Horner's rule for polynomial evaluation
A fast scheme for evaluating a polynomial such as:
You are encouraged to solve this task according to the task description, using any language you may know.
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.
11l
F horner(coeffs, x)
V acc = 0
L(c) reversed(coeffs)
acc = acc * x + c
R acc
print(horner([-19, 7, -4, 6], 3))- Output:
128
360 Assembly
* Horner's rule for polynomial evaluation - 07/10/2015
HORNER CSECT
USING HORNER,R15 set base register
SR R5,R5 accumulator=0
LA R2,N i=number_of_coeff
LOOP M R4,X accumulator=accumulator*x
LR R1,R2 i
SLA R1,2 i*4
L R3,COEF-4(R1) coef(i)
AR R5,R3 accumulator=accumulator+coef(i)
BCT R2,LOOP i=i-1; loop n times
XDECO R5,PG edit accumulator
XPRNT PG,12 print buffer
XR R15,R15 set return code
BR R14 return to caller
COEF DC F'-19',F'7',F'-4',F'6' <== input values
X DC F'3' <== input value
N EQU (X-COEF)/4 number of coefficients
PG DS CL12 buffer
YREGS
END HORNER- Output:
128
ACL2
(defun horner (ps x)
(if (endp ps)
0
(+ (first ps)
(* x (horner (rest ps) x)))))
Action!
INT FUNC Horner(INT ARRAY coeffs INT count,x)
INT v,i
v=0 i=count-1
WHILE i>=0
DO
v=v*x+coeffs(i)
i==-1
OD
RETURN (v)
PROC Main()
INT ARRAY coeffs=[65517 7 65532 6]
INT res,x=[3],i,count=[4]
PrintF("x=%I%E",x)
FOR i=0 TO count-1
DO
PrintI(coeffs(i))
IF i=1 THEN
Print("x")
ELSEIF i>1 THEN
PrintF("x^%I",i)
FI
IF i<count-1 AND coeffs(i+1)>=0 THEN
Print("+")
FI
OD
res=Horner(coeffs,4,x)
PrintF("=%I%E",res)
RETURN- Output:
Screenshot from Atari 8-bit computer
x=3 -19+7x-4x^2+6x^3=128
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;
Output:
128.0
Aime
real
horner(list coeffs, real x)
{
real c, z;
z = 0;
for (, c of coeffs) {
z *= x;
z += c;
}
z;
}
integer
main(void)
{
o_(horner(list(-19r, 7.0, -4r, 6r), 3), "\n");
0;
}ALGOL 68
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) )
)ALGOL W
begin
% Horner's rule for polynominal evaluation %
% returns the value of the polynominal defined by coefficients, %
% at the point x. The number of coefficients must be in ub %
% the coefficients should be in order with x^0 first, x^n last %
real procedure Horner( real array coefficients ( * )
; integer value ub
; real value x
) ;
begin
real xValue;
xValue := 0;
for i := ub step -1 until 1 do xValue := ( xValue * x ) + coefficients( i );
xValue
end Horner ;
% task test case %
begin
real array coefficients ( 1 :: 4 );
integer cPos;
cPos := 1;
for i := -19, 7, -4, 6 do begin
coefficients( cPos ) := i;
cPos := cPos + 1
end for_i ;
write( r_format := "A", r_w := 8, r_d := 2, Horner( coefficients, 4, 3 ) )
end test_cases
end.- Output:
128.00
APL
Works in Dyalog APL
h←⊥∘⌽
- Output:
3 h ¯19 7 ¯4 6 128
ATS
#include
"share/atspre_staload.hats"
fun
horner
(
x: int, cs: List int
) : int = let
//
implement
list_foldright$fopr<int><int> (a, b) = a + b * x
//
in
list_foldright<int><int> (cs, 0)
end // end of [horner]
implement
main0 () = let
val x = 3
val cs = $list{int}(~19, 7, ~4, 6)
val res = horner (x, cs)
in
println! (res)
end // end of [main0]Arturo
horner: function [coeffs, x][
result: 0
loop reverse coeffs 'c ->
result: c + result * x
return result
]
print horner @[neg 19, 7, neg 4, 6] 3
- Output:
128
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
}
Message box shows:
128
AWK
#!/usr/bin/awk -f
function horner(x, A) {
acc = 0;
for (i = length(A); 0<i; i--) {
acc = acc*x + A[i];
}
return acc;
}
BEGIN {
split(p,P);
print horner(x,P);
}
- Output:
awk -v X=3 -v p="-19 7 -4 6" -f horner.awk 128
BASIC
Applesoft BASIC
100 HOME : REM 100 CLS for Chipmunk Basic and GW-BASIC
100 CLS : REM 100 HOME for Applesoft BASIC
110 X = 3
120 DIM COEFFS(3)
130 COEFFS(0) = -19
140 COEFFS(1) = 7
150 COEFFS(2) = -4
160 COEFFS(3) = 6
170 PRINT "Horner's algorithm for the polynomial "
180 PRINT "6*x^3 - 4*x^2 + 7*x - 19 when x = 3 is: ";
190 ACCUM = 0
200 FOR I = 3 TO 0 STEP -1
210 ACCUM = (ACCUM*X)+COEFFS(I)
220 NEXT I
230 PRINT ACCUM
240 END
BASIC256
x = 3
dim coeficientes = {-19, 7, -4, 6}
print "Horner's algorithm for the polynomial ";
print "6*x^3 - 4*x^2 + 7*x - 19 when x = 3: ";
print AlgoritmoHorner(coeficientes, x)
end
function AlgoritmoHorner(coeffs, x)
acumulador = 0
for i = coeffs[?]-1 to 0 step -1
acumulador = (acumulador * x) + coeffs[i]
next i
return acumulador
end function
- Output:
Same as FreeBASIC entry.
Chipmunk Basic
100 CLS
110 x = 3
120 DIM coeffs(3)
130 coeffs(0) = -19
140 coeffs(1) = 7
150 coeffs(2) = -4
160 coeffs(3) = 6
170 PRINT "Horner's algorithm for the polynomial "
180 PRINT "6*x^3 - 4*x^2 + 7*x - 19 when x = 3 is: ";
190 accum = 0
200 FOR i = UBOUND(coeffs,1) TO 0 STEP -1
210 accum = (accum*x)+coeffs(i)
220 NEXT i
230 PRINT accum
240 END
- Output:
Horner's algorithm for the polynomial 6*x^3 - 4*x^2 + 7*x - 19 when x = 3 is: 128
Gambas
Public coeficientes As New Integer[4]
Public Function AlgoritmoHorner(coeficientes As Integer[], x As Integer) As Integer
coeficientes[0] = -19
coeficientes[1] = 7
coeficientes[2] = -4
coeficientes[3] = 6
Dim i As Integer, acumulador As Integer = 0
For i = coeficientes.Count - 1 To 0 Step -1
acumulador = (acumulador * x) + coeficientes[i]
Next
Return acumulador
End Function
Public Sub Main()
Dim x As Integer = 3
Print "Horner's algorithm for the polynomial 6*x^3 - 4*x^2 + 7*x - 19 when x = 3: ";
Print AlgoritmoHorner(coeficientes, x)
End
- Output:
Same as FreeBASIC entry.
GW-BASIC
100 CLS : REM 100 HOME for Applesoft BASIC
110 X = 3
120 DIM COEFFS(3)
130 COEFFS(0) = -19
140 COEFFS(1) = 7
150 COEFFS(2) = -4
160 COEFFS(3) = 6
170 PRINT "Horner's algorithm for the polynomial "
180 PRINT "6*x^3 - 4*x^2 + 7*x - 19 when x = 3 is: ";
190 ACCUM = 0
200 FOR I = 3 TO 0 STEP -1
210 ACCUM = (ACCUM*X)+COEFFS(I)
220 NEXT I
230 PRINT ACCUM
240 END
- Output:
Same as Chipmunk Basic entry.
Minimal BASIC
20 LET X = 3
30 DIM C(3)
40 LET C(0) = -19
50 LET C(1) = 7
60 LET C(2) = -4
70 LET C(3) = 6
80 PRINT "HORNER'S ALGORITHM FOR THE POLYNOMIAL"
90 PRINT "6*X^3 - 4*X^2 + 7*X - 19 WHEN X = 3 : ";
100 LET A = 0
110 FOR I = 3 TO 0 STEP -1
120 LET A = (A*X)+C(I)
130 NEXT I
140 PRINT A
150 END
- Output:
Same as Chipmunk Basic entry.
MSX Basic
The Minimal BASIC solution works without any changes.
QBasic
FUNCTION Horner (coeffs(), x)
acumulador = 0
FOR i = UBOUND(coeffs) TO LBOUND(coeffs) STEP -1
acumulador = (acumulador * x) + coeffs(i)
NEXT i
Horner = acumulador
END FUNCTION
x = 3
DIM coeffs(3)
DATA -19, 7, -4, 6
FOR a = LBOUND(coeffs) TO UBOUND(coeffs)
READ coeffs(a)
NEXT a
PRINT "Horner's algorithm for the polynomial 6*x^3 - 4*x^2 + 7*x - 19 when x = 3 is: ";
PRINT Horner(coeffs(), x)
END
Quite BASIC
The Minimal BASIC solution works without any changes.
Yabasic
x = 3
dim coeffs(4)
coeffs(0) = -19
coeffs(1) = 7
coeffs(2) = -4
coeffs(3) = 6
print "Horner's algorithm for the polynomial ";
print "6*x^3 - 4*x^2 + 7*x - 19 when x = 3: ";
print AlgoritmoHorner(coeffs, x)
end
sub AlgoritmoHorner(coeffs, x)
local acumulador, i
acumulador = 0
for i = arraysize(coeffs(),1) to 0 step -1
acumulador = (acumulador * x) + coeffs(i)
next i
return acumulador
end sub
- Output:
Same as FreeBASIC entry.
Batch File
@echo off
call:horners a:-19 b:7 c:-4 d:6 x:3
call:horners x:3 a:-19 c:-4 d:6 b:7
pause>nul
exit /b
:horners
setlocal enabledelayedexpansion
set a=0
set b=0
set c=0
set d=0
set x=0
for %%i in (%*) do (
for /f "tokens=1,2 delims=:" %%j in ("%%i") do (
set %%j=%%k
)
)
set /a return=((((0)*%x%+%d%)*%x%+(%c%))*%x%+%b%)*%x%+(%a%)
echo %return%
exit /b
- Output:
>a:-19 b:7 c:-4 d:6 x:3 128 >x:3 a:-19 c:-4 d:6 b:7 128
BBC BASIC
DIM coefficients(3)
coefficients() = -19, 7, -4, 6
PRINT FNhorner(coefficients(), 3)
END
DEF FNhorner(coeffs(), x)
LOCAL i%, v
FOR i% = DIM(coeffs(), 1) TO 0 STEP -1
v = v * x + coeffs(i%)
NEXT
= v
Bracmat
( ( Horner
= accumulator coefficients x coeff
. !arg:(?coefficients.?x)
& 0:?accumulator
& whl
' ( !coefficients:?coefficients #%@?coeff
& !accumulator*!x+!coeff:?accumulator
)
& !accumulator
)
& Horner$(-19 7 -4 6.3)
);Output:
128
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;
}
C#
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));
}
}
Output:
128
C++
The same C function works too, but another solution could be:
#include <iostream>
#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;
}
Yet another solution, which is more idiomatic in C++ and works on any bidirectional sequence:
#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;
}
Clojure
(defn horner [coeffs x]
(reduce #(-> %1 (* x) (+ %2)) (reverse coeffs)))
(println (horner [-19 7 -4 6] 3))
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
Common Lisp
(defun horner (coeffs x)
(reduce #'(lambda (coef acc) (+ (* acc x) coef))
coeffs :from-end t :initial-value 0))
Alternate version using LOOP. Coefficients are passed in a vector.
(defun horner (x a)
(loop :with y = 0
:for i :from (1- (length a)) :downto 0
:do (setf y (+ (aref a i) (* y x)))
:finally (return y)))
(horner 1.414 #(-2 0 1))
D
The poly() function of the standard library std.math module uses Horner's rule:
void main() {
void main() {
import std.stdio, std.math;
double x = 3.0;
static real[] pp = [-19,7,-4,6];
poly(x,pp).writeln;
}
}
Basic implementation:
import std.stdio, std.traits;
CommonType!(U, V) horner(U, V)(U[] p, V x) pure nothrow @nogc {
typeof(return) accumulator = 0;
foreach_reverse (c; p)
accumulator = accumulator * x + c;
return accumulator;
}
void main() {
[-19, 7, -4, 6].horner(3.0).writeln;
}
More functional style:
import std.stdio, std.algorithm, std.range;
auto horner(T, U)(in T[] p, in U x) pure nothrow @nogc {
return reduce!((a, b) => a * x + b)(U(0), p.retro);
}
void main() {
[-19, 7, -4, 6].horner(3.0).writeln;
}
Delphi
function EvaluatePolynomial(Args: array of double; X: double): double;
{Evaluate polynomial using Horner's rule }
var I: integer;
begin
Result:=0;
for I:=High(Args) downto 0 do
Result:=(Result * X ) + Args[I];
end;
function GetPolynomialStr(Args: array of double; VarName: string): string;
{Return a string display the polynomial in normal format}
{for example: 6.0 X^3 - 4.0 X^2 + 7.0 X - 19.0}
var I: integer;
begin
Result:='';
for I:=High(Args) downto 0 do
begin
if Args[I]>0 then
begin
if I<>High(Args) then Result:=Result+' + ';
end
else Result:=Result+' - ';
Result:=Result+FloatToStrF(Abs(Args[I]),ffFixed,18,1);
if I>0 then Result:=Result+' '+VarName;
if I>1 then Result:=Result+'^'+IntToStr(I);
end;
end;
procedure ShowHornerPoly(Memo: TMemo; Args: array of double; X: double);
{Evaluate polynomial, show formated polynomal and the result}
var R: double;
begin
R:=EvaluatePolynomial(Args,X);
Memo.Lines.Add(FloatToStrF(R,ffFixed, 18,1));
Memo.Lines.Add(GetPolynomialStr(Args,'X'));
end;
procedure DoHornerPoly(Memo: TMemo);
begin
ShowHornerPoly(Memo,[-19, 7, -4, 6],3)
end;
- Output:
128.0 6.0 X^3 - 4.0 X^2 + 7.0 X - 19.0
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
}
}? makeHornerPolynomial([-19, 7, -4, 6])(3)
# value: 128EasyLang
func horner coeffs[] x .
for i = len coeffs[] downto 1
res = res * x + coeffs[i]
.
return res
.
print horner [ -19 7 -4 6 ] 3
- Output:
128
EchoLisp
Functional version
(define (horner x poly)
(foldr (lambda (coeff acc) (+ coeff (* acc x))) 0 poly))
(horner 3 '(-19 7 -4 6)) → 128
Library
(lib 'math)
Lib: math.lib loaded.
(define P '(-19 7 -4 6))
(poly->string 'x P) → 6x^3 -4x^2 +7x -19
(poly 3 P) → 128
EDSAC order code
[Copyright <2021> <ERIK SARGSYAN>
Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"),
to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense,
and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED,
INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.]
[Horner's rule for polynomial evaluation on EDSAC by Erik Sargsyan]
[EDSAC program, Initial Orders 2]
T 56 K
GK
[0] A 3 F
[1] T 9 F
[2] A 4 F [loading the address of the 1st array element into the accumulator]
[3] A 14 @ [add an instruction code with a zero address field]
[4] T 14 @ [writing the generated instruction, zeroing the accumulator]
[5 loop] A 1 F [load the counter of unprocessed array elements into the accumulator]
[6] S 21 @ [Subtract the constant = 1]
[7] E 10 @ [if (acc >= 0) goto 10 else end]
[8] T 300 F [zeroing the accumulator]
[9] E 0 F [breakpoint, end of program]
[10] T 1 F [update the counter value and reset the accumulator]
[11] V 2 F [multiplying the number in cell 2 by the number in the multiplier register]
[12] L 1024 F [shift the number in the accumulator 12 bits to the left]
[13] L 4 F [shift the number in the accumulator 4 bits to the left]
[14 r1] A 0 F [loading the value from cell 0 into the accumulator]
[15] T 2 F [writing this value to the working cell, zeroing the accumulator]
[16] A 21 @ [loading into accumulator constant value = 1]
[17] L 0 D [shift the number in the accumulator 1 bit to the left]
[18] A 14 @ [add the code of the instruction executed in the previous step]
[19] T 14 @ [write the generated instruction into memory]
[20] E 5 @ [repeat all operations; accumulator zeroed]
[21 const 1] P 0 D [1]
GK
[0] H 10 @ [writing X0 to the multiplier register]
[1] A 11 @ [loading into the accumulator of the degree of the polynomial]
[2] T 1 F [writing the degree of a polynomial in cell 1]
[3] A 13 @ [loading the leading coefficient into the accumulator]
[4] T 2 F [writing the senior coefficient to the working cell 2]
[5] A 12 @ [loading the address of the 1st array element into the accumulator]
[6] T 4 F [writing the address of the 1st element of the array]
[7] A 7 @ [\ call]
[8] G 56 F
[9] Z 0 F
[10] P 3 F [X0 is a fixed value of X, by which we calculate the value of the polynomial]
[11 power] P 2 F [polynomial degree times 2]
[12 addr] P 14 @ [address of the 1st element of the array]
[13] P 3 F [a4 = 6]
[14] P 8 D [a3 = 17]
[15] P 9 F [a2 = 18]
[16] P 12 D [a1 = 25]
[17] P 316 F [a0 = 632]
EZPF- Output:
Cell 2 will contain the number 12878
Elena
ELENA 5.0 :
import extensions;
import system'routines;
horner(coefficients,variable)
{
^ coefficients.clone().sequenceReverse().accumulate(new Real(),(accumulator,coefficient => accumulator * variable + coefficient))
}
public program()
{
console.printLine(horner(new real[]{-19.0r, 7.0r, -4.0r, 6.0r}, 3.0r))
}- Output:
128.0
Elixir
horner = fn(list, x)-> List.foldr(list, 0, fn(c,acc)-> x*acc+c end) end
IO.puts horner.([-19,7,-4,6], 3)
- Output:
128
Emacs Lisp
(require 'cl-lib)
(defun horner (coeffs x)
(cl-reduce #'(lambda (coef acc) (+ (* acc x) coef))
coeffs :from-end t :initial-value 0))
(horner '(-19 7 -4 6) 3)
- Output:
128
Erlang
horner(L,X) ->
lists:foldl(fun(C, Acc) -> X*Acc+C end,0, lists:reverse(L)).
t() ->
horner([-19,7,-4,6], 3).
ERRE
PROGRAM HORNER
! 2 3
! polynomial is -19+7x-4x +6x
!
DIM C[3]
PROCEDURE HORNER(C[],X->RES)
LOCAL I%,V
FOR I%=UBOUND(C,1) TO 0 STEP -1 DO
V=V*X+C[I%]
END FOR
RES=V
END PROCEDURE
BEGIN
C[]=(-19,7,-4,6)
HORNER(C[],3->RES)
PRINT(RES)
END PROGRAMEuler Math Toolbox
>function horner (x,v) ...
$ n=cols(v); res=v{n};
$ loop 1 to n-1; res=res*x+v{n-#}; end;
$ return res
$endfunction
>v=[-19,7,-4,6]
[ -19 7 -4 6 ]
>horner(2,v) // test Horner
27
>evalpoly(2,v) // built-in Horner
27
>horner(I,v) // complex values
-15+1i
>horner(1±0.05,v) // interval values
~-10.9,-9.11~
>function p(x) &= sum(@v[k]*x^(k-1),k,1,4) // Symbolic Polynomial
3 2
6 x - 4 x + 7 x - 19
F#
let horner l x =
List.rev l |> List.fold ( fun acc c -> x*acc+c) 0
horner [-19;7;-4;6] 3
Factor
: horner ( coeff x -- res )
[ <reversed> 0 ] dip '[ [ _ * ] dip + ] reduce ;
( scratchpad ) { -19 7 -4 6 } 3 horner .
128
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.
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 (0:), intent (in) :: coeffs
real, intent (in) :: x
real :: res
integer :: i
res = coeffs(ubound(coeffs,1))
do i = ubound(coeffs,1)-1, 0, -1
res = res * x + coeffs (i)
end do
end function horner
end program test_horner
Output:
128.0
Fortran 77
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
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.
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
FreeBASIC
Function AlgoritmoHorner(coeffs() As Integer, x As Integer) As Integer
Dim As Integer i, acumulador = 0
For i = Ubound(coeffs, 1) To 0 Step -1
acumulador = (acumulador * x) + coeffs(i)
Next i
Return acumulador
End Function
Dim As Integer x = 3
Dim As Integer coeficientes(3) = {-19, 7, -4, 6}
Print "Algoritmo de Horner para el polinomio 6*x^3 - 4*x^2 + 7*x - 19 para x = 3: ";
Print AlgoritmoHorner(coeficientes(), x)
End
- Output:
Algoritmo de Horner para el polinomio 6*x^3 - 4*x^2 + 7*x - 19 para x = 3: 128
FunL
import lists.foldr
def horner( poly, x ) = foldr( \a, b -> a + b*x, 0, poly )
println( horner([-19, 7, -4, 6], 3) )- Output:
128
FutureBasic
include "NSLog.incl"
local fn horner( coeffs as CFArrayRef, x as NSInteger ) as double
CFArrayRef reversedCoeffs
CFNumberRef num
double accumulator = 0.0
// Reverse coeffs array
reversedCoeffs = fn EnumeratorAllObjects( fn ArrayReverseObjectEnumerator( coeffs ) )
// Iterate over CFNumberRefs in reversed array, convert to double values, calculate and add to accumulator
for num in reversedCoeffs
accumulator = ( accumulator * x ) + fn NumberDoubleValue( num )
next
end fn = accumulator
CFArrayRef coeffs
coeffs = @[@-19.0, @7.0, @-4.0, @6.0]
NSLog( @"%7.1f", fn horner( coeffs, 3 ) )
coeffs = @[@4.0, @3.0, @2.0, @1.0]
NSLog( @"%7.1f", fn horner( coeffs, 10 ) )
coeffs = @[@1, @1, @0, @0, @1]
NSLog( @"%7.1f", fn horner( coeffs, 2 ) )
coeffs = @[@1.2, @2.3, @3.4, @4.5, @5.6]
NSLog( @"%7.1f", fn horner( coeffs, 8 ) )
coeffs = @[@1, @0, @1, @1, @1, @0, @0, @1]
NSLog( @"%7.1f", fn horner( coeffs, 2 ) )
HandleEvents- Output:
128.0 1234.0 19.0 25478.8 157.0
GAP
# The idiomatic way to compute with polynomials
x := Indeterminate(Rationals, "x");
# This is a value in a polynomial ring, not a function
p := 6*x^3 - 4*x^2 + 7*x - 19;
Value(p, 3);
# 128
u := CoefficientsOfUnivariatePolynomial(p);
# [ -19, 7, -4, 6 ]
# One may also create the polynomial from coefficients
q := UnivariatePolynomial(Rationals, [-19, 7, -4, 6], x);
# 6*x^3-4*x^2+7*x-19
p = q;
# true
# 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);
# 128
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}))
}
Output:
128
Groovy
Solution:
def hornersRule = { coeff, x -> coeff.reverse().inject(0) { accum, c -> (accum * x) + c } }
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.
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]])
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
horner :: (Num a) => a -> [a] -> a
horner x = foldr (\a b -> a + b*x) 0
main = print $ horner 3 [-19, 7, -4, 6]
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
ENDIcon and Unicon
J
Solution:
horner =: (#."0 _ |.)~ NB. Tacit
horner =: [: +`*/ [: }: ,@,. NB. Alternate tacit (equivalent)
horner =: 4 : ' (+ *&y)/x' NB. Alternate explicit (equivalent)
Example:
_19 7 _4 6 horner 3
128
Note:
The primitive verb p. would normally be used to evaluate polynomials.
_19 7 _4 6 p. 3
128
Java
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;
}
}
Output:
128.0
JavaScript
which includes
function horner(coeffs, x) {
return coeffs.reduceRight( function(acc, coeff) { return(acc * x + coeff) }, 0);
}
console.log(horner([-19,7,-4,6],3)); // ==> 128
jq
# Input: an array of coefficients specifying the polynomial
# to be evaluated at $x, where .[0] is the constant
def horner($x):
. as $coefficients
| reduce range(length-1; -1; -1) as $i (0; . * $x + $coefficients[$i]);
# Example
[-19, 7, -4, 6] | horner(3)Invocation: $JQ -n -f horner.jq
where $JQ is either jq or gojq
- Output:
128
Julia
Imperative:
function horner(coefs, x)
s = coefs[end] * one(x)
for k in length(coefs)-1:-1:1
s = coefs[k] + x * s
end
return s
end
@show horner([-19, 7, -4, 6], 3)
- Output:
horner([-19, 7, -4, 6], 3) = 128
Functional:
horner2(coefs, x) = foldr((u, v) -> u + x * v, coefs, init=zero(promote_type(typeof(x),eltype(coefs))))
@show horner2([-19, 7, -4, 6], 3)
- Output:
horner2([-19, 7, -4, 6], 3) = 128
Note: In Julia 1.4 or later one would normally use the built-in evalpoly function for this purpose:
@show evalpoly(3, [-19, 7, -4, 6])
- Output:
evalpoly(3, [-19, 7, -4, 6]) = 128
K
horner:{y _sv|x}
horner[-19 7 -4 6;3]
128
Kotlin
// version 1.1.2
fun horner(coeffs: DoubleArray, x: Double): Double {
var sum = 0.0
for (i in coeffs.size - 1 downTo 0) sum = sum * x + coeffs[i]
return sum
}
fun main(args: Array<String>) {
val coeffs = doubleArrayOf(-19.0, 7.0, -4.0, 6.0)
println(horner(coeffs, 3.0))
}
- Output:
128.0
Lambdatalk
{def horner
{def horner.r
{lambda {:p :x :r}
{if {A.empty? :p}
then :r
else {horner.r {A.rest :p} :x {+ {A.first :p} {* :x :r}}}}}}
{lambda {:p :x}
{horner.r {A.reverse :p} :x 0}}}
{horner {A.new -19 7 -4 6} 3}
-> 128
{def φ {/ {+ 1 {sqrt 5}} 2}} = 1.618033988749895
{horner {A.new -1 -1 1} φ}
-> 2.220446049250313e-16 ~ 0
Liberty BASIC
src$ = "Hello"
coefficients$ = "-19 7 -4 6" ' list coefficients of all x^0..x^n in order
x = 3
print horner(coefficients$, x) '128
print horner("4 3 2 1", 10) '1234
print horner("1 1 0 0 1", 2) '19
end
function horner(coefficients$, x)
accumulator = 0
'getting length of a list requires extra pass with WORD$.
'So we just started from high above
for index = 100 to 1 step -1
cft$ = word$(coefficients$, index)
if cft$<>"" then accumulator = ( accumulator * x ) + val(cft$)
next
horner = accumulator
end functionLogo
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] ; 128Lua
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 ) )
Maple
applyhorner:=(L::list,x)->foldl((s,t)->s*x+t,op(ListTools:-Reverse(L))):
applyhorner([-19,7,-4,6],x);
applyhorner([-19,7,-4,6],3);Output:
((6 x - 4) x + 7) x - 19
128
Mathematica / Wolfram Language
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
MATLAB
function accumulator = hornersRule(x,coefficients)
accumulator = 0;
for i = (numel(coefficients):-1:1)
accumulator = (accumulator * x) + coefficients(i);
end
end
Output:
>> hornersRule(3,[-19, 7, -4, 6])
ans =
128
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.
>> polyval(fliplr([-19, 7, -4, 6]),3)
ans =
128
Maxima
/* Function horner already exists in Maxima, though it operates on expressions, not lists of coefficients */
horner(5*x^3+2*x+1);
x*(5*x^2+2)+1
/* Here is an implementation */
horner2(p, x) := block([n, y, i],
n: length(p),
y: p[n],
for i: n - 1 step -1 thru 1 do y: y*x + p[i],
y
)$
horner2([-19, 7, -4, 6], 3);
128
/* Another with rreduce */
horner3(p,x):=rreduce(lambda([a,y],x*y+a),p);
horner3([a,b,c,d,e,f],x);
x*(x*(x*(x*(f*x+e)+d)+c)+b)+a
/* Extension to compute also derivatives up to a specified order.
See William Kahan, Roundoff in Polynomial Evaluation, 1986
http://www.cs.berkeley.edu/~wkahan/Math128/Poly.pdf */
poleval(a, x, [m]) := block(
[n: length(a), v, k: 1],
if emptyp(m) then m: 1 else m: 1 + first(m),
v: makelist(0, m),
v[1]: a[n],
for i from n - 1 thru 1 step -1 do (
for j from m thru 2 step -1 do v[j]: v[j] * x + v[j - 1],
v[1]: v[1] * x + a[i]
),
for i from 2 thru m do (
v[i]: v[i] * k,
k: k * i
),
if m = 1 then first(v) else v
)$
poleval([0, 0, 0, 0, 1], x, 4);
[x^4, 4 * x^3, 12 * x^2, 24 * x, 24]
poleval([0, 0, 0, 0, 1], x);
x^4
Mercury
:- module horner.
:- interface.
:- import_module io.
:- pred main(io::di, io::uo) is det.
:- implementation.
:- import_module int, list, string.
main(!IO) :-
io.format("%i\n", [i(horner(3, [-19, 7, -4, 6]))], !IO).
:- func horner(int, list(int)) = int.
horner(X, Cs) = list.foldr((func(C, Acc) = Acc * X + C), Cs, 0).МК-61/52
ИП0 1 + П0
ИПE ИПD * КИП0 + ПE
ИП0 1 - x=0 04
ИПE С/П
Input: Р1:РС - coefficients, Р0 - number of the coefficients, РD - x.
Modula-2
MODULE Horner;
FROM RealStr IMPORT RealToStr;
FROM Terminal IMPORT WriteString,WriteLn,ReadChar;
PROCEDURE Horner(coeff : ARRAY OF REAL; x : REAL) : REAL;
VAR
ans : REAL;
i : CARDINAL;
BEGIN
ans := 0.0;
FOR i:=HIGH(coeff) TO 0 BY -1 DO
ans := (ans * x) + coeff[i];
END;
RETURN ans
END Horner;
TYPE A = ARRAY[0..3] OF REAL;
VAR
buf : ARRAY[0..63] OF CHAR;
coeff : A;
ans : REAL;
BEGIN
coeff := A{-19.0, 7.0, -4.0, 6.0};
ans := Horner(coeff, 3.0);
RealToStr(ans, buf);
WriteString(buf);
WriteLn;
ReadChar
END Horner.
NetRexx
/* NetRexx */
options replace format comments java crossref savelog symbols nobinary
c = [-19, 7, -4, 6] -- # list coefficients of all x^0..x^n in order
n=3
x=3
r=0
loop i=n to 0 by -1
r=r*x+c[i]
End
Say r
Say 6*x**3-4*x**2+7*x-19
Output:
128 128
Nim
# You can also just use `reversed` proc from stdlib `algorithm` module
iterator reversed[T](x: openArray[T]): T =
for i in countdown(x.high, x.low):
yield x[i]
proc horner[T](coeffs: openArray[T], x: T): int =
for c in reversed(coeffs):
result = result * x + c
echo horner([-19, 7, -4, 6], 3)
Oberon-2
MODULE HornerRule;
IMPORT
Out;
TYPE
Coefs = POINTER TO ARRAY OF LONGINT;
VAR
coefs: Coefs;
PROCEDURE Eval(coefs: ARRAY OF LONGINT;size,x: LONGINT): LONGINT;
VAR
i,acc: LONGINT;
BEGIN
acc := 0;
FOR i := LEN(coefs) - 1 TO 0 BY -1 DO
acc := acc * x + coefs[i]
END;
RETURN acc
END Eval;
BEGIN
NEW(coefs,4);
coefs[0] := -19;
coefs[1] := 7;
coefs[2] := -4;
coefs[3] := 6;
Out.Int(Eval(coefs^,4,3),0);Out.Ln
END HornerRule.
- Output:
128
Objeck
class Horner {
function : Main(args : String[]) ~ Nil {
coeffs := Collection.FloatVector->New();
coeffs->AddBack(-19.0);
coeffs->AddBack(7.0);
coeffs->AddBack(-4.0);
coeffs->AddBack(6.0);
PolyEval(coeffs, 3)->PrintLine();
}
function : PolyEval(coefficients : Collection.FloatVector , x : Float) ~ Float {
accumulator := coefficients->Get(coefficients->Size() - 1);
for(i := coefficients->Size() - 2; i > -1; i -= 1;) {
accumulator := (accumulator * x) + coefficients->Get(i);
};
return accumulator;
}
}Objective-C
Using blocks
#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()
{
@autoreleasepool {
NSArray *coeff = @[@-19.0, @7.0, @-4.0, @6.0];
printf("%f\n", [coeff horner: 3.0]);
}
return 0;
}
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>
# let coeffs = [-19; 7; -4; 6] in
horner coeffs 3 ;;
- : int = 128
It's also possible to do fold_right instead of reversing and doing fold_left; but fold_right is not tail-recursive.
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)
ooRexx
/* Rexx ---------------------------------------------------------------
* 04.03.2014 Walter Pachl
*--------------------------------------------------------------------*/
c = .array~of(-19,7,-4,6) -- coefficients of all x^0..x^n in order
n=3
x=3
r=0
loop i=n+1 to 1 by -1
r=r*x+c[i]
End
Say r
Say 6*x**3-4*x**2+7*x-19
Output:
128 128
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}}PARI/GP
Also note that Pari has a polynomial type. Evaluating these is as simple as subst(P,variable(P),x).
horner(v,x)={
my(s=0);
forstep(i=#v,1,-1,s=s*x+v[i]);
s
};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.
Output:
Horner calculated polynomial of 6*x^3 - 4*x^2 + 7*x - 19 for x = 3: 128.0000
PascalABC.NET
function Horner(coeffs: array of real; x: real): real;
begin
Result := 0;
foreach var coeff in coeffs.Reverse do
Result := Result * x + coeff
end;
begin
Print(Horner(|-19.0, 7, -4, 6|, 3))
end.
- Output:
128
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;
Functional version
use strict;
use List::Util qw(reduce);
sub horner($$){
my ($coeff_ref, $x) = @_;
reduce { $a * $x + $b } reverse @$coeff_ref;
}
my @coeff = (-19.0, 7, -4, 6);
my $x = 3;
print horner(\@coeff, $x), "\n";
Recursive version
sub horner {
my ($coeff, $x) = @_;
@$coeff and
$$coeff[0] + $x * horner( [@$coeff[1 .. $#$coeff]], $x )
}
print horner( [ -19, 7, -4, 6 ], 3 );
Phix
with javascript_semantics function horner(atom x, sequence coeff) atom res = 0 for i=length(coeff) to 1 by -1 do res = res*x + coeff[i] end for return res end function ?horner(3,{-19, 7, -4, 6})
- Output:
128
PHP
<?php
function horner($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, $x), "\n";
?>
Functional version
<?php
function horner($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, $x), "\n";
?>
Picat
Recursion
horner([],_X,0).
horner([H|T],X,V) :-
horner(T,X,V1),
V = V1 * X + H.Iterative
horner2(Coeff, X, V) =>
Acc = 0,
foreach(I in Coeff.length..-1..1)
Acc := Acc*X + Coeff[I]
end,
V = Acc.Functional approach
h3(X,A,B) = A+B*X.
horner3(Coeff, X) = fold($h3(X),0,Coeff.reverse()).Test
go =>
horner([-19, 7, -4, 6], 3, V),
println(V),
horner2([-19, 7, -4, 6], 3, V2),
println(V2),
V3 = horner3([-19, 7, -4, 6], 3),
println(V3),
nl.- Output:
128 128 128
PicoLisp
(de horner (Coeffs X)
(let Res 0
(for C (reverse Coeffs)
(setq Res (+ C (* X Res))) ) ) ): (horner (-19.0 7.0 -4.0 6.0) 3.0)
-> 128PL/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;Potion
horner = (x, coef) :
result = 0
coef reverse each (a) :
result = (result * x) + a
.
result
.
horner(3, (-19, 7, -4, 6)) printPowerShell
function horner($coefficients, $x) {
$accumulator = 0
foreach($i in ($coefficients.Count-1)..0){
$accumulator = ( $accumulator * $x ) + $coefficients[$i]
}
$accumulator
}
$coefficients = @(-19, 7, -4, 6)
$x = 3
horner $coefficients $x
Output:
128
Prolog
Tested with SWI-Prolog. Works with other dialects.
horner([], _X, 0).
horner([H|T], X, V) :-
horner(T, X, V1),
V is V1 * X + H.
Output :
?- horner([-19, 7, -4, 6], 3, V).
V = 128.
Functional 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
:- 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).
Functional syntax (Ciao)
Works with Ciao (https://github.com/ciao-lang/ciao) and the fsyntax package:
:- module(_, [horner/3], [fsyntax, hiord]).
:- use_module(library(hiordlib)).
horner(L, X) := ~foldr((''(H,V0,V) :- V is V0*X + H), L, 0).
PureBasic
Procedure Horner(List Coefficients(), b)
Define result
ForEach Coefficients()
result*b+Coefficients()
Next
ProcedureReturn result
EndProcedure
Implemented as
NewList a()
AddElement(a()): a()= 6
AddElement(a()): a()= -4
AddElement(a()): a()= 7
AddElement(a()): a()=-19
Debug Horner(a(),3)
Outputs
128
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
Functional version
>>> 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
>>> import numpy
>>> numpy.polynomial.polynomial.polyval(3, (-19, 7, -4, 6))
128.0
R
Procedural style:
horner <- function(a, x) {
y <- 0
for(c in rev(a)) {
y <- y * x + c
}
y
}
cat(horner(c(-19, 7, -4, 6), 3), "\n")
Functional style:
horner <- function(x, v) {
Reduce(v, right=T, f=function(a, b) {
b * x + a
})
}
- Output:
> v <- c(-19, 7, -4, 6) > horner(3, v) [1] 128
Racket
Translated from Haskell
#lang racket
(define (horner x l)
(foldr (lambda (a b) (+ a (* b x))) 0 l))
(horner 3 '(-19 7 -4 6))
Raku
(formerly Perl 6)
sub horner ( @coeffs, $x ) {
@coeffs.reverse.reduce: { $^a * $x + $^b };
}
say horner( [ -19, 7, -4, 6 ], 3 );
A recursive version would spare us the need for reversing the list of coefficients. However, special care must be taken in order to write it, because the way Raku implements lists is not optimized for this kind of treatment. Lisp-style lists are, and fortunately it is possible to emulate them with Pairs and the reduction meta-operator:
multi horner(Numeric $c, $) { $c }
multi horner(Pair $c, $x) {
$c.key + $x * horner( $c.value, $x )
}
say horner( [=>](-19, 7, -4, 6 ), 3 );
We can also use the composition operator:
sub horner ( @coeffs, $x ) {
([o] map { $_ + $x * * }, @coeffs)(0);
}
say horner( [ -19, 7, -4, 6 ], 3 );
- Output:
128
One advantage of using the composition operator is that it allows for the use of an infinite list of coefficients.
sub horner ( @coeffs, $x ) {
map { .(0) }, [\o] map { $_ + $x * * }, @coeffs;
}
say horner( [ 1 X/ (1, |[\*] 1 .. *) ], i*pi )[20];
- Output:
-0.999999999924349-5.28918515954219e-10i
Rascal
import List;
public int horners_rule(list[int] coefficients, int x){
acc = 0;
for( i <- reverse(coefficients)){
acc = acc * x + i;}
return acc;
}A neater and shorter solution using a reducer:
public int horners_rule2(list[int] coefficients, int x) = (0 | it * x + c | c <- reverse(coefficients));Output:
rascal>horners_rule([-19, 7, -4, 6], 3)
int: 128
rascal>horners_rule2([-19, 7, -4, 6], 3)
int: 128REBOL
REBOL []
horner: func [coeffs x] [
result: 0
foreach i reverse coeffs [
result: (result * x) + i
]
return result
]
print horner [-19 7 -4 6] 3
REXX
version 1
/*REXX program demonstrates using Horner's rule for polynomial evaluation. */
numeric digits 30 /*use extra numeric precision. */
parse arg x poly /*get value of X and the coefficients. */
$= /*start with a clean slate equation. */
do deg=0 until poly=='' /*get the equation's coefficients. */
parse var poly c.deg poly; c.deg=c.deg/1 /*get equation coefficient & normalize.*/
if c.deg>=0 then c.deg= '+'c.deg /*if ¬ negative, then prefix with a + */
$=$ c.deg /*concatenate it to the equation. */
if deg\==0 & c.deg\=0 then $=$'∙x^'deg /*¬1st coefficient & ¬0? Append X pow.*/
$=$ ' ' /*insert some blanks, make it look nice*/
end /*deg*/
say ' x = ' x
say ' degree = ' deg
say ' equation = ' $
a=c.deg /*A: is the accumulator (or answer). */
do j=deg-1 by -1 for deg; a=a*x+c.j /*apply Horner's rule to the equations.*/
end /*j*/
say /*display a blank line for readability.*/
say ' answer = ' a /*stick a fork in it, we're all done. */
output when the following is used for input: 3 -19 7 -4 6
x = 3
degree = 3
equation = -19 +7∙x^1 -4∙x^2 +6∙x^3
answer = 128
version 2
/* REXX ---------------------------------------------------------------
* 27.07.2012 Walter Pachl
* coefficients reversed to descending order of power
* I'm used to x**2+x-3
* equation formatting prettified (coefficients 1 and 0)
*--------------------------------------------------------------------*/
Numeric Digits 30 /* use extra numeric precision. */
Parse Arg x poly /* get value of x and coefficients*/
rpoly=''
Do p=0 To words(poly)-1
rpoly=rpoly word(poly,words(poly)-p)
End
poly=rpoly
equ='' /* start with equation clean slate*/
deg=words(poly)-1
pdeg=deg
Do Until deg<0 /* get the equation's coefficients*/
Parse Var poly c.deg poly /* in descending order of powers */
c.deg=c.deg+0 /* normalize it */
If c.deg>0 & deg<pdeg Then /* positive and not first term */
prefix='+' /* prefix a + sign. */
Else prefix=''
Select
When deg=0 Then term=c.deg
When deg=1 Then
If c.deg=1 Then term='x'
Else term=c.deg'*x'
Otherwise
If c.deg=1 Then term='x^'deg
Else term=c.deg'*x^'deg
End
If c.deg<>0 Then /* build up the equation */
equ=equ||prefix||term
deg=deg-1
End
a=c.pdeg
Do p=pdeg To 1 By -1 /* apply Horner's rule. */
pm1=p-1
a=a*x+c.pm1
End
Say ' x = ' x
Say ' degree = ' pdeg
Say ' equation = ' equ
Say ' '
Say ' result = ' a- Output:
x = 3 degree = 3 equation = 6*x^3-4*x^2+7*x-19 result = 128
Ring
coefficients = [-19, 7, -4, 6]
see "x = 3" + nl +
"degree = 3" + nl +
"equation = 6*x^3-4*x^2+7*x-19" + nl +
"result = " + horner(coefficients, 3) + nl
func horner coeffs, x
w = 0
for n = len(coeffs) to 1 step -1
w = w * x + coeffs[n]
next
return wOutput:
x = 3 degree = 3 equation = 6*x^3-4*x^2+7*x-19 result = 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
>> a = [6, -4, 7, -19]
6 -4 7 -19
>> x=3
3
>> polyval(x, a)
128RPL
Translation of the algorithm
Following the pseudocode given here to the letter:
≪ OVER DUP SIZE GET → a x0 p
≪ a SIZE 1 - 1 FOR j
'a(j)+x0*p' EVAL 'p' STO -1 STEP
p
≫ ≫ ‘HORNR’ STO
Idiomatic one-liner
Reducing the loop to its simplest form: one memory call, one multiplication and one addition.
≪ → x0 ≪ LIST→ 2 SWAP START x0 * + NEXT ≫ ≫ ‘HORNR’ STO
- Input:
{ -19 7 -4 6 } 3 HORNR
- Output:
1: 128
Ruby
def horner(coeffs, x)
coeffs.reverse.inject(0) {|acc, coeff| acc * x + coeff}
end
p horner([-19, 7, -4, 6], 3) # ==> 128Run BASIC
coef$ = "-19 7 -4 6" ' list coefficients of all x^0..x^n in order
x = 3
print horner(coef$,x) '128
print horner("1.2 2.3 3.4 4.5 5.6", 8) '25478.8
print horner("5 4 3 2 1", 10) '12345
print horner("1 0 1 1 1 0 0 1", 2) '157
end
function horner(coef$,x)
while word$(coef$, i + 1) <> ""
i = i + 1 ' count the num of values
wend
for j = i to 1 step -1
accum = ( accum * x ) + val(word$(coef$, j))
next
horner = accum
end functionRust
fn horner(v: &[f64], x: f64) -> f64 {
v.iter().rev().fold(0.0, |acc, coeff| acc*x + coeff)
}
fn main() {
let v = [-19., 7., -4., 6.];
println!("result: {}", horner(&v, 3.0));
}A generic version that works with any number type and much more. So much more, it's hard to imagine what that may be useful for.
extern crate num; // 0.2.0
use num::Zero;
use std::ops::{Add, Mul};
fn horner<Arr, Arg, Out>(v: &[Arr], x: Arg) -> Out
where
Arr: Clone,
Arg: Clone,
Out: Zero + Mul<Arg, Output = Out> + Add<Arr, Output = Out>,
{
v.iter()
.rev()
.fold(Zero::zero(), |acc, coeff| acc * x.clone() + coeff.clone())
}
fn main() {
let v = [-19., 7., -4., 6.];
let output: f64 = horner(&v, 3.0);
println!("result: {}", output);
}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;Scala
def horner(coeffs:List[Double], x:Double)=
coeffs.reverse.foldLeft(0.0){(a,c)=> a*x+c}val coeffs=List(-19.0, 7.0, -4.0, 6.0)
println(horner(coeffs, 3))
-> 128.0Scheme
Works with: Scheme version R RS
(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)Output:
128Seed7
$ 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;Output:
128.0
Sidef
Functional:
func horner(coeff, x) {
coeff.reverse.reduce { |a,b| a*x + b };
}
say horner([-19, 7, -4, 6], 3); # => 128Recursive:
func horner(coeff, x) {
(coeff.len > 0) \
? (coeff[0] + x*horner(coeff.last(-1), x))
: 0
}
say horner([-19, 7, -4, 6], 3) # => 128Smalltalk
OrderedCollection extend [
horner: aValue [
^ self reverse inject: 0 into: [:acc :c | acc * aValue + c].
]
].
(#(-19 7 -4 6) asOrderedCollection horner: 3) displayNl.Standard ML
(* Assuming real type for coefficients and x *)
fun horner coeffList x = foldr (fn (a, b) => a + b * x) (0.0) coeffListSwift
func horner(coefs: [Double], x: Double) -> Double {
return reduce(lazy(coefs).reverse(), 0) { $0 * x + $1 }
}
println(horner([-19, 7, -4, 6], 3))- Output:
128.0
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
}Demonstrating:
puts [horner {-19 7 -4 6} 3]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.
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 FunctionOutput:
print Horner(3, -19, 7, -4, 6) 128
VBScript
Function horners_rule(coefficients,x)
accumulator = 0
For i = UBound(coefficients) To 0 Step -1
accumulator = (accumulator * x) + coefficients(i)
Next
horners_rule = accumulator
End Function
WScript.StdOut.WriteLine horners_rule(Array(-19,7,-4,6),3)- Output:
128
Visual Basic .NET
Module Module1
Function Horner(coefficients As Double(), variable As Double) As Double
Return coefficients.Reverse().Aggregate(Function(acc, coeff) acc * variable + coeff)
End Function
Sub Main()
Console.WriteLine(Horner({-19.0, 7.0, -4.0, 6.0}, 3.0))
End Sub
End Module- Output:
128
Visual FoxPro
Coefficients in ascending order.
LOCAL x As Double
LOCAL ARRAY aCoeffs[1]
CLEAR
CREATE CURSOR coeffs (c1 I, c2 I, c3 I, c4 I)
INSERT INTO coeffs VALUES (-19,7,-4,6)
SCATTER TO aCoeffs
x = VAL(INPUTBOX("Value of x:", "Value"))
? EvalPoly(@aCoeffs, x)
USE IN coeffs
FUNCTION EvalPoly(c, x As Double) As Double
LOCAL s As Double, k As Integer, n As Integer
n = ALEN(c)
s = 0
FOR k = n TO 1 STEP -1
s = s*x + c[k]
ENDFOR
RETURN s
ENDFUNCCoefficients in descending order.
LOCAL x As Double
LOCAL ARRAY aCoeffs[1]
CLEAR
CREATE CURSOR tmp (c1 I, c2 I, c3 I, c4 I)
INSERT INTO tmp VALUES (6,-4,7,-19)
SCATTER TO aCoeffs
x = VAL(INPUTBOX("Value of x:", "Value"))
? EvalPolyDesc(@aCoeffs, x)
USE IN tmp
FUNCTION EvalPolyDesc(c, x As Double) As Double
LOCAL s As Double, e
s = 0
FOR EACH e IN c FOXOBJECT
s = s*x + e
ENDFOR
RETURN s
ENDFUNCV (Vlang)
fn horner(x i64, c []i64) i64 {
mut acc := i64(0)
for i := c.len - 1; i >= 0; i-- {
acc = acc*x + c[i]
}
return acc
}
fn main() {
println(horner(3, [i64(-19), 7, -4, 6]))
}- Output:
128
Wren
var horner = Fn.new { |x, c|
var count = c.count
if (count == 0) return 0
return (count-1..0).reduce(0) { |acc, index| acc*x + c[index] }
}
System.print(horner.call(3, [-19, 7, -4, 6]))- Output:
128
XPL0
code IntOut=11;
func Horner(X, N, C); \Return value of polynomial in X
int X, N, C; \variable, number of terms, coefficient list
int A, I;
[A:= 0;
for I:= N-1 downto 0 do
A:= A*X + C(I);
return A;
];
IntOut(0, Horner(3, 4, [-19, 7, -4, 6]));Output:
128
zkl
fcn horner(coeffs,x)
{ coeffs.reverse().reduce('wrap(a,coeff){ a*x + coeff },0.0) }- Output:
horner(T(-19,7,-4,6), 3).println(); 128