I'm solving the differential equation dy/dx = xy-1 with y(0) = sqrt(pi/2).
This can be used in computing the tail of the normal distribution.
(The actual solution is y(x) = exp(x^2/2) * Integral_x_inf {exp(-t^2/2) dt}
= Integral_0_inf {exp (-xt - t^2/2) dt}. For large x, y ~ 1/x, starting
around x~2.)
I'm testing both lsoda and rk4 from the package odesolve.
rk4 is accurate using step length 10^-2 and probably would be with even
larger steps.
lsoda is pretty accurate out to about x=4, then starts acting strangely. For
step length 10^-3, y suddenly starts to increase after 4, when it should be
strictly decreasing. For step length 10^-4, y instead turns down and start
dropping precipitously.
Any ideas why lsoda would go off the rails when rk4 does so well? I will
soon be using R to solve more complicated systems of ODEs which I don't
understand as well, so I want to know when it can mislead me.
Code:
t4 <- seq(0, 5, by=.0001)
> fn
function(t,y,parms=0){return(list(t*y-1))}
s4 <- lsoda(sqrt(pi/2), t4, fn, 0)
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