[R] LSODA not accurate when RK4 is; what's going on?

[Ravi Varadhan]

John,

I should also mention that by looking at the R source for lsoda() you can
see that the default values for "atol" and "rtol" are 1.e-06.  This should
have been mentioned in the help file, but is not.

Another approach to solving your problem is to restrict "hmax", the maximum
steplength.

s4 <- lsoda(y=sqrt(pi/2), times=t4, func=fn, parms=0, hmax=0.001)

plot(s4, type="l")

Best,
Ravi.

----------------------------------------------------------------------------
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Ravi Varadhan, Ph.D.

Assistant Professor, The Center on Aging and Health

Division of Geriatric Medicine and Gerontology 

Johns Hopkins University

Ph: (410) 502-2619

Fax: (410) 614-9625

Email: rvaradhan at jhmi.edu

Webpage:  http://www.jhsph.edu/agingandhealth/People/Faculty/Varadhan.html

 

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-----Original Message-----
From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org] On
Behalf Of John Tillinghast
Sent: Wednesday, April 09, 2008 7:18 PM
To: r-help at r-project.org
Subject: [R] LSODA not accurate when RK4 is; what's going on?

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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