lsoda creating fictional values? - r

Question

I'm currently developing a model to integrate microbiology and geochemistry that uses lsoda to solve a great number of differential equations. The model is far too large to be posted here because it's made of several modules, but I have some very weird happening.
These are my initial values
enter image description here
I have initialised them as zero because I don't want any kind of microbe, just to check how the chemistry would change without microbes. However after 5 or 6 steps I start seeing values that are different from zero in some of my microbial variables:
enter image description here.
I wonder if maybe lsoda is doing some kind of round and that's why I get these values, because otherwise I cannot explain where these values are popping out from. If this is the case, does anyone know how to stop this kind of round-ups?

Answer 1

Short answer
No, lsoda does not create "fictional" values.
Detailed Answer
As #Ben Bolker stated, "floating point arithmetics is intrinsically imprecise". I want to add that all solvers including lsoda compute approximations, and that nonlinear models can magnify errors.
You can check your model, if all derivatives are exactly zero by calling the model function separately, without the solver, see the following example of a Lorenz system, where all states are set to zero:
library(deSolve)
Lorenz <- function(t, state, parameters) {
with(as.list(c(state, parameters)), {
dX <- a * X + Y * Z
dY <- b * (Y - Z)
dZ <- -X * Y + c * Y - Z
list(c(dX, dY, dZ))
})
}
parameters <- c(a = -8/3, b = -10, c = 28)
state <- c(X = 0, Y = 0, Z = 0)
times <- seq(0, 500, by = 1)
range(Lorenz(0, state, parameters))
that gives
[1] 0 0
Counter example
If we modify the model, so that one derivative is slightly different from zero due to rounding error, e.g. by using Ben's example:
Fancy <- function(t, state, parameters) {
with(as.list(c(state, parameters)), {
dX <- a * X + Y * Z + 27 * (sqrt(X + Y + Z + 2)^2 - 2)
dY <- b * (Y - Z)
dZ <- -X * Y + c * Y - Z
list(c(dX, dY, dZ))
})
}
range(Fancy(0, state, parameters))
then it gives:
[1] 0.000000e+00 1.199041e-14
Depending on tolerances, such a model may then fluctuate or even blow up. Surprisingly, a smaller tolerance may sometimes be worse:
out1 <- ode(y = state, times = times, func = Lorenz, parms = parameters, atol=1e-6)
out2 <- ode(y = state, times = times, func = Fancy, parms = parameters, atol=1e-6)
out3 <- ode(y = state, times = times, func = Fancy, parms = parameters, atol=1e-8)
par(mfrow=c(3,1))
plot(out1[,1:2], type="l", main="Lorenz, derivatives = 0")
plot(out2[,1:2], type="l", main="Fancy, small error in derivatives, atol=1e-6")
plot(out3[,1:2], type="l", main="Fancy, small error in derivatives, atol=1e-8")
Conclusion
lsodaworks well within the limitations of floating point arithmetics.
models should be carefully designed and tested.