Polynomial evaluation

│ English (en) │ français (fr) │

Introduction

It is often useful to evaluate a polynomial at a given point. The method shown below allows to calculate it by using only multiplication and addition, without any power.

Horner's method

Only 1-degree polynomial are evaluated, e.g. 3.x^2 + 2.x + 1, can be written, by means of an astute factorization, ((3.x +2).x + 1).

program PolyTest;

type 
  // coef's rank = it's degree
  TDegres = 0..10; 
  TPolynome = record 
    Deg: TDegres; 
    Coefs: array[TDegres] of Double; 
  end;

function PolyEval(const aPoly: TPolynome; X: Double): Double; 
var 
  i: TDegres; 
begin 
  with aPoly do 
  begin 
    Result := 0; 
    for i := Deg downto Low(Coefs) do 
      Result := Result * X + Coefs[i]; 
  end; 
End; 
  
(* Exemples *) 
var 
  P: TPolynome; 
begin 
  P.Deg := 2; 
  P.Coefs[2] := 3; 
  P.Coefs[1] := 2; 
  P.Coefs[0] := 1;   
  WriteLn('P(',7,') = ', PolyEval(7):10:2);  
  
  P.Deg := 0; 
  P.Coefs[2] := 0; 
  P.Coefs[1] := 0; 
  P.Coefs[0] := 1;   
  WriteLn('P(',7,') = ', PolyEval(7):10:2);  
End

See also

Horner's method (wiki)