A..4 RKF45 (Runge-Kutta-Fehlberg 公式)

(この公式については、次回の黒板前講義で説明するが、余裕があればチャレンジしてみるとよい。)

	$\displaystyle x_{j+1}$	$\displaystyle =$	$\displaystyle x_j + h_j\left( \frac{16}{135}k_1 +\frac{6656}{12825}k_3 +\frac{28561}{56430}k_4 -\frac{9}{50}k_5 +\frac{2}{55}k_6 \right),$
	$\displaystyle x_{j+1}^\ast$	$\displaystyle =$	$\displaystyle x_j+h_j \left( \frac{25}{216}k_1+\frac{1408}{2565}k_3+\frac{2197}{4104}k_4-\frac15k_5 \right),\quad \quad h_j=t_{j+1}-t_j,$

$\displaystyle k_1$	$\displaystyle =$	$\displaystyle f(t_j,x_j),$
$\displaystyle k_2$	$\displaystyle =$	$\displaystyle f\left(t_j+\frac14 h, x_j+\frac14 h k_1\right),$
$\displaystyle k_3$	$\displaystyle =$	$\displaystyle f\left(t_j+\frac38 h, x_j+\frac1{32}h(3 k_1+9k_2)\right),$
$\displaystyle k_4$	$\displaystyle =$	$\displaystyle f\left(t_j+\frac{12}{13}h, x_j+\frac1{2197}h(1932k_1-7200k_2+7296k_3)\right),$
$\displaystyle k_5$	$\displaystyle =$	$\displaystyle f\left(t_j+h, x_j+h \left(\frac{439}{216}k_1-8k_2+\frac{3680}{513}k_3 -\frac{845}{4104}k_4 \right)\right),$
$\displaystyle k_6$	$\displaystyle =$	$\displaystyle f\left(t_j+\frac12 h, x_j+h \left(-\frac8{27}k_1+2k_2-\frac{3544}{2565}k_3 +\frac{1859}{4104}k_4-\frac{11}{40}k_5 \right) \right).$