More functions

2026-01-20

Recursive functions

the prototypical function - similar to proof by induction:

• define values for endings

• get value for f(x) from value of f(x-1)

Factorials

fac <- function(x) {
  stopifnot(is.numeric(x), x >= 0, length(x) == 1)
  if (x == 0) return(1)
  if (x == 1) return(1)
  return(x*fac(x-1))
}

fac(5)

[1] 120

fac(60)

[1] 8.320987e+81

fac(2.5)

Error in `fac()`:
! x >= 0 is not TRUE

Advantage of recursions over loops

Calling functions generally leads to cleaner code (function naming is crucial!)

Focus on “what” is done rather than “how”

… at the cost of memory (function stack)

Your Turn

Use the factorial function to implement the binomial coefficient as a function binom(n, k)

\(\binom{n}{k} = \frac{n!}{k! (n-k)!}\)

What is the result for binom(175,2)?

Rewrite the function to be able to deal with binom(175,2) and binom(175,173)

Floating Point Numbers (FPN)

For base \(\beta\) and precision \(p\) (both \(\beta\) and \(p\) are positive integers), FPNs are represented as

\[ d.dddddd \times \beta^e, \] where \(d.dddddd\) is called the significand and has \(p\) digits.

If the first \(d\) is non-zero, the representation is called normalized. Normalized FP representations are unique.

Floating point numbers have limits

.Machine$double.xmax

[1] 1.797693e+308

Values outside of these limits are encoded as Inf (and -Inf for negative values) in R.

fac(170)

[1] 7.257416e+306

fac(171)

[1] Inf

-fac(171)

[1] -Inf

What happened here?

fac(0.05 + 0.1 - 0.15)

Error in `fac()`:
! x >= 0 is not TRUE

fac

function (x) 
{
    stopifnot(is.numeric(x), x >= 0, length(x) == 1)
    if (x == 0) 
        return(1)
    if (x == 1) 
        return(1)
    return(x * fac(x - 1))
}
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