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Task

Create a program that finds and outputs the roots of a given function, range and (if applicable) step width.

The program should identify whether the root is exact or approximate.

For this task, use: f(x) = x³ - 3x² + 2x

ALGOL 68

Works with: ALGOL 68 version Revision 1 - no extensions to language used

Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny

Finding 3 roots using the secant method: <lang algol68>MODE DBL = LONG REAL; FORMAT dbl = $g(-long real width, long real width-6, -2)$;

MODE XY = STRUCT(DBL x, y); FORMAT xy root = $f(dbl)" ("b("Exactly", "Approximately")")"$;

MODE DBLOPT = UNION(DBL, VOID); MODE XYRES = UNION(XY, VOID);

PROC find root = (PROC (DBL)DBL f, DBLOPT in x1, in x2, in x error, in y error)XYRES:(

 INT limit = ENTIER (long real width / log(2)); # worst case of a binary search) #
 DBL x1 := (in x1|(DBL x1):x1|-5.0), # if x1 is EMPTY then -5.0 #
     x2 := (in x2|(DBL x2):x2|+5.0), 
     x error := (in x error|(DBL x error):x error|small real), 
     y error := (in y error|(DBL y error):y error|small real);
 DBL y1 := f(x1), y2;
 DBL dx := x1 - x2, dy;

 IF y1 = 0 THEN
   XY(x1, y1) # we already have a solution! #
 ELSE
   FOR i WHILE
     y2 := f(x2);
     IF y2 = 0 THEN stop iteration FI;
     IF i = limit THEN value error FI;
     IF y1 = y2 THEN value error FI;
     dy := y1 - y2;
     dx := dx / dy * y2;
     x1 := x2; y1 := y2; # retain for next iteration #
     x2 -:= dx;
 # WHILE # ABS dx > x error AND ABS dy > y error DO 
     SKIP
   OD;
   stop iteration: 
     XY(x2, y2) EXIT
   value error:
     EMPTY
 FI

);

PROC f = (DBL x)DBL: x UP 3 - LONG 3.1 * x UP 2 + LONG 2.0 * x;

DBL first root, second root, third root;

XYRES first result = find root(f, LENG -1.0, LENG 3.0, EMPTY, EMPTY); CASE first result IN

 (XY first result): (
   printf(($"1st root found at x = "f(xy root)l$, x OF first result, y OF first result=0));
   first root := x OF first result
 )
 OUT printf($"No first root found"l$); stop

ESAC;

XYRES second result = find root( (DBL x)DBL: f(x) / (x - first root), EMPTY, EMPTY, EMPTY, EMPTY); CASE second result IN

 (XY second result): (
   printf(($"2nd root found at x = "f(xy root)l$, x OF second result, y OF second result=0));
   second root := x OF second result
 )
 OUT printf($"No second root found"l$); stop

ESAC;

XYRES third result = find root( (DBL x)DBL: f(x) / (x - first root) / ( x - second root ), EMPTY, EMPTY, EMPTY, EMPTY); CASE third result IN

 (XY third result): (
   printf(($"3rd root found at x = "f(xy root)l$, x OF third result, y OF third result=0));
   third root := x OF third result
 )
 OUT printf($"No third root found"l$); stop

ESAC</lang> Output:

1st root found at x =  9.1557112297752398099031e-1 (Approximately)
2nd root found at x =  2.1844288770224760190097e 0 (Approximately)
3rd root found at x =  0.0000000000000000000000e 0 (Exactly)

ATS

include

"share/atspre_staload.hats"

typedef d = double

fun findRoots (

 start: d, stop: d, step: d, f: (d) -> d, nrts: int, A: d

) : void = ( // if start < stop then let

 val A2 = f(start)
 var nrts: int = nrts
 val () =
 if A2 = 0.0
   then (
     nrts := nrts + 1;
     $extfcall(void, "printf", "An exact root is found at %12.9f\n", start)
   ) (* end of [then] *)
 // end of [if]
 val () =
 if A * A2 < 0.0
   then (
     nrts := nrts + 1;
     $extfcall(void, "printf", "An approximate root is found at %12.9f\n", start)
   ) (* end of [then] *)
 // end of [if]

in

 findRoots(start+step, stop, step, f, nrts, A2)

end // end of [then] else (

 if nrts = 0
   then $extfcall(void, "printf", "There are no roots found!\n")
 // end of [if]

) (* end of [else] *) // ) (* end of [findRoots] *)

(* ****** ****** *)

implement main0 () = findRoots (~1.0, 3.0, 0.001, lam (x) => x*x*x - 3.0*x*x + 2.0*x, 0, 0.0) </lang>

Axiom

Using a polynomial solver: <lang Axiom>expr := x^3-3*x^2+2*x solve(expr,x)</lang> Output: <lang Axiom> (1) [x= 2,x= 1,x= 0]

                         Type: List(Equation(Fraction(Polynomial(Integer))))</lang>

Using the secant method in the interpreter: <lang Axiom>digits(30) secant(eq: Equation Expression Float, binding: SegmentBinding(Float)):Float ==

 eps := 1.0e-30
 expr := lhs eq - rhs eq
 x := variable binding
 seg := segment binding
 x1 := lo seg
 x2 := hi seg
 fx1 := eval(expr, x=x1)::Float
 abs(fx1)<eps => return x1
 for i in 1..100 repeat
   fx2 := eval(expr, x=x2)::Float
   abs(fx2)<eps => return x2
   (x1, fx1, x2) := (x2, fx2, x2 - fx2 * (x2 - x1) / (fx2 - fx1))
 error "Function not converging."</lang>

The example can now be called using: <lang Axiom>secant(expr=0,x=-0.5..0.5)</lang>

BBC BASIC

<lang bbcbasic> function$ = "x^3-3*x^2+2*x"

     rangemin = -1
     rangemax = 3
     stepsize = 0.001
     accuracy = 1E-8
     PROCroots(function$, rangemin, rangemax, stepsize, accuracy)
     END
     
     DEF PROCroots(func$, min, max, inc, eps)
     LOCAL x, sign%, oldsign%
     oldsign% = 0
     FOR x = min TO max STEP inc
       sign% = SGN(EVAL(func$))
       IF sign% = 0 THEN
         PRINT "Root found at x = "; x
         sign% = -oldsign%
       ELSE IF sign% <> oldsign% AND oldsign% <> 0 THEN
           IF inc < eps THEN
             PRINT "Root found near x = "; x
           ELSE
             PROCroots(func$, x-inc, x+inc/8, inc/8, eps)
           ENDIF
         ENDIF
       ENDIF
       oldsign% = sign%
     NEXT x
     ENDPROC</lang>

Output:

Root found near x = 2.29204307E-9
Root found near x = 1
Root found at x = 2

C

Secant Method

<lang c>#include <math.h>

include <stdio.h>

double f(double x) {

   return x*x*x-3.0*x*x +2.0*x;

}

double secant( double xA, double xB, double(*f)(double) ) {

   double e = 1.0e-12;
   double fA, fB;
   double d;
   int i;
   int limit = 50;

   fA=(*f)(xA);
   for (i=0; i<limit; i++) {
       fB=(*f)(xB);
       d = (xB - xA) / (fB - fA) * fB;
       if (fabs(d) < e) 
           break;
       xA = xB;
       fA = fB;
       xB -= d;
   }
   if (i==limit) {
       printf("Function is not converging near (%7.4f,%7.4f).\n", xA,xB);
       return -99.0;
   }
   return xB;

}

int main(int argc, char *argv[]) {

   double step = 1.0e-2;
   double e = 1.0e-12;
   double x = -1.032;		// just so we use secant method
   double xx, value;

   int s = (f(x)> 0.0);

   while (x < 3.0) {
       value = f(x);
       if (fabs(value) < e) {
           printf("Root found at x= %12.9f\n", x);
           s = (f(x+.0001)>0.0);
       }
       else if ((value > 0.0) != s) {
           xx = secant(x-step, x,&f);
           if (xx != -99.0)   // -99 meaning secand method failed
               printf("Root found at x= %12.9f\n", xx);
           else
               printf("Root found near x= %7.4f\n", x);
           s = (f(x+.0001)>0.0);
       }
       x += step;
   }
   return 0;

}</lang>

GNU Scientific Library

<lang C>#include <gsl/gsl_poly.h>

include <stdio.h>

int main(int argc, char *argv[]) {

   /* 0 + 2x - 3x^2 + 1x^3 */
   double p[] = {0, 2, -3, 1};
   double z[6];
   gsl_poly_complex_workspace *w = gsl_poly_complex_workspace_alloc(4);
   gsl_poly_complex_solve(p, 4, w, z);
   gsl_poly_complex_workspace_free(w);

   for(int i = 0; i < 3; ++i)
       printf("%.12f\n", z[2 * i]);

   return 0;

}</lang>

One can also use the GNU Scientific Library to find roots of functions. Compile with

gcc roots.c -lgsl -lcblas -o roots

C++

<lang cpp>#include <iostream>

double f(double x) { return (x*x*x - 3*x*x + 2*x); }

int main() { double step = 0.001; // Smaller step values produce more accurate and precise results double start = -1; double stop = 3; double value = f(start); double sign = (value > 0);

// Check for root at start if ( 0 == value ) std::cout << "Root found at " << start << std::endl;

for( double x = start + step; x <= stop; x += step ) { value = f(x);

if ( ( value > 0 ) != sign ) // We passed a root std::cout << "Root found near " << x << std::endl; else if ( 0 == value ) // We hit a root std::cout << "Root found at " << x << std::endl;

// Update our sign sign = ( value > 0 ); } }</lang>

Brent's Method

Brent's Method uses a combination of the bisection method, inverse quadratic interpolation, and the secant method to find roots. It has a guaranteed run time equal to that of the bisection method (which always converges in a known number of steps (log2[(upper_bound-lower_bound)/tolerance] steps to be precise ) unlike the other methods), but the algorithm uses the much faster inverse quadratic interpolation and secant method whenever possible. The algorithm is robust and commonly used in libraries with a roots() function built in.

The algorithm is coded as a function that returns a double value for the root. The function takes an input that requires the function being evaluated, the lower and upper bounds, the tolerance one is looking for before converging (i recommend 0.0001) and the maximum number of iterations before giving up on finding the root (the root will always be found if the root is bracketed and a sufficient number of iterations is allowed).

The implementation is taken from the pseudo code on the wikipedia page for Brent's Method found here: https://en.wikipedia.org/wiki/Brent%27s_method. <lang cpp>#include <iostream>

include <cmath>
include <algorithm>
include <functional>

double brents_fun(std::function<double (double)> f, double lower, double upper, double tol, unsigned int max_iter) { double a = lower; double b = upper; double fa = f(a); // calculated now to save function calls double fb = f(b); // calculated now to save function calls double fs = 0; // initialize

if (!(fa * fb < 0)) { std::cout << "Signs of f(lower_bound) and f(upper_bound) must be opposites" << std::endl; // throws exception if root isn't bracketed return -11; }

if (std::abs(fa) < std::abs(b)) // if magnitude of f(lower_bound) is less than magnitude of f(upper_bound) { std::swap(a,b); std::swap(fa,fb); }

double c = a; // c now equals the largest magnitude of the lower and upper bounds double fc = fa; // precompute function evalutation for point c by assigning it the same value as fa bool mflag = true; // boolean flag used to evaluate if statement later on double s = 0; // Our Root that will be returned double d = 0; // Only used if mflag is unset (mflag == false)

for (unsigned int iter = 1; iter < max_iter; ++iter) { // stop if converged on root or error is less than tolerance if (std::abs(b-a) < tol) { std::cout << "After " << iter << " iterations the root is: " << s << std::endl; return s; } // end if

if (fa != fc && fb != fc) { // use inverse quadratic interopolation s = ( a * fb * fc / ((fa - fb) * (fa - fc)) ) + ( b * fa * fc / ((fb - fa) * (fb - fc)) ) + ( c * fa * fb / ((fc - fa) * (fc - fb)) ); } else { // secant method s = b - fb * (b - a) / (fb - fa); }

// checks to see whether we can use the faster converging quadratic && secant methods or if we need to use bisection if ( ( (s < (3 * a + b) * 0.25) || (s > b) ) || ( mflag && (std::abs(s-b) >= (std::abs(b-c) * 0.5)) ) || ( !mflag && (std::abs(s-b) >= (std::abs(c-d) * 0.5)) ) || ( mflag && (std::abs(b-c) < tol) ) || ( !mflag && (std::abs(c-d) < tol)) ) { // bisection method s = (a+b)*0.5;

mflag = true; } else { mflag = false; }

fs = f(s); // calculate fs d = c; // first time d is being used (wasnt used on first iteration because mflag was set) c = b; // set c equal to upper bound fc = fb; // set f(c) = f(b)

if ( fa * fs < 0) // fa and fs have opposite signs { b = s; fb = fs; // set f(b) = f(s) } else { a = s; fa = fs; // set f(a) = f(s) }

if (std::abs(fa) < std::abs(fb)) // if magnitude of fa is less than magnitude of fb { std::swap(a,b); // swap a and b std::swap(fa,fb); // make sure f(a) and f(b) are correct after swap }

} // end for

std::cout<< "The solution does not converge or iterations are not sufficient" << std::endl;

} // end brents_fun

</lang>

Clojure

Translation of: Haskell

(defn findRoots [f start stop step eps]

     (filter #(-> (f %) Math/abs (< eps)) (range start stop step)))

</lang>

> (findRoots #(+ (* % % %) (* -3 % %) (* 2 %)) -1.0 3.0 0.0001 0.00000001)
(-9.381755897326649E-14 0.9999999999998124 1.9999999999997022)

CoffeeScript

Translation of: Python

<lang coffeescript> print_roots = (f, begin, end, step) ->

 # Print approximate roots of f between x=begin and x=end,
 # using sign changes as an indicator that a root has been
 # encountered. 
 x = begin
 y = f(x)
 last_y = y
 
 cross_x_axis = ->
   (last_y < 0 and y > 0) or (last_y > 0 and y < 0)
 
 console.log '-----'
 while x <= end
   y = f(x)
   if y == 0
     console.log "Root found at", x
   else if cross_x_axis()
     console.log "Root found near", x
   x += step
   last_y = y

do ->

 # Smaller steps produce more accurate/precise results in general,
 # but for many functions we'll never get exact roots, either due
 # to imperfect binary representation or irrational roots.
 step = 1 / 256

 f1 = (x) -> x*x*x - 3*x*x + 2*x
 print_roots f1, -1, 5, step
 f2 = (x) -> x*x - 4*x + 3
 print_roots f2, -1, 5, step
 f3 = (x) -> x - 1.5
 print_roots f3, 0, 4, step
 f4 = (x) -> x*x - 2
 print_roots f4, -2, 2, step

</lang>

output

<lang> > coffee roots.coffee

Root found at 0 Root found at 1 Root found at 2

Root found at 1 Root found at 3

Root found at 1.5

Root found near -1.4140625 Root found near 1.41796875 </lang>

Common Lisp

Translation of: Perl

find-roots prints roots (and values near roots) and returns a list of root designators, each of which is either a number n, in which case (zerop (funcall function n)) is true, or a cons whose car and cdr are such that the sign of function at car and cdr changes.

<lang lisp>(defun find-roots (function start end &optional (step 0.0001))

 (let* ((roots '())
        (value (funcall function start))
        (plusp (plusp value)))
   (when (zerop value)
     (format t "~&Root found at ~W." start))
   (do ((x (+ start step) (+ x step)))
       ((> x end) (nreverse roots))
     (setf value (funcall function x))
     (cond
      ((zerop value)
       (format t "~&Root found at ~w." x)
       (push x roots))
      ((not (eql plusp (plusp value)))
       (format t "~&Root found near ~w." x)
       (push (cons (- x step) x) roots)))
     (setf plusp (plusp value)))))</lang>

> (find-roots #'(lambda (x) (+ (* x x x) (* -3 x x) (* 2 x))) -1 3)
Root found near 5.3588345E-5.
Root found near 1.0000072.
Root found near 2.000073.
((-4.6411653E-5 . 5.3588345E-5)
 (0.99990714 . 1.0000072)
 (1.9999729 . 2.000073))

D

<lang d>import std.stdio, std.math, std.algorithm;

bool nearZero(T)(in T a, in T b = T.epsilon * 4) pure nothrow {

   return abs(a) <= b;

}

T[] findRoot(T)(immutable T function(in T) pure nothrow fi,

               in T start, in T end, in T step=T(0.001L),
               T tolerance = T(1e-4L)) {
   if (step.nearZero)
       writefln("WARNING: step size may be too small.");

   /// Search root by simple bisection.
   T searchRoot(T a, T b) pure nothrow {
       T root;
       int limit = 49;
       T gap = b - a;

       while (!nearZero(gap) && limit--) {
           if (fi(a).nearZero)
               return a;
           if (fi(b).nearZero)
               return b;
           root = (b + a) / 2.0L;
           if (fi(root).nearZero)
               return root;
           ((fi(a) * fi(root) < 0) ? b : a) = root;
           gap = b - a;
       }

       return root;
   }

   immutable dir = T(end > start ? 1.0 : -1.0);
   immutable step2 = (end > start) ? abs(step) : -abs(step);
   T[T] result;
   for (T x = start; (x * dir) <= (end * dir); x += step2)
       if (fi(x) * fi(x + step2) <= 0) {
           immutable T r = searchRoot(x, x + step2);
           result[r] = fi(r);
       }

   return result.keys.sort().release;

}

void report(T)(in T[] r, immutable T function(in T) pure f,

              in T tolerance = T(1e-4L)) {
   if (r.length) {
       writefln("Root found (tolerance = %1.4g):", tolerance);

       foreach (const x; r) {
           immutable T y = f(x);

           if (nearZero(y))
               writefln("... EXACTLY at %+1.20f, f(x) = %+1.4g",x,y);
           else if (nearZero(y, tolerance))
               writefln(".... MAY-BE at %+1.20f, f(x) = %+1.4g",x,y);
           else
               writefln("Verify needed, f(%1.4g) = " ~
                        "%1.4g > tolerance in magnitude", x, y);
       }
   } else
       writefln("No root found.");

}

void main() {

   static real f(in real x) pure nothrow {
       return x ^^ 3 - (3 * x ^^ 2) + 2 * x;
   }

   findRoot(&f, -1.0L, 3.0L, 0.001L).report(&f);

}</lang>

Output:

Root found (tolerance = 0.0001):
.... MAY-BE at -0.00000000000000000080, f(x) = -1.603e-18
... EXACTLY at +1.00000000000000000020, f(x) = -2.168e-19
.... MAY-BE at +1.99999999999999999950, f(x) = -8.674e-19

NB: smallest increment for real type in D is real.epsilon = 1.0842e-19.

DWScript

Translation of: C

<lang delphi>type TFunc = function (x : Float) : Float;

function f(x : Float) : Float; begin

  Result := x*x*x-3.0*x*x +2.0*x;

end;

const e = 1.0e-12;

function Secant(xA, xB : Float; f : TFunc) : Float; const

  limit = 50;

var

  fA, fB : Float;
  d : Float;
  i : Integer;

begin

  fA := f(xA);
  for i := 0 to limit do begin
     fB := f(xB);
     d := (xB-xA)/(fB-fA)*fB;
     if Abs(d) < e then
        Exit(xB);
     xA := xB;
     fA := fB;
     xB -= d;
  end;
  PrintLn(Format('Function is not converging near (%7.4f,%7.4f).', [xA, xB]));
  Result := -99.0;

end;

const fstep = 1.0e-2;

var x := -1.032; // just so we use secant method var xx, value : Float; var s := f(x)>0.0;

while (x < 3.0) do begin

  value := f(x);
  if Abs(value)<e then begin
     PrintLn(Format("Root found at x= %12.9f", [x]));
     s := (f(x+0.0001)>0.0);
  end else if (value>0.0) <> s then begin
     xx := Secant(x-fstep, x, f);
     if xx <> -99.0 then   // -99 meaning secand method failed
        PrintLn(Format('Root found at x = %12.9f', [xx]))
     else PrintLn(Format('Root found near x= %7.4f', [xx]));
     s := (f(x+0.0001)>0.0);
  end;
  x += fstep;

end;</lang>

EchoLisp

We use the 'math' library, and define f(x) as the polynomial : x³ -3x² +2x

<lang lisp> (lib 'math.lib) Lib: math.lib loaded. (define fp ' ( 0 2 -3 1)) (poly->string 'x fp) → x^3 -3x^2 +2x (poly->html 'x fp) → x³ -3x² +2x (define (f x) (poly x fp)) (math-precision 1.e-6) → 0.000001

(root f -1000 1000) → 2.0000000133245677 ;; 2 (root f -1000 (- 2 epsilon)) → 1.385559938161431e-7 ;; 0 (root f epsilon (- 2 epsilon)) → 1.0000000002190812 ;; 1 </lang>

Elixir

Translation of: Ruby

<lang elixir>defmodule RC do

 def find_roots(f, range, step \\ 0.001) do
   first .. last = range
   max = last + step / 2
   Stream.iterate(first, &(&1 + step))
   |> Stream.take_while(&(&1 < max))
   |> Enum.reduce(sign(first), fn x,sn ->
        value = f.(x)
        cond do
          abs(value) < step / 100 ->
            IO.puts "Root found at #{x}"
            0
          sign(value) == -sn ->
            IO.puts "Root found between #{x-step} and #{x}"
            -sn
          true -> sign(value)
        end
      end)
 end
 
 defp sign(x) when x>0, do: 1
 defp sign(x) when x<0, do: -1
 defp sign(0)         , do: 0

end

f = fn x -> x*x*x - 3*x*x + 2*x end RC.find_roots(f, -1..3)</lang>

Output:

Root found at 8.81239525796218e-16
Root found at 1.0000000000000016
Root found at 1.9999999999998914

Erlang

<lang erlang>% Implemented by Arjun Sunel -module(roots). -export([main/0]). main() -> F = fun(X)->X*X*X - 3*X*X + 2*X end, Step = 0.001, % Using smaller steps will provide more accurate results Start = -1, Stop = 3, Sign = F(Start) > 0, X = Start, while(X, Step, Start, Stop, Sign,F).

while(X, Step, Start, Stop, Sign,F) -> Value = F(X), if Value == 0 -> % We hit a root

       	io:format("Root found at ~p~n",[X]),
       	while(X+Step, Step, Start, Stop,  Value > 0,F);

(Value < 0) == Sign -> % We passed a root io:format("Root found near ~p~n",[X]), while(X+Step , Step, Start, Stop, Value > 0,F);

X > Stop -> io:format("") ; true ->

       	while(X+Step, Step, Start, Stop,  Value > 0,F)

end. </lang>

Output:

Root found near 8.81239525796218e-16
Root found near 1.0000000000000016
Root found near 2.0009999999998915
ok

ERRE

<lang ERRE> PROGRAM ROOTS_FUNCTION

!VAR E,X,STP,VALUE,S%,I%,LIMIT%,X1,X2,D

FUNCTION F(X)

   F=X*X*X-3*X*X+2*X

END FUNCTION

BEGIN

 X=-1
 STP=1.0E-6
 E=1.0E-9
 S%=(F(X)>0)

 PRINT("VERSION 1: SIMPLY STEPPING X")
 WHILE X<3.0 DO
   VALUE=F(X)
   IF ABS(VALUE)<E THEN
       PRINT("ROOT FOUND AT X =";X)
       S%=NOT S%
     ELSE
       IF ((VALUE>0)<>S%) THEN
         PRINT("ROOT FOUND AT X =";X)
         S%=NOT S%
       END IF
   END IF
   X=X+STP
 END WHILE

 PRINT
 PRINT("VERSION 2: SECANT METHOD")
 X1=-1.0
 X2=3.0
 E=1.0E-15
 I%=1
 LIMIT%=300
 LOOP
   IF I%>LIMIT% THEN
      PRINT("ERROR: FUNCTION NOT CONVERGING")
      EXIT
   END IF
   D=(X2-X1)/(F(X2)-F(X1))*F(X2)
   IF ABS(D)<E THEN
     IF D=0 THEN
        PRINT("EXACT ";)
      ELSE
        PRINT("APPROXIMATE ";)
     END IF
     PRINT("ROOT FOUND AT X =";X2)
     EXIT
   END IF
   X1=X2
   X2=X2-D
   I%=I%+1
 END LOOP

END PROGRAM </lang> Note: Outputs are calculated in single precision.

Output:

VERSION 1: SIMPLY STEPPING X
ROOT FOUND AT X = 8.866517E-07
ROOT FOUND AT X = 1.000001
ROOT FOUND AT X = 2

VERSION 2: SECANT METHOD
EXACT ROOT FOUND AT X = 1

Fortran

Works with: Fortran version 90 and later

<lang fortran>PROGRAM ROOTS_OF_A_FUNCTION

 IMPLICIT NONE

 INTEGER, PARAMETER :: dp = SELECTED_REAL_KIND(15)
 REAL(dp) :: f, e, x, step, value
 LOGICAL :: s 
  
 f(x) = x*x*x - 3.0_dp*x*x + 2.0_dp*x
  
 x = -1.0_dp ; step = 1.0e-6_dp ; e = 1.0e-9_dp

 s = (f(x) > 0)
 DO WHILE (x < 3.0)
   value = f(x)
   IF(ABS(value) < e) THEN
     WRITE(*,"(A,F12.9)") "Root found at x =", x
     s = .NOT. s
   ELSE IF ((value > 0) .NEQV. s) THEN
     WRITE(*,"(A,F12.9)") "Root found near x = ", x
     s = .NOT. s
   END IF
   x = x + step
 END DO

END PROGRAM ROOTS_OF_A_FUNCTION</lang> The following approach uses the Secant Method to numerically find one root. Which root is found will depend on the start values x1 and x2 and if these are far from a root this method may not converge. <lang fortran>INTEGER, PARAMETER :: dp = SELECTED_REAL_KIND(15) INTEGER :: i=1, limit=100 REAL(dp) :: d, e, f, x, x1, x2

f(x) = x*x*x - 3.0_dp*x*x + 2.0_dp*x

x1 = -1.0_dp ; x2 = 3.0_dp ; e = 1.0e-15_dp

DO

 IF (i > limit) THEN
   WRITE(*,*) "Function not converging"
   EXIT
 END IF
 d = (x2 - x1) / (f(x2) - f(x1)) * f(x2)
 IF (ABS(d) < e) THEN
   WRITE(*,"(A,F18.15)") "Root found at x = ", x2
   EXIT    
 END IF
 x1 = x2
 x2 = x2 - d
 i = i + 1

END DO</lang>

Go

Secant method. No error checking. <lang go>package main

import ( "fmt" "math" )

func main() { example := func(x float64) float64 { return x*x*x - 3*x*x + 2*x } findroots(example, -.5, 2.6, 1) }

func findroots(f func(float64) float64, lower, upper, step float64) { for x0, x1 := lower, lower+step; x0 < upper; x0, x1 = x1, x1+step { x1 = math.Min(x1, upper) r, status := secant(f, x0, x1) if status != "" && r >= x0 && r < x1 { fmt.Printf(" %6.3f %s\n", r, status) } } }

func secant(f func(float64) float64, x0, x1 float64) (float64, string) { var f0 float64 f1 := f(x0) for i := 0; i < 100; i++ { f0, f1 = f1, f(x1) switch { case f1 == 0: return x1, "exact" case math.Abs(x1-x0) < 1e-6: return x1, "approximate" } x0, x1 = x1, x1-f1*(x1-x0)/(f1-f0) } return 0, "" }</lang> Output:

   0.000 approximate
   1.000 exact
   2.000 approximate

Haskell

<lang haskell>f x = x^3-3*x^2+2*x

findRoots start stop step eps =

 [x | x <- [start, start+step .. stop], abs (f x) < eps]</lang>

Executed in GHCi: <lang haskell>*Main> findRoots (-1.0) 3.0 0.0001 0.000000001 [-9.381755897326649e-14,0.9999999999998124,1.9999999999997022]</lang>

Or using package hmatrix from HackageDB. <lang haskell>import Numeric.GSL.Polynomials import Data.Complex

Main> mapM_ print $ polySolve [0,2,-3,1]

(-5.421010862427522e-20) :+ 0.0 2.000000000000001 :+ 0.0 0.9999999999999996 :+ 0.0</lang> No complex roots, so: <lang haskell>*Main> mapM_ (print.realPart) $ polySolve [0,2,-3,1] -5.421010862427522e-20 2.000000000000001 0.9999999999999996</lang>

HicEst

HicEst's SOLVE function employs the Levenberg-Marquardt method: <lang HicEst>OPEN(FIle='test.txt')

1 DLG(NameEdit=x0, DNum=3)

x = x0 chi2 = SOLVE(NUL=x^3 - 3*x^2 + 2*x, Unknown=x, I=iterations, NumDiff=1E-15) EDIT(Text='approximate exact ', Word=(chi2 == 0), Parse=solution)

WRITE(FIle='test.txt', LENgth=6, Name) x0, x, solution, chi2, iterations GOTO 1</lang> <lang HicEst>x0=0.5; x=1; solution=exact; chi2=79E-32 iterations=65; x0=0.4; x=2E-162 solution=exact; chi2=0; iterations=1E4; x0=0.45; x=1; solution=exact; chi2=79E-32 iterations=67; x0=0.42; x=2E-162 solution=exact; chi2=0; iterations=1E4; x0=1.5; x=1.5; solution=approximate; chi2=0.1406; iterations=14: x0=1.54; x=1; solution=exact; chi2=44E-32 iterations=63; x0=1.55; x=2; solution=exact; chi2=79E-32 iterations=55; x0=1E10; x=2; solution=exact; chi2=18E-31 iterations=511; x0=-1E10; x=0; solution=exact; chi2=0; iterations=1E4;</lang>

Icon and Unicon

Translation of: Java

Works in both languages: <lang unicon>procedure main()

   showRoots(f, -1.0, 4, 0.002)

end

procedure f(x)

   return x^3 - 3*x^2 + 2*x

end

procedure showRoots(f, lb, ub, step)

   ox := x := lb
   oy := f(x)
   os := sign(oy)
   while x <= ub do {
       if (s := sign(y := f(x))) = 0 then write(x)
       else if s ~= os then {
           dx := x-ox
           dy := y-oy
           cx := x-dx*(y/dy)
           write("~",cx)
           }
       (ox := x, oy := y, os := s)
       x +:= step
       }

end

procedure sign(x)

   return (x<0, -1) | (x>0, 1) | 0

end</lang>

Output:

->roots
~2.616794878713638e-18
~1.0
~2.0
->

J

J has builtin a root-finding operator, p., whose input is the coeffiecients of the polynomial (where the exponent of the indeterminate variable matches the index of the coefficient: 0 1 2 would be 0 + x + (2 times x squared)). Hence:

We can determine whether the roots are exact or approximate by evaluating the polynomial at the candidate roots, and testing for zero:

As you can see, p. is also the operator which evaluates polynomials. This is not a coincidence.

That said, we could also implement the technique used by most others here. Specifically: we can implement the function as a black box and check every 1 millionth of a unit between minus one and three, and we can test that result for exactness.

<lang J> blackbox=: 0 2 _3 1&p.

  (#~ (=<./)@:|@blackbox) i.&.(1e6&*)&.(1&+) 3

0 1 2

  0=blackbox 0 1 2

1 1 1</lang>

Here, we see that each of the results (0, 1 and 2) are as accurate as we expect our computer arithmetic to be. (The = returns 1 where paired values are equal and 0 where they are not equal).

Java

<lang java>public class Roots {

   public interface Function {

public double f(double x);

   private static int sign(double x) {

return (x < 0.0) ? -1 : (x > 0.0) ? 1 : 0;

   public static void printRoots(Function f, double lowerBound,

double upperBound, double step) { double x = lowerBound, ox = x; double y = f.f(x), oy = y; int s = sign(y), os = s;

for (; x <= upperBound ; x += step) { s = sign(y = f.f(x)); if (s == 0) { System.out.println(x); } else if (s != os) { double dx = x - ox; double dy = y - oy; double cx = x - dx * (y / dy); System.out.println("~" + cx); } ox = x; oy = y; os = s; }

   public static void main(String[] args) {

Function poly = new Function () { public double f(double x) { return x*x*x - 3*x*x + 2*x; } }; printRoots(poly, -1.0, 4, 0.002);

}</lang> Produces this output:

~2.616794878713638E-18
~1.0000000000000002
~2.000000000000001

JavaScript

Translation of: Java

Works with: SpiderMonkey version 22

Works with: Firefox version 22

<lang javascript> // This function notation is sorta new, but useful here // Part of the EcmaScript 6 Draft // developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Functions_and_function_scope var poly = (x => x*x*x - 3*x*x + 2*x);

function sign(x) { return (x < 0.0) ? -1 : (x > 0.0) ? 1 : 0; }

function printRoots(f, lowerBound, upperBound, step) { var x = lowerBound, ox = x, y = f(x), oy = y, s = sign(y), os = s;

for (; x <= upperBound ; x += step) { s = sign(y = f(x)); if (s == 0) { console.log(x); } else if (s != os) { var dx = x - ox; var dy = y - oy; var cx = x - dx * (y / dy); console.log("~" + cx); } ox = x; oy = y; os = s; } }

printRoots(poly, -1.0, 4, 0.002); </lang>

jq

printRoots(f; lower; upper; step) finds approximations to the roots of an arbitrary continuous real-valued function, f, in the range [lower, upper], assuming step is small enough.

The algorithm is similar to that used for example in the Javascript section on this page, except that a bug has been removed at the point when the previous and current signs are compared.

The function, f, may be an expression (as in the example below) or a defined filter.

printRoots/3 emits an array of results, each of which is either a number (representing an exact root within the limits of machine arithmetic) or a string consisting of "~" followed by an approximation to the root. <lang jq>def sign:

 if . < 0 then -1 elif . > 0 then 1 else 0 end;

def printRoots(f; lowerBound; upperBound; step):

lowerBound as $x
| ($x|f) as $y
| ($y|sign) as $s
| reduce range($x; upperBound+step; step) as $x
  # state: [ox, oy, os, roots]
  ( [$x, $y, $s, [] ];
    .[0] as $ox | .[1] as $oy | .[2] as $os
    | ($x|f) as $y
    | ($y | sign) as $s
    | if $s == 0 then  [$x, $y, $s, (.[3] + [$x] )]
      elif $s != $os and $os != 0 then

($x - $ox) as $dx

         | ($y - $oy) as $dy
   	  | ($x - ($dx *  $y / $dy)) as $cx       # by geometry
         | [$x, $y, $s, (.[3] + [ "~\($cx)" ])]  # an approximation
      else [$x, $y, $s, .[3] ] 
      end )
 | .[3] ;

</lang> We present two examples, one where step is a power of 1/2, and one where it is not:

Output:

<lang jq>printRoots( .*.*. - 3*.*. + 2*.; -1.0; 4; 1/256)

[

 0,
 1,
 2

]

printRoots( .*.*. - 3*.*. + 2*.; -1.0; 4; .001) [

 "~1.320318770141425e-18",
 "~1.0000000000000002",
 "~1.9999999999999993"

]</lang>

Julia

Assuming that one has the Roots package installed:

<lang Julia>using Roots

println(fzeros(x -> x^3 - 3x^2 + 2x))</lang>

Output:

[0.0,1.0,2.0]

Without the Roots package, newton's method may be defined in this manner: <lang Julia>function newton(f, fp, x,tol=1e-14,maxsteps=100)

        #f: the function of x
        #fp: the derivative of f

xnew, xold = x, Inf fn, fo = f(xnew), Inf

counter = 1

while (counter < maxsteps) && (abs(xnew - xold) > tol) && ( abs(fn - fo) > tol ) x = xnew - f(xnew)/fp(xnew) # update step xnew, xold = x, xnew

          fn, fo = f(xnew), fn

counter = counter + 1 end

if counter == maxsteps error("Did not converge in ", string(maxsteps), " steps")

        else

xnew, counter

end

end </lang>

Finding the roots of f(x) = x3 - 3x2 + 2x:

<lang Julia> f(x) = x^3 - 3*x^2 + 2*x fp(x) = 3*x^2-6*x+2

x_s, count = newton(f,fp,1.00) </lang>

Output:

(1.0,2)

Liberty BASIC

<lang lb>' Finds and output the roots of a given function f(x), ' within a range of x values.

' [RC]Roots of an function

   mainwin 80 12

   xMin  =-1
   xMax  = 3
   y     =f( xMin) '   Since Liberty BASIC has an 'eval(' function the fn
   '                       and limits would be better entered via 'input'.
   LastY =y

   eps  =1E-12 '   closeness acceptable

   bigH=0.01

   print
   print " Checking for roots of x^3 -3 *x^2 +2 *x =0 over range -1 to +3"
   print

   x=xMin: dx = bigH
   do
       x=x+dx
       y = f(x)
       'print x, dx, y
       if y*LastY <0 then 'there is a root, should drill deeper
           if dx < eps then    'we are close enough
               print " Just crossed axis, solution f( x) ="; y; " at x ="; using( "#.#####", x)
               LastY = y
               dx = bigH   'after closing on root, continue with big step
           else
               x=x-dx  'step back
               dx = dx/10  'repeat with smaller step
           end if
       end if
   loop while x<xMax

   print
   print " Finished checking in range specified."

end

   function f( x)
       f =x^3 -3 *x^2 +2 *x
   end function</lang>

Lua

<lang Lua>-- Function to have roots found function f (x) return x^3 - 3*x^2 + 2*x end

-- Finds roots of f within range x=[start, stop] or approximations thereof function root (f, start, stop, step)

   local roots, x, sign, foundExact, value = {}, start, f(start) > 0
   while x <= stop do
       value = f(x)
       if value == 0 then
           table.insert(roots, {val = x, err = 0})
           foundExact = true
       end
       if value > 0 ~= sign then
           if foundExact then
               foundExact = false
           else
               table.insert(roots, {val = x, err = step})
           end
       end
       sign = value > 0
       x = x + step
   end
   return roots

end

-- Main procedure print("Root (to 12DP)\tMax. Error\n") for _, r in pairs(root(f, -1, 3, 10^-6)) do

   print(string.format("%0.12f", r.val), r.err)

end</lang>

Output:

Root (to 12DP)  Max. Error

0.000000000008  1e-06
1.000000000016  1e-06
2.000000999934  1e-06

Note that the roots found are all near misses because fractional numbers that seem nice and 'round' in decimal (such as 10^-6) often have some rounding error when represented in binary. To increase the chances of finding exact integer roots, try using an integer start value with a step value that is a power of two. <lang Lua>-- Main procedure print("Root (to 12DP)\tMax. Error\n") for _, r in pairs(root(f, -1, 3, 2^-10)) do

   print(string.format("%0.12f", r.val), r.err)

end</lang>

Output:

Root (to 12DP)  Max. Error

0.000000000000  0
1.000000000000  0
2.000000000000  0

Maple

<lang maple>f := x^3-3*x^2+2*x; roots(f,x);</lang>

outputs:

which means there are three roots. Each root is named as a pair where the first element is the value (0, 1, and 2), the second one the multiplicity (=1 for each means none of the three are degenerate).

By itself (i.e. unless specifically asked to do so), Maple will only perform exact (symbolic) operations and not attempt to do any kind of numerical approximation.

Mathematica

There are multiple obvious ways to do this in Mathematica.

Solve

This requires a full equation and will perform symbolic operations only: <lang mathematica>Solve[x^3-3*x^2+2*x==0,x]</lang> Output

 {{x->0},{x->1},{x->2}}

NSolve

This requires merely the polynomial and will perform numerical operations if needed: <lang mathematica> NSolve[x^3 - 3*x^2 + 2*x , x]</lang> Output

 {{x->0.},{x->1.},{x->2.}}

(note that the results here are floats)

FindRoot

This will numerically try to find one(!) local root from a given starting point: <lang mathematica>FindRoot[x^3 - 3*x^2 + 2*x , {x, 1.5}]</lang> Output

 {x->0.}

From a different start point: <lang mathematica>FindRoot[x^3 - 3*x^2 + 2*x , {x, 1.1}]</lang> Output

{x->1.}

(note that there is no guarantee which one is found).

FindInstance

This finds a value (optionally out of a given domain) for the given variable (or a set of values for a set of given variables) that satisfy a given equality or inequality: <lang mathematica> FindInstance[x^3 - 3*x^2 + 2*x == 0, x]</lang> Output

{{x->0}}

Reduce

This will (symbolically) reduce a given expression to the simplest possible form, solving equations and performing substitutions in the process: <lang mathematica>Reduce[x^3 - 3*x^2 + 2*x == 0, x]</lang>

 x==0||x==1||x==2

(note that this doesn't yield a "solution" but a different expression that expresses the same thing as the original)

Maxima

<lang maxima>e: x^3 - 3*x^2 + 2*x$

/* Number of roots in a real interval, using Sturm sequences */ nroots(e, -10, 10); 3

solve(e, x); [x=1, x=2, x=0]

/* 'solve sets the system variable 'multiplicities */

solve(x^4 - 2*x^3 + 2*x - 1, x); [x=-1, x=1]

multiplicities; [1, 3]

/* Rational approximation of roots using Sturm sequences and bisection */

realroots(e); [x=1, x=2, x=0]

/* 'realroots also sets the system variable 'multiplicities */

multiplicities; [1, 1, 1]

/* Numerical root using Brent's method (here with another equation) */

find_root(sin(t) - 1/2, t, 0, %pi/2); 0.5235987755983

fpprec: 60$

bf_find_root(sin(t) - 1/2, t, 0, %pi/2); 5.23598775598298873077107230546583814032861566562517636829158b-1

/* Numerical root using Newton's method */

load(newton1)$ newton(e, x, 1.1, 1e-6); 1.000000017531147

/* For polynomials, Jenkins–Traub algorithm */

allroots(x^3 + x + 1); [x=1.161541399997252*%i+0.34116390191401,

x=0.34116390191401-1.161541399997252*%i,
x=-0.68232780382802]

bfallroots(x^3 + x + 1); [x=1.16154139999725193608791768724717407484314725802151429063617b0*%i + 3.41163901914009663684741869855524128445594290948999288901864b-1,

x=3.41163901914009663684741869855524128445594290948999288901864b-1 - 1.16154139999725193608791768724717407484314725802151429063617b0*%i,
x=-6.82327803828019327369483739711048256891188581897998577803729b-1]</lang>

Objeck

Translation of: C++

<lang objeck> bundle Default {

 class Roots {
   function : f(x : Float) ~ Float
   {
     return (x*x*x - 3.0*x*x + 2.0*x);
   }
    
   function : Main(args : String[]) ~ Nil 
   {
     step := 0.001;
     start := -1.0;
     stop := 3.0;
     value := f(start);
     sign := (value > 0);
    
     if(0.0 = value) {
       start->PrintLine();
     };
     
     for(x := start + step; x <= stop;  x += step;)  {
       value := f(x);
       
       if((value > 0) <> sign) {
         IO.Console->Instance()->Print("~")->PrintLine(x);
       }
       else if(0 = value) {
         IO.Console->Instance()->Print("~")->PrintLine(x);
       };
       
       sign := (value > 0);
     };
   }
 }

} </lang>

OCaml

A general root finder using the False Position (Regula Falsi) method, which will find all simple roots given a small step size.

<lang ocaml>let bracket u v =

 ((u > 0.0) && (v < 0.0)) || ((u < 0.0) && (v > 0.0));;

let xtol a b = (a = b);; (* or use |a-b| < epsilon *)

let rec regula_falsi a b fa fb f =

 if xtol a b then (a, fa) else
 let c = (fb*.a -. fa*.b) /. (fb -. fa) in
 let fc = f c in
 if fc = 0.0 then (c, fc) else
 if bracket fa fc then
   regula_falsi a c fa fc f
 else
   regula_falsi c b fc fb f;;

let search lo hi step f =

 let rec next x fx =
   if x > hi then [] else
     let y = x +. step in
     let fy = f y in
     if fx = 0.0 then
       (x,fx) :: next y fy
     else if bracket fx fy then
       (regula_falsi x y fx fy f) :: next y fy
     else
       next y fy in
 next lo (f lo);;

let showroot (x,fx) =

 Printf.printf "f(%.17f) = %.17f [%s]\n" 
   x fx (if fx = 0.0 then "exact" else "approx") in

let f x = ((x -. 3.0)*.x +. 2.0)*.x in List.iter showroot (search (-5.0) 5.0 0.1 f);;</lang>

Output:

f(0.00000000000000000) = 0.00000000000000000 [exact]
f(1.00000000000000022) = 0.00000000000000000 [exact]
f(1.99999999999999978) = 0.00000000000000000 [exact]

Note these roots are exact solutions with floating-point calculation.

Oforth

<lang Oforth>: findRoots(f, a, b, st) | x y lasty |

  a f perform dup ->y ->lasty

  a b st step: x [
     x f perform -> y
     y ==0 ifTrue: [ System.Out "Root found at " << x << cr ]
     else: [ y lasty * sgn -1 == ifTrue: [ System.Out "Root near " << x << cr ] ]
     y ->lasty
     ] ;

f(x) x 3 pow x sq 3 * - x 2 * + ; </lang>

Output:

findRoots(#f, -1, 3, 0.0001)
Root found at 0
Root found at 1
Root found at 2

findRoots(#f, -1.000001, 3, 0.0001)
Root near 9.90000000000713e-005
Root near 1.000099
Root near 2.000099

Octave

If the equation is a polynomial, we can put the coefficients in a vector and use roots:

<lang octave>a = [ 1, -3, 2, 0 ]; r = roots(a); % let's print it for i = 1:3

 n = polyval(a, r(i));
 printf("x%d = %f (%f", i, r(i), n);
 if (n != 0.0)
   printf(" not");
 endif
 printf(" exact)\n");

endfor</lang>

Otherwise we can program our (simple) method:

Translation of: Python

<lang octave>function y = f(x)

 y = x.^3 -3.*x.^2 + 2.*x;

endfunction

step = 0.001; tol = 10 .* eps; start = -1; stop = 3; se = sign(f(start));

x = start; while (x <= stop)

 v = f(x);
 if ( (v < tol) && (v > -tol) )
   printf("root at %f\n", x);
 elseif ( sign(v) != se )
   printf("root near %f\n", x);
 endif
 se = sign(v);
 x = x + step;

endwhile</lang>

PARI/GP

Gourdon–Schönhage algorithm

<lang parigp>polroots(x^3-3*x^2+2*x)</lang>

Newton's method

This uses a modified version of the Newton–Raphson method. <lang parigp>polroots(x^3-3*x^2+2*x,1)</lang>

Brent's method

<lang parigp>solve(x=-.5,.5,x^3-3*x^2+2*x) solve(x=.5,1.5,x^3-3*x^2+2*x) solve(x=1.5,2.5,x^3-3*x^2+2*x)</lang>

Factorization to linear factors

<lang parigp>findRoots(P)={

 my(f=factor(P),t);
 for(i=1,#f[,1],
   if(poldegree(f[i,1]) == 1,
     for(j=1,f[i,2],
       print(-polcoeff(f[i,1], 0), " (exact)")
     )
   );
   if(poldegree(f[i,1]) > 1,
     t=polroots(f[i,1]);
     for(j=1,#t,
       for(k=1,f[i,2],
         print(if(imag(t[j]) == 0.,real(t[j]),t[j]), " (approximate)")
       )
     )
   )
 )

}; findRoots(x^3-3*x^2+2*x)</lang>

Factorization to quadratic factors

Of course this process could be continued to degrees 3 and 4 with sufficient additional work. <lang parigp>findRoots(P)={

 my(f=factor(P),t);
 for(i=1,#f[,1],
   if(poldegree(f[i,1]) == 1,
     for(j=1,f[i,2],
       print(-polcoeff(f[i,1], 0), " (exact)")
     )
   );
   if(poldegree(f[i,1]) == 2,
     t=solveQuadratic(polcoeff(f[i,1],2),polcoeff(f[i,1],1),polcoeff(f[i,1],0));
     for(j=1,f[i,2],
       print(t[1]" (exact)\n"t[2]" (exact)")
     )
   );
   if(poldegree(f[i,1]) > 2,
     t=polroots(f[i,1]);
     for(j=1,#t,
       for(k=1,f[i,2],
         print(if(imag(t[j]) == 0.,real(t[j]),t[j]), " (approximate)")
       )
     )
   )
 )

}; solveQuadratic(a,b,c)={

 my(t=-b/2/a,s=b^2/4/a^2-c/a,inner=core(numerator(s))/core(denominator(s)),outer=sqrtint(s/inner));
 if(inner < 0,
   outer *= I;
   inner *= -1
 );
 s=if(inner == 1,
   outer
 ,
   if(outer == 1,
     Str("sqrt(", inner, ")")
   ,
     Str(outer, " * sqrt(", inner, ")")
   )
 );
 if (t,
   [Str(t, " + ", s), Str(t, " - ", s)]
 ,
   [s, Str("-", s)]
 )

}; findRoots(x^3-3*x^2+2*x)</lang>

Pascal

Translation of: Fortran

<lang pascal>Program RootsFunction;

var

 e, x, step, value: double;
 s: boolean;
 i, limit: integer;
 x1, x2, d: double;

function f(const x: double): double;

 begin
   f := x*x*x - 3*x*x + 2*x;
 end;

begin

 x    := -1;
 step := 1.0e-6;
 e    := 1.0e-9;
 s    := (f(x) > 0);

 writeln('Version 1: simply stepping x:');
 while x < 3.0 do
 begin
   value := f(x);
   if abs(value) < e then
   begin
     writeln ('root found at x = ', x);
     s := not s;
   end
   else if ((value > 0) <> s) then
   begin
     writeln ('root found at x = ', x);
     s := not s;
   end;
   x := x + step;
 end;
 
 writeln('Version 2: secant method:');
 x1 := -1.0;
 x2 :=  3.0;
 e  :=  1.0e-15;
 i  :=  1;
 limit := 300;
 while true do
 begin
   if i > limit then
   begin
     writeln('Error: function not converging');
     exit;
   end;
   d := (x2 - x1) / (f(x2) - f(x1)) * f(x2);
   if abs(d) < e then
   begin
     if d = 0 then
       write('Exact ')
     else
       write('Approximate ');
     writeln('root found at x = ', x2);
     exit;
   end;
   x1 := x2;
   x2 := x2 - d;
   i  := i + 1;
 end;

end. </lang> Output:

Version 1: simply stepping x:
root found at x =  7.91830063542152E-012
root found at x =  1.00000000001584E+000
root found at x =  1.99999999993357E+000
Version 2: secant method:
Exact root found at x =  1.00000000000000E+000

Perl

<lang perl>sub f {

       my $x = shift;

       return ($x * $x * $x - 3*$x*$x + 2*$x);

}

my $step = 0.001; # Smaller step values produce more accurate and precise results my $start = -1; my $stop = 3; my $value = &f($start); my $sign = $value > 0;

Check for root at start

print "Root found at $start\n" if ( 0 == $value );

for( my $x = $start + $step;

       $x <= $stop;
       $x += $step )

{

       $value = &f($x);

       if ( 0 == $value )
       {
               # We hit a root
               print "Root found at $x\n";
       }
       elsif ( ( $value > 0 ) != $sign )
       {
               # We passed a root
               print "Root found near $x\n";
       }

       # Update our sign
       $sign = ( $value > 0 );

}</lang>

Perl 6

Uses exact arithmetic. <lang perl6>sub f($x) { $x*$x*$x - 3*$x*$x + 2*$x }

my $start = -1; my $stop = 3; my $step = 0.001;

for $start, * + $step ... $stop -> $x {

   state $sign = 0;
   given f($x) {
       my $next = .sign;
       when 0.0 {
           say "Root found at $x";
       }
       when $sign and $next != $sign {
           say "Root found near $x";
       }
       NEXT $sign = $next;
   }

}</lang>

Output:

Root found at 0
Root found at 1
Root found at 2

PicoLisp

Translation of: Clojure

<lang PicoLisp>(de findRoots (F Start Stop Step Eps)

  (filter
     '((N) (> Eps (abs (F N))))
     (range Start Stop Step) ) )

(scl 12)

(mapcar round

  (findRoots
     '((X) (+ (*/ X X X `(* 1.0 1.0)) (*/ -3 X X 1.0) (* 2 X)))
     -1.0 3.0 0.0001 0.00000001 ) )</lang>

Output:

-> ("0.000" "1.000" "2.000")

PL/I

<lang PL/I> f: procedure (x) returns (float (18));

  declare x float (18);
  return (x**3 - 3*x**2 + 2*x );

end f;

declare eps float, (x, y) float (18); declare dx fixed decimal (15,13);

eps = 1e-12;

do dx = -5.03 to 5 by 0.1;

  x = dx;
  if sign(f(x)) ^= sign(f(dx+0.1)) then
     call locate_root;

end;

locate_root: procedure;

  declare (left, mid, right) float (18);

  put skip list ('Looking for root in [' || x, x+0.1 || ']' );
  left = x; right = dx+0.1;
  PUT SKIP LIST (F(LEFT), F(RIGHT) );
  if abs(f(left) ) < eps then
     do; put skip list ('Found a root at x=', left); return; end;
  else if abs(f(right) ) < eps then
     do; put skip list ('Found a root at x=', right); return; end;
  do forever;
     mid = (left+right)/2;
     if sign(f(mid)) = 0 then
        do; put skip list ('Root found at x=', mid); return; end;
     else if sign(f(left)) ^= sign(f(mid)) then
        right = mid;
     else
        left = mid;
     /* put skip list (left || right); */
     if abs(right-left) < eps then
        do; put skip list ('There is a root near ' ||
           (left+right)/2); return;
        end;
  end;

end locate_root; </lang>

PureBasic

Translation of: C++

<lang PureBasic>Procedure.d f(x.d)

 ProcedureReturn x*x*x-3*x*x+2*x

EndProcedure

Procedure main()

 OpenConsole()
 Define.d StepSize= 0.001
 Define.d Start=-1, stop=3
 Define.d value=f(start), x=start
 Define.i oldsign=Sign(value)
 
 If value=0
   PrintN("Root found at "+StrF(start))
 EndIf
 
 While x<=stop
   value=f(x)
   If Sign(value) <> oldsign
     PrintN("Root found near "+StrF(x))
   ElseIf value = 0
     PrintN("Root found at "+StrF(x))
   EndIf
   oldsign=Sign(value)
   x+StepSize
 Wend

EndProcedure

main()</lang>

Python

Translation of: Perl

<lang python>f = lambda x: x * x * x - 3 * x * x + 2 * x

step = 0.001 # Smaller step values produce more accurate and precise results start = -1 stop = 3

sign = f(start) > 0

x = start while x <= stop:

   value = f(x)

   if value == 0:
       # We hit a root
       print "Root found at", x
   elif (value > 0) != sign:
       # We passed a root
       print "Root found near", x

   # Update our sign
   sign = value > 0

   x += step</lang>

R

Translation of: Octave

<lang R>f <- function(x) x^3 -3*x^2 + 2*x

findroots <- function(f, begin, end, tol = 1e-20, step = 0.001) {

 se <- ifelse(sign(f(begin))==0, 1, sign(f(begin)))
 x <- begin
 while ( x <= end ) {
   v <- f(x)
   if ( abs(v) < tol ) {
     print(sprintf("root at %f", x))
   } else if ( ifelse(sign(v)==0, 1, sign(v)) != se )  {
     print(sprintf("root near %f", x))
   }
   se <- ifelse( sign(v) == 0 , 1, sign(v))
   x <- x + step
 }

}

findroots(f, -1, 3)</lang>

Racket

lang racket

Attempts to find all roots of a real-valued function f in a given interval [a b] by dividing the interval into N parts and using the root-finding method on each subinterval which proves to contain a root.

(define (find-roots f a b

                   #:divisions [N 10]
                   #:method [method secant])
 (define h (/ (- b a) N))
 (for*/list ([x1 (in-range a b h)] 
             [x2 (in-value (+ x1 h))]
             #:when (or (root? f x1)
                        (includes-root? f x1 x2)))
   (find-root f x1 x2 #:method method)))

Finds a root of a real-valued function f in a given interval [a b].

(define (find-root f a b #:method [method secant])

 (cond 
   [(root? f a) a]
   [(root? f b) b]
   [else (and (includes-root? f a b) (method f a b))]))

Returns #t if x is a root of a real-valued function f with absolute accuracy (tolerance).

(define (root? f x) (almost-equal? 0 (f x)))

Returns #t if interval (a b) contains a root (or the odd number of roots) of a real-valued function f.

(define (includes-root? f a b) (< (* (f a) (f b)) 0))

Returns #t if a and b are equal with respect to the relative accuracy (tolerance).

(define (almost-equal? a b)

 (or (< (abs (+ b a)) (tolerance))      
     (< (abs (/ (- b a) (+ b a))) (tolerance))))

(define tolerance (make-parameter 5e-16)) </lang>

Different root-finding methods

<lang racket> (define (secant f a b)

 (let next ([x1 a] [y1 (f a)] [x2 b] [y2 (f b)] [n 50])
   (define x3 (/ (- (* x1 y2) (* x2 y1)) (- y2 y1)))
   (cond
     ; if the method din't converge within given interval
     ; switch to more robust bisection method
     [(or (not (< a x3 b)) (zero? n)) (bisection f a b)] 
     [(almost-equal? x3 x2) x3]
     [else (next x2 y2 x3 (f x3) (sub1 n))])))

(define (bisection f x1 x2)

 (let divide ([a x1] [b x2])
   (and (<= (* (f a) (f b)) 0)
        (let ([c (* 0.5 (+ a b))])
          (if (almost-equal? a b) 
              c
              (or (divide a c) (divide c b)))))))

</lang>

Examples: <lang racket> -> (find-root (λ (x) (- 2. (* x x))) 1 2) 1.414213562373095 -> (sqrt 2) 1.4142135623730951

-> (define (f x) (+ (* x x x) (* -3.0 x x) (* 2.0 x))) -> (find-roots f -3 4 #:divisions 50) '(2.4932181969624796e-33 1.0 2.0) </lang>

In order to provide a comprehensive code the given solution does not optimize the number of function calls. The functional nature of Racket allows to perform the optimization without changing the main code using memoization.

Simple memoization operator <lang racket> (define (memoized f)

 (define tbl (make-hash))
 (λ x
   (cond [(hash-ref tbl x #f) => values]
         [else (define res (apply f x))
               (hash-set! tbl x res)
               res])))

</lang>

To use memoization just call <lang racket> -> (find-roots (memoized f) -3 4 #:divisions 50) '(2.4932181969624796e-33 1.0 2.0) </lang>

The profiling shows that memoization reduces the number of function calls in this example from 184 to 67 (50 calls for primary interval division and about 6 calls for each point refinement).

REXX

The bisection method is used. <lang rexx>/*REXX program finds the roots of a specific function: x^3 - 3*x^2 + 2*x via bisection*/ parse arg bot top inc . /*obtain optional arguments from the CL*/ if bot== | bot=="," then bot= -5 /*Not specified? Then use the default.*/ if top== | top=="," then top= +5 /* " " " " " " */ if inc== | inc=="," then inc= .0001 /* " " " " " " */ z=f(bot-inc); !=sign(z) /*use these values for initial compare.*/

     do j=bot  to top  by  inc                  /*traipse through the specified range. */
     z=f(j);              $=sign(z)             /*compute new value;  obtain the sign. */
     if z=0  then                                 say  'found an exact root at'   j/1
             else if  !\==$  then  if !\==0  then say  'passed a root at'         j/1
     !=$                                        /*use the new sign for the next compare*/
     end   /*j*/                                /*dividing by unity normalizes J  [↑]  */

exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ f: parse arg x; return x*(x*(x-3)+2) /*formula used ──► x^3 - 3x^2 + 2x */

                                                /*with factoring ──► x{ x^2 -3x  + 2 } */
                                                /*more     "     ──► x{ x( x-3 ) + 2 } */</lang>

output when using the defaults for input:

found an exact root at 0
found an exact root at 1
found an exact root at 2

Ring

<lang ring> load "stdlib.ring" function = "return pow(x,3)-3*pow(x,2)+2*x" rangemin = -1 rangemax = 3 stepsize = 0.001 accuracy = 0.1 roots(function, rangemin, rangemax, stepsize, accuracy)

func roots funct, min, max, inc, eps

    oldsign = 0
    for x = min to max step inc
        num = sign(eval(funct))
        if num = 0 
           see "root found at x = " + x + nl
           num = -oldsign
        else if num != oldsign and oldsign != 0
             if inc < eps 
                see "root found near x = " + x + nl
             else roots(funct, x-inc, x+inc/8, inc/8, eps) ok ok ok
        oldsign = num
    next

</lang> Output:

root found near x = 0.00
root found near x = 1.00
root found near x = 2.00

RLaB

RLaB implements a number of solvers from the GSL and the netlib that find the roots of a real or vector function of a real or vector variable. The solvers are grouped with respect whether the variable is a scalar, findroot, or a vector, findroots. Furthermore, for each group there are two types of solvers, one that does not require the derivative of the objective function (which root(s) are being sought), and one that does.

The script that finds a root of a scalar function $f(x)=x^{3}-3\,x^{2}+2\,x$ of a scalar variable x using the bisection method on the interval -5 to 5 is, <lang RLaB> f = function(x) {

 rval = x .^ 3 - 3 * x .^ 2 + 2 * x;
 return rval;

};

>> findroot(f, , [-5,5])

</lang>

For a detailed description of the solver and its parameters interested reader is directed to the rlabplus manual.

Ruby

Translation of: Python

<lang ruby>def sign(x)

 x <=> 0

end

def find_roots(f, range, step=0.001)

 sign = sign(f[range.begin])
 range.step(step) do |x|
   value = f[x]
   if value == 0
     puts "Root found at #{x}"
   elsif sign(value) == -sign
     puts "Root found between #{x-step} and #{x}"
   end
   sign = sign(value)
 end

end

f = lambda { |x| x**3 - 3*x**2 + 2*x } find_roots(f, -1..3)</lang>

Output:

Root found at 0.0
Root found at 1.0
Root found at 2.0

Or we could use Enumerable#inject, monkey patching and block:

<lang ruby>class Numeric

 def sign
   self <=> 0
 end

end

def find_roots(range, step = 1e-3)

 range.step( step ).inject( yield(range.begin).sign ) do |sign, x|
   value = yield(x)
   if value == 0
     puts "Root found at #{x}"
   elsif value.sign == -sign
     puts "Root found between #{x-step} and #{x}"
   end
   value.sign
 end

end

find_roots(-1..3) { |x| x**3 - 3*x**2 + 2*x }</lang>

Scala

This example is in need of improvement.

<lang Scala>object RootsOfAFunction extends App {

   def findRoots(fn: Double => Double, start: Double, stop: Double, step: Double, epsilon: Double) = {
   for {
     x <- start to stop by step
     if fn(x).abs < epsilon
   } yield x
 }

 def fn(x: Double) = x * x * x - 3 * x * x + 2 * x

 println(findRoots(fn, -1.0, 3.0, 0.0001, 0.000000001))

}</lang>

Output:

Vector(-9.381755897326649E-14, 0.9999999999998124, 1.9999999999997022)

Sidef

<lang ruby>func f(x) {

   x*x*x - 3*x*x + 2*x;

}

var step = 0.001; var start = -1; var stop = 3;

range(start+step, stop, step).each { |x|

   static sign = false;
   given (var value = f(x)) {
       when (0) {
           say "Root found at #{x}";
       }
       case (sign && ((value > 0) != sign)) {
           say "Root found near #{x}";
       }
   }
   sign = value>0;

}</lang>

Output:

Root found at 0
Root found at 1
Root found at 2

Tcl

This simple brute force iteration marks all results, with a leading "~", as approximate. This version always reports its results as approximate because of the general limits of computation using fixed-width floating-point numbers (i.e., IEEE double-precision floats). <lang Tcl>proc froots {lambda {start -3} {end 3} {step 0.0001}} {

   set res {}
   set lastsign [sgn [apply $lambda $start]]
   for {set x $start} {$x <= $end} {set x [expr {$x + $step}]} {
       set sign [sgn [apply $lambda $x]]
       if {$sign != $lastsign} {
           lappend res [format ~%.11f $x]
       }
       set lastsign $sign
   }
   return $res

} proc sgn x {expr {($x>0) - ($x<0)}}

puts [froots {x {expr {$x**3 - 3*$x**2 + 2*$x}}}]</lang> Result and timing:

/Tcl $ time ./froots.tcl
~0.00000000000 ~1.00000000000 ~2.00000000000

real    0m0.368s
user    0m0.062s
sys     0m0.030s

A more elegant solution (and faster, because you can usually make the initial search coarser) is to use brute-force iteration and then refine with Newton-Raphson, but that requires the differential of the function with respect to the search variable. <lang Tcl>proc frootsNR {f df {start -3} {end 3} {step 0.001}} {

   set res {}
   set lastsign [sgn [apply $f $start]]
   for {set x $start} {$x <= $end} {set x [expr {$x + $step}]} {
       set sign [sgn [apply $f $x]]
       if {$sign != $lastsign} {
           lappend res [format ~%.15f [nr $x $f $df]]
       }
       set lastsign $sign
   }
   return $res

} proc sgn x {expr {($x>0) - ($x<0)}} proc nr {x1 f df} {

   # Newton's method converges very rapidly indeed
   for {set iters 0} {$iters < 10} {incr iters} {
       set x1 [expr {
           [set x0 $x1] - [apply $f $x0]/[apply $df $x0]
       }]
       if {$x0 == $x1} {
           break
       }
   }
   return $x1

}

puts [frootsNR \

   {x {expr {$x**3 - 3*$x**2 + 2*$x}}} \
   {x {expr {3*$x**2 - 6*$x + 2}}}]</lang>

TI-89 BASIC

Finding roots is a built-in function: zeros(x^3-3x^2+2x, x) returns {0,1,2}.

In this case, the roots are exact; inexact results are marked by decimal points.

zkl

Translation of: Haskell

<lang zkl>fcn findRoots(f,start,stop,step,eps){

  [start..stop,step].filter('wrap(x){ f(x).closeTo(0.0,eps) })

}</lang> <lang zkl>fcn f(x){ x*x*x - 3.0*x*x + 2.0*x } findRoots(f, -1.0, 3.0, 0.0001, 0.00000001).println();</lang>

Output:

L(-9.38176e-14,1,2)

Translation of: C

<lang zkl>fcn secant(f,xA,xB){

  reg e=1.0e-12;

  fA:=f(xA); if(fA.closeTo(0.0,e)) return(xA);

  do(50){
     fB:=f(xB);
     d:=(xB - xA) / (fB - fA) * fB;
     if(d.closeTo(0,e)) break;
     xA = xB; fA = fB; xB -= d;
  }
  if(f(xB).closeTo(0.0,e)) xB
  else "Function is not converging near (%7.4f,%7.4f).".fmt(xA,xB);

}</lang> <lang zkl>step:=0.1; xs:=findRoots(f, -1.032, 3.0, step, 0.1); xs.println(" --> ",xs.apply('wrap(x){ secant(f,x-step,x+step) }));</lang>

Output:

L(-0.032,0.968,1.068,1.968) --> L(1.87115e-19,1,1,2)