Horner rule for two variables of a polynomial

Question

The Horner rule is used to simplify the process of computing a polynomial based on specific values ​​of variables. https://rosettacode.org/wiki/Horner%27s_rule_for_polynomial_evaluation#Standard_ML

I easily applied the method using SML to one variable polynomial represented as an int list:

fun horner coeffList x = foldr (fn (a, b) => a + b * x) (0.0) coeffList

It works great. Then we can call it using:

- val test = horner [1.0, 2.0, 3.0] 2.0;
> val test = 17.0 : real

Where [1.0, 2.0, 3.0]is the list representing the coefficients of the polynomial, 2.0is the value of the variable x, and 17.0is the result of computing the polynomial.

My problem is this: We have two polynomial variables represented (list of list of objects). The nth element in the high-level list will represent all polynomial members containing y ^ n, and the mth element in the low-level list will represent all polynomial members containing x ^ m.

For example: [[2],[3,0,0,3],[1,2]]- polynomial

(2 (x ^ 0) (y ^ 0)) +
(3 (x ^ 0) (y ^ 1) + 0 (x ^ 1) (y ^ 1) + 0 (x ^ 2) (y ^ 1) + 3 (x ^ 3) (y ^ 1)) +
(1 (x ^ 0) (y ^ 2) + 2 (x ^ 1) (y ^ 2))

The function should return the polynomial value for the given x and y.

I tried various methods using the mlton compiler.

• First I tried the foldr nested function:

  fun evalXY (z::zs) x y = 
          foldr 
          (fn (s, li:list) => 
              s + ((foldr (fn(a, b) => a + b*x) 0 li)*y)
          )
          0 
          z:zs
  

, "s" , , "a" . , foldr, "" , foldr , foldr. , , . * , , , , foldr, . > , li , . *

2. . :

   fun evalXY (z::zs) x y = 
           map 
           (foldr (fn(a, b) => a + b*x) 0 ???)
           z:zs
   

   * , ints. int, int foldr. , y . : fn evalXY: (int list list) * int * int) → ... → int list *

SML, , , - . , , ,

Answer 1

. . C , ,

, s ,

. x , . SML horner

fun sumj Ci = horner Ci x

SML val D = map sumj C. D:

, horner, . SML

horner D y

... !

---

:

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

? map.

Answer 2

, , . horner, horner horner applied to inner list x , :

fun evalXY coeffLists x y = horner (map (fn coeffList => horner coeffList x) coeffLists) y

horner , .

, horner, evalXY:

fun horner x coeffList = foldr (fn (a, b) => a + b * x) (0.0) coeffList
fun evalXY x y coeffLists = horner y (map (horner x) coeffLists)

, currying , , horner x coeffList, fn coeffList => horner coeffList x. , curries , , .

, SML . horner , horner list 2. horner list 2.0. , 0, 0.0 .