Content deleted Content added
Line 74:
:<math> p(t) = \sum_{j=0}^{s-1} \frac{(-1)^{s-j-1}f(t_{n+j}, y_{n+j})}{j!(s-j-1)!h^{s-1}} \prod_{i=0 \atop i\ne j}^{s-1} (t-t_{n+i}). </math>
The polynomial ''p'' is locally a good approximation of the right-hand side of the differential equation <math> y' = f(t,y) </math> that is to be solved, so consider the equation <math> y' = p(t) </math> instead. This equation can be solved exactly; the solution is simply the integral of ''p''. This suggests taking
:<math> y_{n+s} = y_{n+s-1} + \int_{t_{n+s-1}}^{t_{n+s}} p(t)\,\mathrm dt. </math>
The Adams–Bashforth method arises when the formula for ''p'' is substituted. The coefficients <math> b_j </math> turn out to be given by
:<math> b_{s-j-1} = \frac{(-1)^j}{j!(s-j-1)!} \int_0^1 \prod_{i=0 \atop i\ne j}^{s-1} (u+i) \,\mathrm du, \qquad \text{for } j=0,\ldots,s-1. </math>
Replacing <math> f(t, y) </math> by its interpolant ''p'' incurs an error of order ''h''<sup>''s''</sup>, and it follows that the ''s''-step Adams–Bashforth method has indeed order ''s'' {{harv|Iserles|1996|loc=§2.1}}
| |||