Horner's rule for two-variable polynomial

Question

Horner\'s rule is used to simplify the process of evaluating a polynomial at specific variable values. https://rosettacode.org/wiki/Horner%27s_rule_for_polynomial_evaluation

Answer 1

You're very close. Let's begin by formalizing the problem. Given coefficients C as a nested list like you indicated, you want to evaluate

Notice that you can pull out the s from the inner sum, to get

Look closely at the inner sum. This is just a polynomial on variable x with coefficients given by . In SML, we can write the inner sum in terms of your horner function as

fun sumj Ci = horner Ci x

Let's go a step further and define

In SML, this is val D = map sumj C. We can now write the original polynomial in terms of D:

It should be clear that this is just another instance of horner, since we have a polynomial with coefficients . In SML, the value of this polynomial is

horner D y

...and we're done!

---

Here's the final code:

fun horner2 C x y =
  let
    fun sumj Ci = horner Ci x
    val D = map sumj C
  in
    horner D y
  end

Isn't that nice? All we need is multiple applications of Horner's method, and map.