Integration in R language - r

Question

I'm trying to compute the integral between 1 and some cutoff 'cut' of the function given in the R-code below as 'int'. It depends on 2 parameters dM[i] and dLambda[j] defined before I make the integration and for each pair I save the results in vector 'vec':
vec = c() #vector for INT values: this is our goal
dM = seq(from = 0, to = 3, by = 0.01) #vector for mass density parameter
dLambda = seq(from = -1.5, to = 3, by = 0.01) #vector for vacuum energy density parameter
for (i in 1:length(dM)) {
for (j in 1:length(dLambda)) {
int = function(x) ((dM[i]*x^4*(x - 1) + dLambda[j]*x^2*(1 - x^2) + x^4)^(-1/2))
cut = 30
INT_data = integrate(int, 1, cut)
INT = INT_data$value
vec = c(vec, INT)
}
}
But when I run the script I get the error: "Error in integrate(int, 1, cut) : non-finite function value
". Nonetheless, if I tried the following code
int = function(x) ((0*x^4*(x - 1) -1.5*x^2*(1 - x^2) + x^4)^(-1/2))
cut = 30
INT_data = integrate(int, 1, cut)
INT = INT_data$value
vec = c(vec, INT)
I get the correct result without any error. So the error above is not true, it can calculate the integral but it seems that R cannot work it out if I use the 2 'for'-loops. How can I re-write the code so I can compute all the different values for dM[i] and dLambda[j] I want?

Answer 1

Your function is only defined for some values of dM and dLambda. You can use the try() function to attempt evaluation, but not stop in case an error occurs.
It's also a lot more efficient to pre-allocate the object to hold the results; running vec = c(vec, INT) gradually grows it, and that's very slow, because R needs to keep creating new vectors just one element longer than the last one.
This code fixes both issues, and then plots the result:
dM <- seq(from = 0, to = 3, by = 0.01) #vector for mass density parameter
dLambda <- seq(from = -1.5, to = 3, by = 0.01) #vector for vacuum energy density parameter
result <- matrix(NA, length(dM), length(dLambda))
for (i in 1:length(dM)) {
for (j in 1:length(dLambda)) {
int <- function(x) ((dM[i]*x^4*(x - 1) + dLambda[j]*x^2*(1 - x^2) + x^4)^(-1/2))
cut <- 30
INT_data <- try(integrate(int, 1, cut), silent = TRUE)
if (!inherits(INT_data, "try-error"))
result[i, j] <- INT_data$value
}
}
image(dM, dLambda, result)
Edited to add: Here's how this works. If integrate signals an error in your original code, the loop will stop. try() prevents that. If there's no error, it returns the result of the integrate call. If there is an error, it returns an object with information about the error. That object has class "try-error", so the check if (!inherits(INT_data, "try-error")) is basically asking "Was there an error?" If there was an error, nothing happens, and that entry of the result is left as NA, as it was initialized. The loop then goes on to try the next dM, dLambda pair.

Answer 2

The problem is mathematical rather than being related to coding. The function is not defined for the whole domain you are integrating. With dM[1] = 0 and dLambda > 1, your expression
(dM[i]*x^4*(x - 1) + dLambda[j]*x^2*(1 - x^2) + x^4)^(-1/2)
simplifies to
(dLambda[j] * x^2 * (1 - x^2) + x^4)^(-1/2)
so let's take dLambda[j] at 1.01, which is where your calculation stops:
(1.01 * x^2 * (1 - x^2) + x^4)^(-1/2)
which is
(1.01 * x^2 - 1.01 * x^4 + x^4)^(-1/2)
or
(1.01 * x^2 - 0.01 x^4)^(-1/2)
Now, you are evaluating x between 1 and 30. So what happens when x = 11?
(1.01 * 121 - 0.01 * 14641)^(-1/2)
This leaves you
(122.21 - 146.41)^(-1/2)
which is equivalent to
1/sqrt(-24.2)
So the reason for the error is that you are integrating a function in a domain in which it is undefined.
The function is badly behaved for other values of dM too, with infinite peaks in the midst of the range, so even using the integrate(..., stop.on.error = F) option won't allow you to keep calculating because you will get an infinite sum.