Horner's rule for polynomial evaluation

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Task
Horner's rule for polynomial evaluation
You are encouraged to solve this task according to the task description, using any language you may know.

A fast scheme for evaluating a polynomial such as:

when

.

is to arrange the computation as follows:

And compute the result from the innermost brackets outwards as in this pseudocode: 

coefficients := [-19, 7, -4, 6] # list coefficients of all x^0..x^n in order
x := 3
accumulator := 0
for i in length(coefficients) downto 1 do
    # Assumes 1-based indexing for arrays
    accumulator := ( accumulator * x ) + coefficients[i]
done
# accumulator now has the answer

Task Description

Create a routine that takes a list of coefficients of a polynomial in order of increasing powers of x; together with a value of x to compute its value at, and return the value of the polynomial at that value using Horner's rule.

Cf. Formal power series

Ada

<lang Ada>with Ada.Float_Text_IO; use Ada.Float_Text_IO;

procedure Horners_Rule is 

  type Coef is array(Positive range <>) of Float;

  function Horner(Coeffs: Coef; Val: Float) return Float is
     Res : Float := 0.0;
  begin
     for P in reverse Coeffs'Range loop
        Res := Res*Val + Coeffs(P);
     end loop;
     return Res;
  end Horner;

begin 

  Put(Horner(Coeffs => (-19.0, 7.0, -4.0, 6.0), Val => 3.0), Aft=>1, Exp=>0);

end Horners_Rule;</lang> Output: 

128.0

Aime

<lang aime>real horner(list coeffs, real x) { 

   integer i;
   real z;

   z = 0;

   i = l_length(coeffs);
   while (i) {

i -= 1; z *= x; z += l_q_real(coeffs, i); 

   }

   return z;

}


integer main(void) { 

   list l;

   l_append(l, -19r);
   l_append(l, 7.0);
   l_append(l, -real(4));
   l_append(l, 6r);

   o_real(1, horner(l, 3));
   o_byte('\n');

   return 0;

}</lang>

ALGOL 68

Works with: ALGOL 68G

<lang algol68>PROC horner = ([]REAL c, REAL x)REAL : ( 

 REAL res := 0.0;
 FOR i FROM UPB c BY -1 TO LWB c DO
   res := res * x + c[i]
 OD;
 res

);

main:( 

 [4]REAL coeffs := (-19.0, 7.0, -4.0, 6.0);
 print( horner(coeffs, 3.0) )

)</lang>

AutoHotkey

<lang autohotkey>Coefficients = -19, 7, -4, 6 x := 3

MsgBox, % EvalPolynom(Coefficients, x)


---------------------------------------------------------------------------

EvalPolynom(Coefficients, x) { ; using Horner's rule

---------------------------------------------------------------------------
   StringSplit, Co, coefficients, `,, %A_Space%
   Result := 0
   Loop, % Co0
       i := Co0 - A_Index + 1, Result := Result * x + Co%i%
   Return, Result

}</lang> Message box shows: 

128

C

Translation of: Fortran

<lang c>#include <stdio.h>

double horner(double *coeffs, int s, double x) { 

 int i;
 double res = 0.0;
 
 for(i=s-1; i >= 0; i--)
 {
   res = res * x + coeffs[i];
 }
 return res;

}


int main() { 

 double coeffs[] = { -19.0, 7.0, -4.0, 6.0 };
 
 printf("%5.1f\n", horner(coeffs, sizeof(coeffs)/sizeof(double), 3.0));
 return 0;

}</lang>

C#

<lang csharp>using System; using System.Linq;

class Program { 

   static double Horner(double[] coefficients, double variable)
   {
       return coefficients.Reverse().Aggregate(
               (accumulator, coefficient) => accumulator * variable + coefficient);
   }

   static void Main()
   {
       Console.WriteLine(Horner(new[] { -19.0, 7.0, -4.0, 6.0 }, 3.0));
   }

}</lang> Output: 

128

C++

The same C function works too, but another solution could be:

<lang cpp>#include <iostream>

  1. include <vector>

using namespace std;

double horner(vector<double> v, double x) { 

 double s = 0;
 
 for( vector<double>::const_reverse_iterator i = v.rbegin(); i != v.rend(); i++ )
   s = s*x + *i;
 return s;

}

int main() { 

 double c[] = { -19, 7, -4, 6 };
 vector<double> v(c, c + sizeof(c)/sizeof(double));
 cout << horner(v, 3.0) << endl;
 return 0;

}</lang>

Yet another solution, which is more idiomatic in C++ and works on any bidirectional sequence:

<lang cpp>

  1. include <iostream>

template<typename BidirIter> 

double horner(BidirIter begin, BidirIter end, double x)

{ 

 double result = 0;
 while (end != begin)
   result = result*x + *--end;
 return result;

}

int main() { 

 double c[] = { -19, 7, -4, 6 };
 std::cout << horner(c, c + 4, 3) << std::endl;

} </lang>

Clojure

<lang clojure>(defn horner [coeffs x] 

 (reduce (fn [acc coeff] (+ (* acc x) coeff)) (reverse coeffs)))

(println (horner [-19 7 -4 6] 3))</lang>

CoffeeScript

<lang coffeescript> eval_poly = (x, coefficients) -> 

 # coefficients are for ascending powers
 return 0 if coefficients.length == 0
 ones_place = coefficients.shift()
 x * eval_poly(x, coefficients) + ones_place

console.log eval_poly 3, [-19, 7, -4, 6] # 128 console.log eval_poly 10, [4, 3, 2, 1] # 1234 console.log eval_poly 2, [1, 1, 0, 0, 1] # 19 </lang>

Common Lisp

<lang lisp>(defun horner (coeffs x) 

 (reduce #'(lambda (coef acc) (+ (* acc x) coef))

coeffs :from-end t :initial-value 0))</lang>

D

The poly() function of the standard library std.math module uses Horner's rule: <lang d>import std.stdio, std.math;

void main() { 

   writeln(poly(3.0, [-19, 7, -4, 6]));

}</lang> Basic implementation: <lang d>import std.stdio, std.traits;

CommonType!(U, V) horner(U, V)(U[] p, V x) { 

   typeof(return) accumulator = 0;
   foreach_reverse (c; p)
       accumulator = accumulator * x + c;
   return accumulator;

}

void main() { 

   writeln([-19, 7, -4, 6].horner(3.0));

}</lang> More functional style: <lang d>import std.stdio, std.algorithm, std.range;

auto horner(U, V)(U[] p, V x) { 

   return reduce!((a, b) => a * x + b)(cast(V)0, retro(p));

}

void main() { 

   writeln([-19, 7, -4, 6].horner(3.0));

}</lang>

E

<lang e>def makeHornerPolynomial(coefficients :List) { 

   def indexing := (0..!coefficients.size()).descending()
   return def hornerPolynomial(x) {
       var acc := 0
       for i in indexing {
           acc := acc * x + coefficients[i]
       }
       return acc
   }

}</lang>

<lang e>? makeHornerPolynomial([-19, 7, -4, 6])(3)

  1. value: 128</lang>

Erlang

<lang erlang> horner(L,X) -> 

 lists:foldl(fun(C, Acc) -> X*Acc+C end,0, lists:reverse(L)).

t() -> 

 horner([-19,7,-4,6], 3).

</lang>

F#

<lang fsharp> let horner l x = 

   List.rev l |> List.fold ( fun acc c -> x*acc+c) 0

horner [-19;7;-4;6] 3 </lang>

Factor

<lang factor>: horner ( coeff x -- res ) 

   [ <reversed> 0 ] dip '[ [ _ * ] dip + ] reduce ;</lang>

( scratchpad ) { -19 7 -4 6 } 3 horner .
128

Forth

<lang forth>: fhorner ( coeffs len F: x -- F: val ) 

 0e
 floats bounds ?do
   fover f*  i f@ f+
 1 floats +loop
 fswap fdrop ;

create coeffs 6e f, -4e f, 7e f, -19e f,

coeffs 4 3e fhorner f. \ 128.</lang>

Fortran

Works with: Fortran version 90 and later

<lang fortran>program test_horner 

 implicit none

 write (*, '(f5.1)') horner ((/-19.0, 7.0, -4.0, 6.0/), 3.0)

contains 

 function horner (coeffs, x) result (res)

   implicit none
   real, dimension (:), intent (in) :: coeffs
   real, intent (in) :: x
   real :: res
   integer :: i

   res = 0.0
   do i = size (coeffs), 1, -1
     res = res * x + coeffs (i)
   end do

 end function horner

end program test_horner</lang> Output: 

128.0

Fortran 77

<lang fortran> FUNCTION HORNER(N,A,X) 

     IMPLICIT NONE
     INTEGER I,N
     DOUBLE PRECISION A(N),X,Y,HORNER
     Y=A(N)
     DO I=N-1,1,-1
     Y=Y*X+A(I)
     END DO
     HORNER=Y
     END</lang>

As a matter of fact, computing the derivative is not much more difficult (see Roundoff in Polynomial Evaluation, W. Kahan, 1986). The following subroutine computes both polynomial value and derivative for argument x.

<lang fortran> SUBROUTINE HORNER2(N,A,X,Y,Z) C COMPUTE POLYNOMIAL VALUE AND DERIVATIVE C SEE "ROUNDOFF IN POLYNOMIAL EVALUATION", W. KAHAN, 1986 C POLY: A(1)+A(2)*X+...+A(N)*X**(N-1) C Y: VALUE, Z: DERIVATIVE 

     IMPLICIT NONE
     INTEGER I,N
     DOUBLE PRECISION A(N),X,Y,Z
     Z=0.0D0
     Y=A(N)
     DO 10 I=N-1,1,-1
     Z=Z*X+Y
  10 Y=Y*X+A(I)
     END</lang>

GAP

<lang gap># The idiomatic way to compute with polynomials

x := Indeterminate(Rationals, "x");

  1. This is a value in a polynomial ring, not a function

p := 6*x^3 - 4*x^2 + 7*x - 19;

Value(p, 3);

  1. 128

u := CoefficientsOfUnivariatePolynomial(p);

  1. [ -19, 7, -4, 6 ]
  1. One may also create the polynomial from coefficients

q := UnivariatePolynomial(Rationals, [-19, 7, -4, 6], x);

  1. 6*x^3-4*x^2+7*x-19

p = q;

  1. true
  1. Now a Horner implementation

Horner := function(coef, x) local v, c; v := 0; for c in Reversed(coef) do v := x*v + c; od; return v; end;

Horner(u, 3);

  1. 128</lang>

Go

<lang go>package main

import "fmt"

func horner(x int64, c []int64) (acc int64) { 

   for i := len(c) - 1; i >= 0; i-- {
       acc = acc*x + c[i]
   }
   return

}

func main() { 

   fmt.Println(horner(3, []int64{-19, 7, -4, 6}))

}</lang> Output: 

128

Groovy

Solution: <lang groovy>def hornersRule = { coeff, x -> coeff.reverse().inject(0) { accum, c -> (accum * x) + c } }</lang>

Test includes demonstration of currying to create polynomial functions of one variable from generic Horner's rule calculation. Also demonstrates constructing the derivative function for the given polynomial. And finally demonstrates in the Newton-Raphson method to find one of the polynomial's roots using the polynomial and derivative functions constructed earlier. <lang groovy>def coefficients = [-19g, 7g, -4g, 6g] println (["p coefficients":coefficients])

def testPoly = hornersRule.curry(coefficients) println (["p(3)":testPoly(3g)]) println (["p(0)":testPoly(0g)])

def derivativeCoefficients = { coeff -> (1..<(coeff.size())).collect { coeff[it] * it } } println (["p' coefficients":derivativeCoefficients(coefficients)])

def testDeriv = hornersRule.curry(derivativeCoefficients(coefficients)) println (["p'(3)":testDeriv(3g)]) println (["p'(0)":testDeriv(0g)])

def newtonRaphson = { x, f, fPrime -> 

   while (f(x).abs() > 0.0001) {
       x -= f(x)/fPrime(x)
   }
   x

}

def root = newtonRaphson(3g, testPoly, testDeriv) println ([root:root.toString()[0..5], "p(root)":testPoly(root).toString()[0..5], "p'(root)":testDeriv(root).toString()[0..5]])</lang>

Output: 

[p coefficients:[-19, 7, -4, 6]]
[p(3):128]
[p(0):-19]
[p' coefficients:[7, -8, 18]]
[p'(3):145]
[p'(0):7]
[root:1.4183, p(root):0.0000, p'(root):31.862]

Haskell

<lang haskell>horner :: (Num a) => a -> [a] -> a horner x = foldr (\a b -> a + b*x) 0

main = print $ horner 3 [-19, 7, -4, 6]</lang>

HicEst

<lang HicEst>REAL :: x=3, coeffs(4) DATA coeffs/-19.0, 7.0, -4.0, 6.0/

WRITE(Messagebox) Horner(coeffs, x) ! shows 128

FUNCTION Horner(c, x) 

  DIMENSION c(1)
  Horner = 0
  DO i = LEN(c), 1, -1
     Horner = x*Horner + c(i)
  ENDDO

END</lang>

Icon and Unicon

<lang Icon> procedure poly_eval (x, coeffs) 

 accumulator := 0
 every index := *coeffs to 1 by -1 do 
   accumulator := accumulator * x + coeffs[index]
 return accumulator

end

procedure main () 

 write (poly_eval (3, [-19, 7, -4, 6]))

end </lang>

J

Solution:<lang j> 

  horner =: (#."0 _ |.)~

  horner2=: [: +`*/ [: }: ,@,.    NB. Alternate

</lang> Example:<lang j> _19 7 _4 6 horner 3 128</lang> Note:
The primitive verb p. would normally be used to evaluate polynomials. <lang j> _19 7 _4 6 p. 3 128</lang>

Java

Works with: Java version 1.5+

<lang java5>import java.util.ArrayList; import java.util.Collections; import java.util.List;

public class Horner { 

   public static void main(String[] args){
       List<Double> coeffs = new ArrayList<Double>();
       coeffs.add(-19.0);
       coeffs.add(7.0);
       coeffs.add(-4.0);
       coeffs.add(6.0);
       System.out.println(polyEval(coeffs, 3));
   }

   public static double polyEval(List<Double> coefficients, double x) {
       Collections.reverse(coefficients);
       Double accumulator = coefficients.get(0);
       for (int i = 1; i < coefficients.size(); i++) {
           accumulator = (accumulator * x) + (Double) coefficients.get(i);
       }
       return accumulator;
   }

}</lang> Output: 

128.0

JavaScript

Works with: JavaScript version 1.8

which includes

Works with: Firefox version 3
Translation of: Haskell

<lang javascript>function horner(coeffs, x) { 

   return coeffs.reduceRight(function(acc, coeff) {return(acc * x + coeff)}, 0);

} print(horner([-19,7,-4,6],3)); // ==> 128 </lang>

K

<lang K> 

 horner:{y _sv|x}
 horner[-19 7 -4 6;3]

128 </lang>

<lang logo>to horner :x :coeffs 

 if empty? :coeffs [output 0]
 output (first :coeffs) + (:x * horner :x bf :coeffs)

end

show horner 3 [-19 7 -4 6]  ; 128</lang>

Lua

<lang lua>function horners_rule( coeff, x ) 

   local res = 0    
   for i = #coeff, 1, -1 do
       res = res * x + coeff[i]
   end
   return res

end

x = 3 coefficients = { -19, 7, -4, 6 } print( horners_rule( coefficients, x ) )</lang>

Mathematica

<lang Mathematica>Horner[l_List, x_] := Fold[x #1 + #2 &, 0, l] Horner[{6, -4, 7, -19}, x] -> -19 + x (7 + x (-4 + 6 x))

-19 + x (7 + x (-4 + 6 x)) /. x -> 3 -> 128</lang>

MATLAB

<lang MATLAB>function accumulator = hornersRule(x,coefficients) 

   accumulator = 0;
   
   for i = (numel(coefficients):-1:1)
       accumulator = (accumulator * x) + coefficients(i);
   end

end</lang> Output: <lang MATLAB>>> hornersRule(3,[-19, 7, -4, 6])

ans = 

  128</lang>

Matlab also has a built-in function "polyval" which uses Horner's Method to evaluate polynomials. The list of coefficients is in descending order of power, where as to task spec specifies ascending order. <lang MATLAB>>> polyval(fliplr([-19, 7, -4, 6]),3)

ans = 

  128</lang>

Objective-C

Works with: Mac OS X version 10.6+

Using blocks

<lang objc>#import <Foundation/Foundation.h>

typedef double (^mfunc)(double, double);

@interface NSArray (HornerRule) - (double)horner: (double)x; - (NSArray *)reversedArray; - (double)injectDouble: (double)s with: (mfunc)op; @end

@implementation NSArray (HornerRule) - (NSArray *)reversedArray { 

 return [[self reverseObjectEnumerator] allObjects];

}


- (double)injectDouble: (double)s with: (mfunc)op { 

 double sum = s;
 for(NSNumber* el in self) {
   sum = op(sum, [el doubleValue]);
 }
 return sum;

}

- (double)horner: (double)x { 

 return [[self reversedArray] injectDouble: 0.0 with: ^(double s, double a) { return s * x + a; } ];

} @end

int main() { 

 NSAutoreleasePool *pool = [[NSAutoreleasePool alloc] init];

 NSArray *coeff = [NSArray

arrayWithObjects: [NSNumber numberWithDouble: -19.0], 

      		       [NSNumber numberWithDouble: 7.0],

[NSNumber numberWithDouble: -4.0], [NSNumber numberWithDouble: 6.0], nil]; 

 printf("%f\n", [coeff horner: 3.0]);

 [pool release];
 return 0;

}</lang>

OCaml

<lang ocaml># let horner coeffs x = 

   List.fold_left (fun acc coef -> acc * x + coef) 0 (List.rev coeffs) ;;

val horner : int list -> int -> int = <fun>

  1. let coeffs = [-19; 7; -4; 6] in
 horner coeffs 3 ;;

- : int = 128</lang> It's also possible to do fold_right instead of reversing and doing fold_left; but fold_right is not tail-recursive.

Octave

<lang octave>function r = horner(a, x) 

 r = 0.0;
 for i = length(a):-1:1
   r = r*x + a(i);
 endfor

endfunction

horner([-19, 7, -4, 6], 3)</lang>

Oz

<lang oz>declare 

 fun {Horner Coeffs X}
    {FoldL1 {Reverse Coeffs} 
     fun {$ Acc Coeff}
        Acc*X + Coeff
     end}
 end
 
 fun {FoldL1 X|Xr Fun}
    {FoldL Xr Fun X}
 end

in 

 {Show {Horner [~19 7 ~4 6] 3}}</lang>

PARI/GP

Also note that Pari has a polynomial type. Evaluating these is as simple as subst(P,variable(P),x). <lang parigp>horner(v,x)={ 

 my(s=0);
 forstep(i=#v,1,-1,s=s*x+v[i]);
 s

};</lang>

Pascal

<lang pascal>Program HornerDemo(output);

function horner(a: array of double; x: double): double; 

 var
   i: integer;
 begin
   horner := a[high(a)];
   for i := high(a) - 1 downto low(a) do
     horner := horner * x + a[i];
 end;

const 

 poly: array [1..4] of double = (-19.0, 7.0, -4.0, 6.0);

begin 

 write ('Horner calculated polynomial of 6*x^3 - 4*x^2 + 7*x - 19 for x = 3: ');
 writeln (horner (poly, 3.0):8:4);

end.</lang> Output: 

Horner calculated polynomial of 6*x^3 - 4*x^2 + 7*x - 19 for x = 3: 128.0000

Perl

<lang Perl>use 5.10.0; use strict; use warnings;

sub horner(\@$){ my ($coef, $x) = @_; my $result = 0; $result = $result * $x + $_ for reverse @$coef; return $result; }

my @coeff = (-19.0, 7, -4, 6); my $x = 3; say horner @coeff, $x;</lang>

Functional version

<lang perl>use strict; use List::Util qw(reduce);

sub horner($$){ my ($coeff_ref, $x) = @_; reduce { $a * $x + $b } reverse @$coeff_ref; }

my @coeff = (-19.0, 7, -4, 6); my $x = 3; print horner(\@coeff, $x), "\n";</lang>

Perl 6

<lang perl6>sub horner ( @coeffs, $x ) { 

   @coeffs.reverse.reduce: { $^a * $x + $^b };

}

say horner( [ -19, 7, -4, 6 ], 3 );</lang>

PHP

<lang php><?php function horner($coeff, $x) { 

   $result = 0;
   foreach (array_reverse($coeff) as $c)
       $result = $result * $x + $c;
   return $result;

}

$coeff = array(-19.0, 7, -4, 6); $x = 3; echo horner($coeff, $x), "\n"; ?></lang>

Functional version

Works with: PHP version 5.3+

<lang php><?php function horner($coeff, $x) { 

   return array_reduce(array_reverse($coeff), function ($a, $b) use ($x) { return $a * $x + $b; }, 0);

}

$coeff = array(-19.0, 7, -4, 6); $x = 3; echo horner($coeff, $x), "\n"; ?></lang>

PicoLisp

<lang PicoLisp>(de horner (Coeffs X) 

  (let Res 0
     (for C (reverse Coeffs)
        (setq Res (+ C (* X Res))) ) ) )</lang>

<lang PicoLisp>: (horner (-19.0 7.0 -4.0 6.0) 3.0) -> 128</lang>

PL/I

<lang PL/I> declare (i, n) fixed binary, (x, value) float; /* 11 May 2010 */ get (x); get (n); begin; 

  declare a(0:n) float;
  get list (a);
  value = a(n);
  do i = n to 1 by -1;
     value = value*x + a(i-1);
  end;
  put (value);

end; </lang>

Prolog

Tested with SWI-Prolog. Works with other dialects. <lang Prolog>horner([], _X, 0).

horner([H|T], X, V) :- horner(T, X, V1), V is V1 * X + H. </lang> Output : <lang Prolog> ?- horner([-19, 7, -4, 6], 3, V). V = 128.</lang>

Functionnal approach

Works with SWI-Prolog and module lambda, written by Ulrich Neumerkel found there http://www.complang.tuwien.ac.at/ulrich/Prolog-inedit/lambda.pl <lang Prolog>:- use_module(library(lambda)).


% foldr(Pred, Init, List, R). % foldr(_Pred, Val, [], Val). foldr(Pred, Val, [H | T], Res) :- foldr(Pred, Val, T, Res1), call(Pred, Res1, H, Res).

f_horner(L, V, R) :- foldr(\X^Y^Z^(Z is X * V + Y), 0, L, R). </lang>

PureBasic

<lang PureBasic>Procedure Horner(List Coefficients(), b) 

 Define result
 ForEach Coefficients()
   result*b+Coefficients()
 Next
 ProcedureReturn result

EndProcedure</lang>

Implemented as <lang PureBasic>NewList a() AddElement(a()): a()= 6 AddElement(a()): a()= -4 AddElement(a()): a()= 7 AddElement(a()): a()=-19 Debug Horner(a(),3)</lang> Outputs 

128

Python

<lang python>>>> def horner(coeffs, x): acc = 0 for c in reversed(coeffs): acc = acc * x + c return acc

>>> horner( (-19, 7, -4, 6), 3) 128</lang>

Functional version

<lang python>>>> try: from functools import reduce except: pass

>>> def horner(coeffs, x): return reduce(lambda acc, c: acc * x + c, reversed(coeffs), 0)

>>> horner( (-19, 7, -4, 6), 3) 128</lang>

R

<lang r>horner <- function(a, x) { 

 iv <- 0
 for(i in length(a):1) {
   iv <- iv * x + a[i]
 }
 iv

}

cat(horner(c(-19, 7, -4, 6), 3), "\n")</lang>

REXX

<lang rexx>/*REXX program using Horner's rule for polynomial evaulation. */

arg 'X=' x poly /*get value of X and coefficients*/ equ= /*start with equation clean slate*/ 

                                      /*works for any degree equation. */
 do deg=0 until poly==              /*get the equation's coefficents.*/
 parse var poly c.deg poly            /*get  a  equation   coefficent. */
 c.deg=(c.deg+0)/1                    /*normalize it  (add 1, div by 1)*/
 if c.deg>=0 then c.deg='+'c.deg      /*if positive, then preprend a + */
 equ=equ c.deg                        /*concatenate it to the equation.*/
 if deg\==0 &,                        /*if not the first coefficient & */
    c.deg\=0 then equ=equ 'x^'deg     /* not 0,  append power (^) of X.*/
 equ=equ '  '                         /*insert some blanks, look pretty*/
 end

say ' x=' x say ' degree=' deg say 'equation=' equ a=c.deg 

 do j=deg by -1 to 1                  /*apply Horner's rule to evaulate*/
 jm1=j-1
 a=a*x + c.jm1
 end

say ' answer=' a</lang>

Output when the following is used for input: x=3 -19 7 -4 6 

       x= 3
  degree= 3
equation=  -19    +7 x^1    -4 x^2    +6 x^3
  answer= 128

RLaB

RLaB implements horner's scheme for polynomial evaluation in its built-in function polyval. What is important is that RLaB stores the polynomials as row vectors starting from the highest power just as matlab and octave do.

This said, solution to the problem is <lang RLaB> >> a = [6, -4, 7, -19] 

          6            -4             7           -19

>> x=3 

          3

>> polyval(x, a) 

        128

</lang>

Ruby

<lang ruby>def horner(coeffs, x) 

 coeffs.reverse.inject(0) {|acc, coeff| acc * x + coeff}

end p horner([-19, 7, -4, 6], 3) # ==> 128</lang>

Sather

<lang sather>class MAIN is 

 action(s, e, x:FLT):FLT is
   return s*x + e;
 end;

 horner(v:ARRAY{FLT}, x:FLT):FLT is
   rv ::= v.reverse;
   return rv.reduce(bind(action(_, _, x)));
 end;

 main is
   #OUT + horner(|-19.0, 7.0, -4.0, 6.0|, 3.0) + "\n";
 end;

end;</lang>

Scala

<lang scala>def horner(coeffs:List[Double], x:Double)= 

  coeffs.reverse.foldLeft(0.0){(a,c)=> a*x+c}

</lang> <lang scala>val coeffs=List(-19.0, 7.0, -4.0, 6.0) println(horner(coeffs, 3)) 

  -> 128.0

</lang>

Scheme

Works with: Scheme version RRS

<lang scheme>(define (horner lst x) 

 (define (*horner lst x acc)
   (if (null? lst)
       acc
       (*horner (cdr lst) x (+ (* acc x) (car lst)))))
 (*horner (reverse lst) x 0))

(display (horner (list -19 7 -4 6) 3)) (newline)</lang> Output: <lang>128</lang>

Seed7

<lang seed7>$ include "seed7_05.s7i"; 

 include "float.s7i";

const type: coeffType is array float;

const func float: horner (in coeffType: coeffs, in float: x) is func 

 result
   var float: res is 0.0;
 local
   var integer: i is 0;
 begin
   for i range length(coeffs) downto 1 do
     res := res * x + coeffs[i];
   end for;
 end func;

const proc: main is func 

 local
   const coeffType: coeffs is [] (-19.0, 7.0, -4.0, 6.0);
 begin
   writeln(horner(coeffs, 3.0) digits 1);
 end func;</lang>

Output: 

128.0

Smalltalk

Works with: GNU Smalltalk

<lang smalltalk>OrderedCollection extend [ 

 horner: aValue [
   ^ self reverse inject: 0 into: [:acc :c | acc * aValue + c].
 ]

].

(#(-19 7 -4 6) asOrderedCollection horner: 3) displayNl.</lang>

Standard ML

<lang sml>(* Assuming real type for coefficients and x *) fun horner coeffList x = foldr (fn (a, b) => a + b * x) (0.0) coeffList</lang>

Tcl

<lang tcl>package require Tcl 8.5 proc horner {coeffs x} { 

   set y 0
   foreach c [lreverse $coeffs] {
       set y [expr { $y*$x+$c }]
   }
   return $y

}</lang> Demonstrating: <lang tcl>puts [horner {-19 7 -4 6} 3]</lang> Output: 

128

VBA

Note: this function, "Horner", gets its coefficients as a ParamArray which has no specified length. This array collect all arguments after the first one(s). This means you must specify x first, then the coefficients.

<lang VBA> Public Function Horner(x, ParamArray coeff()) Dim result As Double Dim ncoeff As Integer

result = 0 ncoeff = UBound(coeff())

For i = ncoeff To 0 Step -1 

 result = (result * x) + coeff(i)

Next i Horner = result End Function </lang>

Output: 

print Horner(3, -19, 7, -4, 6)
 128
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