[R] How to formulate an analytical gradient?

[francogrex]

How to formulate an analytical gradient?

Suppose I have the following function/expression:

fr<-function(x){
x1=x[1]
x2=x[2]
x3=x[3]
x4=x[4]
x5=x[5]
z<-((gamma(x1+n)))/((gamma(x1)*factorial(n))*((1+(e/x2))^x1)*((1+(x2/e))^n))
v<-((gamma(x3+n)))/((gamma(x3)*factorial(n))*((1+(e/x4))^x3)*((1+(x4/e))^n))

sum(log( (x5*z)+ ((1-x5)*v) ))
}

These are a mix of two negative binomial distributions, where n and e are
know vectors, and I would like to calculate the maxiumum likelihood
estimates of the parameters x1,x2,x3,x4 and X5
I am relying on numerical gradients but I think if I use an analytical one
it will be more accurate especially when number of parameters is more than
4.

Thanks.
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