Does the construction of the dataframe change the values of the list?

If I write this:

f <- function(lambda, a_0, n) {
  a <- numeric(n)
  a[1] <- a_0
  for (i in 2:n) {
    a[i] <- lambda * a[i-1] * (1 - a[i-1])
  }
  return(a)
}
f(4, 0.75, 100)

then a list is produced that contains only the number 0.75. This is expected.

However, if I write:

lambda <- 4
n <- 100
a_0_values <- seq(0.05, 0.95, by = 0.05)
df <- data.frame(matrix(ncol = length(a_0_values), nrow = n))
for (i in 1:length(a_0_values)) {
  df[, i] <- f(lambda, a_0_values[i], n)
}
df[,15]

then a list is produced that initially contains 0.75, but after a while, it starts changing (see the table I provide at the end).

I tried increasing the length of the generated list by writing up to f(4, 0.75, 1000000), and it still worked as expected. It only produced 0.75, as it should. Of course, even if it did change after a certain point, it still doesn't explain why f(4, 0.75, 100) and df[,15] produce different results.

Why is that?

df[,15]
  [1] 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01
  [7] 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01
 [13] 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01
 [19] 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01
 [25] 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01 7.499999e-01
 [31] 7.500001e-01 7.499998e-01 7.500005e-01 7.499990e-01 7.500019e-01 7.499962e-01
 [37] 7.500076e-01 7.499847e-01 7.500305e-01 7.499390e-01 7.501221e-01 7.497558e-01
 [43] 7.504881e-01 7.490228e-01 7.519506e-01 7.460836e-01 7.577714e-01 7.342156e-01
 [49] 7.805721e-01 6.851171e-01 8.629266e-01 4.731372e-01 9.971136e-01 1.151244e-02
 [55] 4.551962e-02 1.737903e-01 5.743490e-01 9.778889e-01 8.648884e-02 3.160341e-01
 [61] 8.646262e-01 4.681910e-01 9.959528e-01 1.612345e-02 6.345395e-02 2.377102e-01
 [67] 7.248162e-01 7.978306e-01 6.451876e-01 9.156822e-01 3.088332e-01 8.538210e-01
 [73] 4.992429e-01 9.999977e-01 9.170494e-06 3.668164e-05 1.467212e-04 5.867986e-04
 [79] 2.345817e-03 9.361256e-03 3.709449e-02 1.428740e-01 4.898440e-01 9.995874e-01
 [85] 1.649634e-03 6.587650e-03 2.617701e-02 1.019671e-01 3.662792e-01 9.284750e-01
 [91] 2.656366e-01 7.802952e-01 6.857385e-01 8.620049e-01 4.758098e-01 9.976593e-01
 [97] 9.340721e-03 3.701389e-02 1.425754e-01 4.889907e-01

Solution

I think it might be the precision of a_0 value, since you can see that

> a_0_values[15] - 0.75
[1] 1.110223e-16

If you apply a_0_values[15] to f

> f(lambda, a_0_values[15], n)
  [1] 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01
  [6] 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01
 [11] 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01
 [16] 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01
 [21] 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01
 [26] 7.500000e-01 7.500000e-01 7.500000e-01 7.500000e-01 7.499999e-01
 [31] 7.500001e-01 7.499998e-01 7.500005e-01 7.499990e-01 7.500019e-01
 [36] 7.499962e-01 7.500076e-01 7.499847e-01 7.500305e-01 7.499390e-01
 [41] 7.501221e-01 7.497558e-01 7.504881e-01 7.490228e-01 7.519506e-01
 [46] 7.460836e-01 7.577714e-01 7.342156e-01 7.805721e-01 6.851171e-01
 [51] 8.629266e-01 4.731372e-01 9.971136e-01 1.151244e-02 4.551962e-02
 [56] 1.737903e-01 5.743490e-01 9.778889e-01 8.648884e-02 3.160341e-01
 [61] 8.646262e-01 4.681910e-01 9.959528e-01 1.612345e-02 6.345395e-02
 [66] 2.377102e-01 7.248162e-01 7.978306e-01 6.451876e-01 9.156822e-01
 [71] 3.088332e-01 8.538210e-01 4.992429e-01 9.999977e-01 9.170494e-06
 [76] 3.668164e-05 1.467212e-04 5.867986e-04 2.345817e-03 9.361256e-03
 [81] 3.709449e-02 1.428740e-01 4.898440e-01 9.995874e-01 1.649634e-03
 [86] 6.587650e-03 2.617701e-02 1.019671e-01 3.662792e-01 9.284750e-01
 [91] 2.656366e-01 7.802952e-01 6.857385e-01 8.620049e-01 4.758098e-01
 [96] 9.976593e-01 9.340721e-03 3.701389e-02 1.425754e-01 4.889907e-01

If you round it by 2 digits

> f(lambda, round(a_0_values[15], 2), n)
  [1] 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75
 [16] 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75
 [31] 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75
 [46] 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75
 [61] 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75
 [76] 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75
 [91] 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75