Roots of a function
Roots of a function
You are encouraged to solve this task according to the task description, using any language you may know.
You are encouraged to solve this task according to the task description, using any language you may know.
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 example, use f(x)=x^3-3x^2+2x.
Ada
with Ada.Text_Io; use Ada.Text_Io; procedure Roots_Of_Function is package Real_Io is new Ada.Text_Io.Float_Io(Long_Float); use Real_Io; function F(X : Long_Float) return Long_Float is begin return (X**3 - 3.0*X*X + 2.0*X); end F; Step : constant Long_Float := 1.0E-6; Start : constant Long_Float := -1.0; Stop : constant Long_Float := 3.0; Value : Long_Float := F(Start); Sign : Boolean := Value > 0.0; X : Long_Float := Start + Step; begin if Value = 0.0 then Put("Root found at "); Put(Item => Start, Fore => 1, Aft => 6, Exp => 0); New_Line; end if; while X <= Stop loop Value := F(X); if (Value > 0.0) /= Sign then Put("Root found near "); Put(Item => X, Fore => 1, Aft => 6, Exp => 0); New_Line; elsif Value = 0.0 then Put("Root found at "); Put(Item => X, Fore => 1, Aft => 6, Exp => 0); New_Line; end if; Sign := Value > 0.0; X := X + Step; end loop; end Roots_Of_Function;
C++
#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 ); } }
D
module findroot ; import std.stdio ; import std.math ; void report(T)(T[] r, T function(T) f, T tolerance = cast(T) 1e-4L) { if (r.length) { writefln("Root found (tolerance = %1.4g) :", tolerance) ; foreach(x ; r) { 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.") ; } bool nearZero(T)(T a, T b = T.epsilon * 4) { return abs(a) <= b ; } T[] findroot(T)(T function(T) f, T start, T end, T step = cast(T) 0.001L, T tolerance = cast(T) 1e-4L) { T[T] result ; if (nearZero(step)) writefln("WARNING: step size may be too small.") ; T searchRoot(T a, T b) { // search root by simple bisection T root ; int limit = 49 ; T gap = b - a ; while (!nearZero(gap) && limit--) { if (nearZero(f(a))) return a ; if (nearZero(f(b))) return b ; root = (b + a)/2.0L ; if (nearZero(f(root))) return root ; if (f(a) * f(root) < 0) b = root ; else a = root ; gap = b - a ; } return root ; } T dir = cast(T) (end > start ? 1.0 : -1.0) ; step = (end > start) ? abs(step) : - abs(step) ; for(T x = start ; x*dir <= end*dir ; x = x + step) if (f(x)*f(x + step) <= 0) { T r = searchRoot(x, x+ step) ; result[r] = f(r) ; } return result.keys.sort ; // reduce duplacated root, if any } real f(real x){ return x*x*x - 3*x*x + 2*x ; } void main(){ findroot(&f, -1.0L, 3.0L, 0.001L).report(&f) ; }
Output ( NB:smallest increment for real type in D is real.epsilon = 1.0842e-19 ):
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
Maple
f := x^3-3*x^2+2*x; roots(f,x);
outputs:
[[0, 1], [1, 1], [2, 1]]
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.
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 ); }