# [R] How to find when a value is reached given a function?

Duncan Murdoch murdoch@dunc@n @end|ng |rom gm@||@com
Sun Jan 24 21:40:06 CET 2021

```On 24/01/2021 2:57 p.m., Luigi Marongiu wrote:
> Hello
> I am trying to simulate a PCR by running a logistic equation. So I set
> the function:
> ```
> PCR <- function(initCopy, dupRate, Carry) {
>    ROI_T = initCopy
>    A = array()
>    for (i in 1:45) {
>      ROI_TplusOne <- ROI_T * dupRate * (1 - ROI_T/Carry)
>      A[i] <- ROI_TplusOne
>      ROI_T <- ROI_TplusOne
>    }
>    return(A)
> }
> ```
> Which returns an array that follows the logistic shape, for instance
> ```
> d <- 2
> K <- 10^13
> A_0 <- 10000
> PCR_array <- PCR(A_0, d, K)
> plot(PCR_array)
> ```
> Given the formula `ROI_TplusOne <- ROI_T * dupRate * (1 -
> ROI_T/Carry)`, is it possible to determine at what time point `i` a
> given threshold is reached? For instance, what fractional value of i
> returns 1000 000 copies?

The brute force answer is just to try it and count how far you need to
go.  This is really simple, but really inefficient.

The faster and more elegant way is to solve the recursive relation for
an explicit solution.  You've got a quadratic recurrence relation;
there's no general solution to those, but there are solutions in special
cases.  See https://math.stackexchange.com/q/3179834 and links therein
for some hints.

Duncan Murdoch

```