Friday, March 26, 2004
The whole thing misses me...
So it's about time I made an update. What makes it time? Well, it's just been too long. I can barely think of anything to post so I suppose that sort of defeats the purpose of writing anything. I think I'm just grumpy. Someones on their period and I'm getting stuck with the mood swings. But I guess I will go one with this post.
Now I want to talk about something that is near to my heart. Please bear with me, here goes...
Euler's method uses a fixed step (h) to approximate the exact solution of an equation. The value of the dependent variable (y) is chosen in relation to the iterations of h. The graph of the function y(t) is approximated by its tangent line at each step. The tangent line at the point (t0, y(t0)) is the function y0(t0) + y'(t0)(t - t0), with y'(t0) = f(t0, y0). Additionally, y0(t0) = y0, so the function at (t0, y(t0)) is y0 + f(t0, y0)(t - t0). Since t is increasing in iterations of h, it follows that t = t1 = t0 + h (and h = t1 - t0). Thus, y1 = y0 + f(t0, y0)h. This formula extends, each time building on itself to develop into the formula yn = yn-1 + f(tn-1, yn-1)h with tn = tn-1 + h.
I've been waiting to get that off of my chest for quite some time now. Thank you for listening. Now, it is time for bed.
(Partial credit to Andiphone.)