法、微分近似と積分近似について書き留めます。
数列$Y$で微分近似と積分近似を使った記事
定義しておくこと
本記事では以下を数列$a$と定義しておきます。
積分近似
積分近似の概要
積分近似では、その名から察せる通り、数列を面積的な比較により近似をします。
ここで、$y = f(x)$ のグラフを考えます。$x=a_k$ から次の項に進むとき、次の項は$ x + f(x)$ で表されますので、これをグラフに表すと以下のようになります。

この時、$y=0, y=f(x), x=a_k, x=a_{k+1}$で囲まれる図形と$y=0, x=a_k, x=a_{k+1}$また$(a_k, f(a_k))$と$(a_{k+1}, f(a_{k+1}))$を結んだ線分で囲まれる図形の面積を比べる。
図から双方が近い値を取る事が分かるので、これらを比べると \begin{eqnarray*} \int_{a_k}^{a_{k+1}} f(x)dx &\simeq& \frac{1}{2} \left( a_{k+1} - a_k \right)\left( f(k+1) + f(k) \right) \\ \int_{a_k}^{a_{k+1}} f(x)dx &\simeq& \frac{1}{2} f(a_k)\left( f(k+1) + f(k) \right) \\ \int_{a_s}^{a_{t}} f(x)dx &\simeq& \frac{1}{2} \sum_{k=s}^{t-1} f(a_k)\left( f(k+1) + f(k) \right) \end{eqnarray*} である。
このように求める事が出来ました。また、積分近似では$f''(x)$の絶対値が小さいほど、傾きの変化が少なくなることから近似が一致していきます。
解決済み数列での積分近似の利用
積分近似で近似をすると漸化式(の近似式)が複雑になりますので、積分近似を漸化式の近似解解法として利用するのは難しいです。
そこで、あらかじめ解決された数列において積分近似を使います。
\[ f(a_n) = a_{n+1} - a_n = \sqrt{n+1} - \sqrt{n} \] である。また、 \begin{eqnarray*} a_{n+1}^2 &=& a_n^2 + 1 \\ a_{n+1} &=& \sqrt{a_n^2+1} \end{eqnarray*} より \[ f(a_n) = a_{n+1} - a_n = \sqrt{a_n^2 + 1} - a_n \] なので、 \[ f(x) = \sqrt{x^2 + 1} - x \] である。この数列で積分近似を用いると \begin{eqnarray*} \int_{\sqrt{s}}^{\sqrt{t}} \left( \sqrt{x^2 + 1} - x \right)dx &\simeq& \frac{1}{2} \sum_{k=s}^{t - 1} \left(\sqrt{k+1} - \sqrt{k} \right)\left( \sqrt{k+2} - \sqrt{k} \right) \\ 2\int_{\sqrt{s}}^{\sqrt{t}} \sqrt{x^2 + 1}dx - \left[x^2\right]_{\sqrt{s}}^{\sqrt{t}} &\simeq& \sum_{k=s}^{t-1} \left(\sqrt{k+1} - \sqrt{k} \right)\left( \sqrt{k+2} - \sqrt{k} \right) \\ 2\int_{\sqrt{s}}^{\sqrt{t}} \sqrt{x^2 + 1}dx - t + s &\simeq& \sum_{k=s}^{t-1} \left(\sqrt{k+1} - \sqrt{k} \right)\left( \sqrt{k+2} - \sqrt{k} \right) \end{eqnarray*} ここで、 \begin{equation} \int \sqrt{x^2+1} dx = \frac{1}{2} \left( x\sqrt{x^2+1} + \log\left(x+\sqrt{x^2+1}\right) \right) + C \end{equation} なので、 \begin{eqnarray*} \left[ x\sqrt{x^2+1} + \log\left(x+\sqrt{x^2+1}\right) \right]_{\sqrt{s}}^{\sqrt{t}} - t + s &\simeq& \sum_{k=s}^{t-1} \left(\sqrt{k+1} - \sqrt{k} \right)\left( \sqrt{k+2} - \sqrt{k} \right) \\ \sqrt{t^2 + t} - t - \sqrt{s^2 + s} + s + \log\left(\frac{\sqrt{t}+\sqrt{t+1}}{\sqrt{s}+\sqrt{s+1}}\right)&\simeq& \sum_{k=s}^{t-1} \left(\sqrt{k+1} - \sqrt{k} \right)\left( \sqrt{k+2} - \sqrt{k} \right) \end{eqnarray*} である。
このように、不等式を算出する事ができました。左辺右辺を入れ替えると、シグマの計算の近似となっている事が分かります。
この近似は$s$が大きくなるほど精度が上がりますが、$s=1, t=10000$の時ですら、左辺または右辺な値が$4.5$程度なのに対して誤差は$0.005$に満たないほど精度が高いです。
また、この数列において、$n$を無限に飛ばす事を考えます。
このように積分近似では、シグマ計算の極限を求める事も可能となります。
以上が積分近似でした。
微分近似
微分近似の概要
微分近似では、積分近似のようにグラフの形を見るのではなく、式に着目します。
となる関数$g$について考えます($h$は定数)。
このように関数$g$が満たす微分方程式の近似式を得る事ができます。これを逆に考えると、以下の事が言えます。
この時、$h \to 0$ で完全一致するように見えますが、実はそうはなりません。
$g$が満たす微分方程式において $t = xh$ である事を考慮すると、右辺は$x$の関数となり、左辺は$g(xh)$の関数となるため、$g(xh)$は$h$に依存しない事が分かります。
すなわち、$g(nh)$は$h$に依存しないので、$h=1$の時と$h \to 0$の時の近似も変わらない事が分かります。よって一致するという事はあり得ません。
また、これを活かすと以下のように言い換えられます。
また、この微分方程式を見ると分かりますが、この近似方法は$\frac{1}{f(u)}$の積分が可能であれば$g(x)$を求める事が出来ます。
微分近似を用いて漸化式の解にアプローチした例ば以下の記事にあります。
【数列】数列Yとの闘い p4:近似式【シリーズ記事】
微分近似の特徴
微分近似は視覚的な近似方法ではなく式からの近似方法ですので、どういった場合に近似の精度が上がるのか分かりずらいです。
そこで、微分近似の式と漸化式を、差分記号(定義はここを参照)を用いて表して視覚的に比較できるようにします。
ここから、微分近似は$g'(x)$の変化が少ない方が近似の精度が向上するという事が分かります。$g'(x)$は$f(g(x))$と等しいですので、
以上の事から、$|f(x)|$と$|f'(x)|$が小さい時に精度が向上するという事が分かります。
以上、微分近似でした。
微分近似や積分近似は、離散的なものを連続的なものと比べて近似するというものでした。是非皆さんもこのような近似を探してみてると面白いと思います。
それでは、お疲れさまでした。

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