Roots of a quadratic function: Difference between revisions
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procedure solve(a,b,c) |
procedure solve(a,b,c) |
||
d := sqrt(b*b - 4.0*a*c) |
d := sqrt(b*b - 4.0*a*c) |
||
roots := if b < 0 then [r1 := (-b+d)/2.0*a, c/(a*r1)] |
roots := if b < 0 then [r1 := (-b+d)/(2.0*a), c/(a*r1)] |
||
else [r1 := (-b-d)/2.0*a, c/(a*r1)] |
else [r1 := (-b-d)/(2.0*a), c/(a*r1)] |
||
write(a,"*x^2 + ",b,"*x + ",c," has roots ",roots[1]," and ",roots[2]) |
write(a,"*x^2 + ",b,"*x + ",c," has roots ",roots[1]," and ",roots[2]) |
||
end</lang> |
end</lang> |
||
Revision as of 17:32, 13 August 2020
You are encouraged to solve this task according to the task description, using any language you may know.
Write a program to find the roots of a quadratic equation, i.e., solve the equation . Your program must correctly handle non-real roots, but it need not check that .
The problem of solving a quadratic equation is a good example of how dangerous it can be to ignore the peculiarities of floating-point arithmetic. The obvious way to implement the quadratic formula suffers catastrophic loss of accuracy when one of the roots to be found is much closer to 0 than the other. In their classic textbook on numeric methods Computer Methods for Mathematical Computations, George Forsythe, Michael Malcolm, and Cleve Moler suggest trying the naive algorithm with , , and . (For double-precision floats, set .) Consider the following implementation in Ada: <lang ada>with Ada.Text_IO; use Ada.Text_IO; with Ada.Numerics.Elementary_Functions; use Ada.Numerics.Elementary_Functions;
procedure Quadratic_Equation is
type Roots is array (1..2) of Float;
function Solve (A, B, C : Float) return Roots is
SD : constant Float := sqrt (B**2 - 4.0 * A * C);
AA : constant Float := 2.0 * A;
begin
return ((- B + SD) / AA, (- B - SD) / AA);
end Solve;
R : constant Roots := Solve (1.0, -10.0E5, 1.0);
begin
Put_Line ("X1 =" & Float'Image (R (1)) & " X2 =" & Float'Image (R (2)));
end Quadratic_Equation;</lang>
- Output:
X1 = 1.00000E+06 X2 = 0.00000E+00
As we can see, the second root has lost all significant figures. The right answer is that X2 is about . The naive method is numerically unstable.
Suggested by Middlebrook (D-OA), a better numerical method: to define two parameters and
and the two roots of the quardratic are: and
Task: do it better. This means that given , , and , both of the roots your program returns should be greater than . Or, if your language can't do floating-point arithmetic any more precisely than single precision, your program should be able to handle . Either way, show what your program gives as the roots of the quadratic in question. See page 9 of
"What Every Scientist Should Know About Floating-Point Arithmetic" for a possible algorithm.
Ada
<lang ada>with Ada.Text_IO; use Ada.Text_IO; with Ada.Numerics.Elementary_Functions; use Ada.Numerics.Elementary_Functions;
procedure Quadratic_Equation is
type Roots is array (1..2) of Float;
function Solve (A, B, C : Float) return Roots is
SD : constant Float := sqrt (B**2 - 4.0 * A * C);
X : Float;
begin
if B < 0.0 then
X := (- B + SD) / (2.0 * A);
return (X, C / (A * X));
else
X := (- B - SD) / (2.0 * A);
return (C / (A * X), X);
end if;
end Solve;
R : constant Roots := Solve (1.0, -10.0E5, 1.0);
begin
Put_Line ("X1 =" & Float'Image (R (1)) & " X2 =" & Float'Image (R (2)));
end Quadratic_Equation;</lang> Here precision loss is prevented by checking signs of operands. On errors, Constraint_Error is propagated on numeric errors and when roots are complex.
- Output:
X1 = 1.00000E+06 X2 = 1.00000E-06
ALGOL 68
<lang algol68>quadratic equation: BEGIN
MODE ROOTS = UNION([]REAL, []COMPL); MODE QUADRATIC = STRUCT(REAL a,b,c);
PROC solve = (QUADRATIC q)ROOTS:
BEGIN
REAL a = a OF q, b = b OF q, c = c OF q;
REAL sa = b**2 - 4*a*c;
IF sa >=0 THEN # handle the +ve case as REAL #
REAL sqrt sa = ( b<0 | sqrt(sa) | -sqrt(sa));
REAL r1 = (-b + sqrt sa)/(2*a),
r2 = (-b - sqrt sa)/(2*a);
[]REAL((r1,r2))
ELSE # handle the -ve case as COMPL conjugate pairs #
COMPL compl sqrt sa = ( b<0 | complex sqrt(sa) | -complex sqrt(sa));
COMPL r1 = (-b + compl sqrt sa)/(2*a),
r2 = (-b - compl sqrt sa)/(2*a);
[]COMPL (r1, r2)
FI
END # solve #;
PROC real evaluate = (QUADRATIC q, REAL x )REAL: (a OF q*x + b OF q)*x + c OF q;
PROC compl evaluate = (QUADRATIC q, COMPL x)COMPL: (a OF q*x + b OF q)*x + c OF q;
# only a very tiny difference between the 2 examples # []QUADRATIC test = ((1, -10e5, 1), (1, 0, 1), (1,-3,2), (1,3,2), (4,0,4), (3,4,5)); FORMAT real fmt = $g(-0,8)$; FORMAT compl fmt = $f(real fmt)"+"f(real fmt)"i"$; FORMAT quadratic fmt = $f(real fmt)" x**2 + "f(real fmt)" x + "f(real fmt)" = 0"$;
FOR index TO UPB test DO QUADRATIC quadratic = test[index]; ROOTS r = solve(quadratic);
- Output the two different scenerios #
printf(($"Quadratic: "$, quadratic fmt, quadratic, $l$));
CASE r IN
([]REAL r):
printf(($"REAL x1 = "$, real fmt, r[1],
$", x2 = "$, real fmt, r[2], $"; "$,
$"REAL y1 = "$, real fmt, real evaluate(quadratic,r[1]),
$", y2 = "$, real fmt, real evaluate(quadratic,r[2]), $";"ll$
)),
([]COMPL c):
printf(($"COMPL x1,x2 = "$, real fmt, re OF c[1], $"+/-"$,
real fmt, ABS im OF c[1], $"; "$,
$"COMPL y1 = "$, compl fmt, compl evaluate(quadratic,c[1]),
$", y2 = "$, compl fmt, compl evaluate(quadratic,c[2]), $";"ll$
))
ESAC
OD
END # quadratic_equation #</lang>
- Output:
Quadratic: 1.00000000 x**2 + -1000000.00000000 x + 1.00000000 = 0 REAL x1 = 999999.99999900, x2 = .00000100; REAL y1 = -.00000761, y2 = -.00000761; Quadratic: 1.00000000 x**2 + .00000000 x + 1.00000000 = 0 COMPL x1,x2 = .00000000+/-1.00000000; COMPL y1 = .00000000+.00000000i, y2 = .00000000+.00000000i; Quadratic: 1.00000000 x**2 + -3.00000000 x + 2.00000000 = 0 REAL x1 = 2.00000000, x2 = 1.00000000; REAL y1 = .00000000, y2 = .00000000; Quadratic: 1.00000000 x**2 + 3.00000000 x + 2.00000000 = 0 REAL x1 = -2.00000000, x2 = -1.00000000; REAL y1 = .00000000, y2 = .00000000; Quadratic: 4.00000000 x**2 + .00000000 x + 4.00000000 = 0 COMPL x1,x2 = .00000000+/-1.00000000; COMPL y1 = .00000000+.00000000i, y2 = .00000000+.00000000i; Quadratic: 3.00000000 x**2 + 4.00000000 x + 5.00000000 = 0 COMPL x1,x2 = -.66666667+/-1.10554160; COMPL y1 = .00000000+.00000000i, y2 = .00000000+-.00000000i;
AutoHotkey
ahk forum: discussion <lang AutoHotkey>MsgBox % quadratic(u,v, 1,-3,2) ", " u ", " v MsgBox % quadratic(u,v, 1,3,2) ", " u ", " v MsgBox % quadratic(u,v, -2,4,-2) ", " u ", " v MsgBox % quadratic(u,v, 1,0,1) ", " u ", " v SetFormat FloatFast, 0.15e MsgBox % quadratic(u,v, 1,-1.0e8,1) ", " u ", " v
quadratic(ByRef x1, ByRef x2, a,b,c) { ; -> #real roots {x1,x2} of ax²+bx+c
If (a = 0)
Return -1 ; ERROR: not quadratic
d := b*b - 4*a*c
If (d < 0) {
x1 := x2 := ""
Return 0
}
If (d = 0) {
x1 := x2 := -b/2/a
Return 1
}
x1 := (-b - (b<0 ? -sqrt(d) : sqrt(d)))/2/a
x2 := c/a/x1
Return 2
}</lang>
BBC BASIC
<lang bbcbasic> FOR test% = 1 TO 7
READ a$, b$, c$
PRINT "For a = " ; a$ ", b = " ; b$ ", c = " ; c$ TAB(32) ;
PROCsolvequadratic(EVAL(a$), EVAL(b$), EVAL(c$))
NEXT
END
DATA 1, -1E9, 1
DATA 1, 0, 1
DATA 2, -1, -6
DATA 1, 2, -2
DATA 0.5, SQR(2), 1
DATA 1, 3, 2
DATA 3, 4, 5
DEF PROCsolvequadratic(a, b, c)
LOCAL d, f
d = b^2 - 4*a*c
CASE SGN(d) OF
WHEN 0:
PRINT "the single root is " ; -b/2/a
WHEN +1:
f = (1 + SQR(1-4*a*c/b^2))/2
PRINT "the real roots are " ; -f*b/a " and " ; -c/b/f
WHEN -1:
PRINT "the complex roots are " ; -b/2/a " +/- " ; SQR(-d)/2/a "*i"
ENDCASE
ENDPROC</lang>
- Output:
For a = 1, b = -1E9, c = 1 the real roots are 1E9 and 1E-9 For a = 1, b = 0, c = 1 the complex roots are 0 +/- 1*i For a = 2, b = -1, c = -6 the real roots are 2 and -1.5 For a = 1, b = 2, c = -2 the real roots are -2.73205081 and 0.732050808 For a = 0.5, b = SQR(2), c = 1 the single root is -1.41421356 For a = 1, b = 3, c = 2 the real roots are -2 and -1 For a = 3, b = 4, c = 5 the complex roots are -0.666666667 +/- 1.1055416*i
C
Code that tries to avoid floating point overflow and other unfortunate loss of precissions: (compiled with gcc -std=c99 for complex, though easily adapted to just real numbers)
<lang C>#include <stdio.h>
- include <stdlib.h>
- include <complex.h>
- include <math.h>
typedef double complex cplx;
void quad_root (double a, double b, double c, cplx * ra, cplx *rb) { double d, e; if (!a) { *ra = b ? -c / b : 0; *rb = 0; return; } if (!c) { *ra = 0; *rb = -b / a; return; }
b /= 2; if (fabs(b) > fabs(c)) { e = 1 - (a / b) * (c / b); d = sqrt(fabs(e)) * fabs(b); } else { e = (c > 0) ? a : -a; e = b * (b / fabs(c)) - e; d = sqrt(fabs(e)) * sqrt(fabs(c)); }
if (e < 0) { e = fabs(d / a); d = -b / a; *ra = d + I * e; *rb = d - I * e; return; }
d = (b >= 0) ? d : -d; e = (d - b) / a; d = e ? (c / e) / a : 0; *ra = d; *rb = e; return; }
int main() { cplx ra, rb; quad_root(1, 1e12 + 1, 1e12, &ra, &rb); printf("(%g + %g i), (%g + %g i)\n", creal(ra), cimag(ra), creal(rb), cimag(rb));
quad_root(1e300, -1e307 + 1, 1e300, &ra, &rb); printf("(%g + %g i), (%g + %g i)\n", creal(ra), cimag(ra), creal(rb), cimag(rb));
return 0; }</lang>
- Output:
(-1e+12 + 0 i), (-1 + 0 i) (1.00208e+07 + 0 i), (9.9792e-08 + 0 i)
<lang c>#include <stdio.h>
- include <math.h>
- include <complex.h>
void roots_quadratic_eq(double a, double b, double c, complex double *x) {
double delta;
delta = b*b - 4.0*a*c; x[0] = (-b + csqrt(delta)) / (2.0*a); x[1] = (-b - csqrt(delta)) / (2.0*a);
}</lang>
<lang c>void roots_quadratic_eq2(double a, double b, double c, complex double *x) {
b /= a;
c /= a;
double delta = b*b - 4*c;
if ( delta < 0 ) {
x[0] = -b/2 + I*sqrt(-delta)/2.0;
x[1] = -b/2 - I*sqrt(-delta)/2.0;
} else {
double root = sqrt(delta);
double sol = (b>0) ? (-b - root)/2.0 : (-b + root)/2.0;
x[0] = sol;
x[1] = c/sol;
}
}</lang>
<lang c>int main() {
complex double x[2];
roots_quadratic_eq(1, -1e20, 1, x);
printf("x1 = (%.20le, %.20le)\nx2 = (%.20le, %.20le)\n\n",
creal(x[0]), cimag(x[0]), creal(x[1]), cimag(x[1]));
roots_quadratic_eq2(1, -1e20, 1, x);
printf("x1 = (%.20le, %.20le)\nx2 = (%.20le, %.20le)\n\n",
creal(x[0]), cimag(x[0]), creal(x[1]), cimag(x[1]));
return 0;
}</lang>
x1 = (1.00000000000000000000e+20, 0.00000000000000000000e+00) x2 = (0.00000000000000000000e+00, 0.00000000000000000000e+00) x1 = (1.00000000000000000000e+20, 0.00000000000000000000e+00) x2 = (9.99999999999999945153e-21, 0.00000000000000000000e+00)
C#
<lang csharp>using System; using System.Numerics;
class QuadraticRoots {
static Tuple<Complex, Complex> Solve(double a, double b, double c)
{
var q = -(b + Math.Sign(b) * Complex.Sqrt(b * b - 4 * a * c)) / 2;
return Tuple.Create(q / a, c / q);
}
static void Main()
{
Console.WriteLine(Solve(1, -1E20, 1));
}
}</lang>
- Output:
((1E+20, 0), (1E-20, 0))
C++
<lang cpp>#include <iostream>
- include <utility>
- include <complex>
typedef std::complex<double> complex;
std::pair<complex, complex>
solve_quadratic_equation(double a, double b, double c)
{
b /= a;
c /= a;
double discriminant = b*b-4*c;
if (discriminant < 0)
return std::make_pair(complex(-b/2, std::sqrt(-discriminant)/2),
complex(-b/2, -std::sqrt(-discriminant)/2));
double root = std::sqrt(discriminant);
double solution1 = (b > 0)? (-b - root)/2
: (-b + root)/2;
return std::make_pair(solution1, c/solution1);
}
int main() {
std::pair<complex, complex> result = solve_quadratic_equation(1, -1e20, 1); std::cout << result.first << ", " << result.second << std::endl;
}</lang>
- Output:
(1e+20,0), (1e-20,0)
Clojure
<lang clojure>(defn quadratic
"Compute the roots of a quadratic in the form ax^2 + bx + c = 1.
Returns any of nil, a float, or a vector."
[a b c]
(let [sq-d (Math/sqrt (- (* b b) (* 4 a c)))
f #(/ (% b sq-d) (* 2 a))]
(cond
(neg? sq-d) nil
(zero? sq-d) (f +)
(pos? sq-d) [(f +) (f -)]
:else nil))) ; maybe our number ended up as NaN</lang>
- Output:
<lang clojure>user=> (quadratic 1.0 1.0 1.0) nil user=> (quadratic 1.0 2.0 1.0) 1.0 user=> (quadratic 1.0 3.0 1.0) [2.618033988749895 0.3819660112501051] </lang>
Common Lisp
<lang lisp>(defun quadratic (a b c)
(list (/ (+ (- b) (sqrt (- (expt b 2) (* 4 a c)))) (* 2 a)) (/ (- (- b) (sqrt (- (expt b 2) (* 4 a c)))) (* 2 a))))</lang>
D
<lang d>import std.math, std.traits;
CommonType!(T1, T2, T3)[] naiveQR(T1, T2, T3)
(in T1 a, in T2 b, in T3 c)
pure nothrow if (isFloatingPoint!T1) {
alias ReturnT = typeof(typeof(return).init[0]);
if (a == 0)
return [ReturnT(c / b)]; // It's a linear function.
immutable ReturnT det = b ^^ 2 - 4 * a * c;
if (det < 0)
return []; // No real number root.
immutable SD = sqrt(det);
return [(-b + SD) / 2 * a, (-b - SD) / 2 * a];
}
CommonType!(T1, T2, T3)[] cautiQR(T1, T2, T3)
(in T1 a, in T2 b, in T3 c)
pure nothrow if (isFloatingPoint!T1) {
alias ReturnT = typeof(typeof(return).init[0]);
if (a == 0)
return [ReturnT(c / b)]; // It's a linear function.
immutable ReturnT det = b ^^ 2 - 4 * a * c;
if (det < 0)
return []; // No real number root.
immutable SD = sqrt(det);
if (b * a < 0) {
immutable x = (-b + SD) / 2 * a;
return [x, c / (a * x)];
} else {
immutable x = (-b - SD) / 2 * a;
return [c / (a * x), x];
}
}
void main() {
import std.stdio;
writeln("With 32 bit float type:");
writefln(" Naive: [%(%g, %)]", naiveQR(1.0f, -10e5f, 1.0f));
writefln("Cautious: [%(%g, %)]", cautiQR(1.0f, -10e5f, 1.0f));
writeln("\nWith 64 bit double type:");
writefln(" Naive: [%(%g, %)]", naiveQR(1.0, -10e5, 1.0));
writefln("Cautious: [%(%g, %)]", cautiQR(1.0, -10e5, 1.0));
writeln("\nWith real type:");
writefln(" Naive: [%(%g, %)]", naiveQR(1.0L, -10e5L, 1.0L));
writefln("Cautious: [%(%g, %)]", cautiQR(1.0L, -10e5L, 1.0L));
}</lang>
- Output:
With 32 bit float type: Naive: [1e+06, 0] Cautious: [1e+06, 1e-06] With 64 bit double type: Naive: [1e+06, 1.00001e-06] Cautious: [1e+06, 1e-06] With real type: Naive: [1e+06, 1e-06] Cautious: [1e+06, 1e-06]
Elixir
<lang elixir>defmodule Quadratic do
def roots(a, b, c) do
IO.puts "Roots of a quadratic function (#{a}, #{b}, #{c})"
d = b * b - 4 * a * c
a2 = a * 2
cond do
d > 0 ->
sd = :math.sqrt(d)
IO.puts " the real roots are #{(- b + sd) / a2} and #{(- b - sd) / a2}"
d == 0 ->
IO.puts " the single root is #{- b / a2}"
true ->
sd = :math.sqrt(-d)
IO.puts " the complex roots are #{- b / a2} +/- #{sd / a2}*i"
end
end
end
Quadratic.roots(1, -2, 1) Quadratic.roots(1, -3, 2) Quadratic.roots(1, 0, 1) Quadratic.roots(1, -1.0e10, 1) Quadratic.roots(1, 2, 3) Quadratic.roots(2, -1, -6)</lang>
- Output:
Roots of a quadratic function (1, -2, 1) the single root is 1.0 Roots of a quadratic function (1, -3, 2) the real roots are 2.0 and 1.0 Roots of a quadratic function (1, 0, 1) the complex roots are 0.0 +/- 1.0*i Roots of a quadratic function (1, -1.0e10, 1) the real roots are 1.0e10 and 0.0 Roots of a quadratic function (1, 2, 3) the complex roots are -1.0 +/- 1.4142135623730951*i Roots of a quadratic function (2, -1, -6) the real roots are 2.0 and -1.5
ERRE
<lang>PROGRAM QUADRATIC
PROCEDURE SOLVE_QUADRATIC
D=B*B-4*A*C
IF ABS(D)<1D-6 THEN D=0 END IF
CASE SGN(D) OF
0->
PRINT("the single root is ";-B/2/A)
END ->
1->
F=(1+SQR(1-4*A*C/(B*B)))/2
PRINT("the real roots are ";-F*B/A;"and ";-C/B/F)
END ->
-1->
PRINT("the complex roots are ";-B/2/A;"+/-";SQR(-D)/2/A;"*i")
END ->
END CASE
END PROCEDURE
BEGIN
PRINT(CHR$(12);) ! CLS
FOR TEST%=1 TO 7 DO
READ(A,B,C)
PRINT("For a=";A;",b=";B;",c=";C;TAB(32);)
SOLVE_QUADRATIC
END FOR
DATA(1,-1E9,1)
DATA(1,0,1)
DATA(2,-1,-6)
DATA(1,2,-2)
DATA(0.5,1.4142135,1)
DATA(1,3,2)
DATA(3,4,5)
END PROGRAM</lang>
- Output:
For a= 1 ,b=-1E+09 ,c= 1 the real roots are 1E+09 and 1E-09 For a= 1 ,b= 0 ,c= 1 the complex roots are 0 +/- 1 *i For a= 2 ,b=-1 ,c=-6 the real roots are 2 and -1.5 For a= 1 ,b= 2 ,c=-2 the real roots are -2.732051 and .7320508 For a= .5 ,b= 1.414214 ,c= 1 the single root is -1.414214 For a= 1 ,b= 3 ,c= 2 the real roots are -2 and -1 For a= 3 ,b= 4 ,c= 5 the complex roots are -.6666667 +/- 1.105542 *i
Factor
<lang factor>:: quadratic-equation ( a b c -- x1 x2 )
b sq a c * 4 * - sqrt :> sd b 0 < [ b neg sd + a 2 * / ] [ b neg sd - a 2 * / ] if :> x x c a x * / ;</lang>
<lang factor>( scratchpad ) 1 -1.e20 1 quadratic-equation --- Data stack: 1.0e+20 9.999999999999999e-21</lang>
Middlebrook method <lang factor>:: quadratic-equation2 ( a b c -- x1 x2 )
a c * sqrt b / :> q 1 4 q sq * - sqrt 0.5 * 0.5 + :> f b neg a / f * c neg b / f / ;
</lang>
<lang factor>( scratchpad ) 1 -1.e20 1 quadratic-equation
--- Data stack:
1.0e+20
1.0e-20</lang>
Forth
Without locals: <lang forth>: quadratic ( fa fb fc -- r1 r2 )
frot frot ( c a b ) fover 3 fpick f* -4e f* fover fdup f* f+ ( c a b det ) fdup f0< if abort" imaginary roots" then fsqrt fover f0< if fnegate then f+ fnegate ( c a b-det ) 2e f/ fover f/ ( c a r1 ) frot frot f/ fover f/ ;</lang>
With locals: <lang forth>: quadratic { F: a F: b F: c -- r1 r2 }
b b f* 4e a f* c f* f- fdup f0< if abort" imaginary roots" then fsqrt b f0< if fnegate then b f+ fnegate 2e f/ a f/ c a f/ fover f/ ;
\ test 1 set-precision 1e -1e6 1e quadratic fs. fs. \ 1e-6 1e6</lang>
Fortran
Fortran 90
<lang fortran>PROGRAM QUADRATIC
IMPLICIT NONE INTEGER, PARAMETER :: dp = SELECTED_REAL_KIND(15) REAL(dp) :: a, b, c, e, discriminant, rroot1, rroot2 COMPLEX(dp) :: croot1, croot2
WRITE(*,*) "Enter the coefficients of the equation ax^2 + bx + c" WRITE(*, "(A)", ADVANCE="NO") "a = " READ *, a WRITE(*,"(A)", ADVANCE="NO") "b = " READ *, b WRITE(*,"(A)", ADVANCE="NO") "c = " READ *, c WRITE(*,"(3(A,E23.15))") "Coefficients are: a = ", a, " b = ", b, " c = ", c e = 1.0e-9_dp discriminant = b*b - 4.0_dp*a*c IF (ABS(discriminant) < e) THEN rroot1 = -b / (2.0_dp * a) WRITE(*,*) "The roots are real and equal:" WRITE(*,"(A,E23.15)") "Root = ", rroot1 ELSE IF (discriminant > 0) THEN rroot1 = -(b + SIGN(SQRT(discriminant), b)) / (2.0_dp * a) rroot2 = c / (a * rroot1) WRITE(*,*) "The roots are real:" WRITE(*,"(2(A,E23.15))") "Root1 = ", rroot1, " Root2 = ", rroot2 ELSE croot1 = (-b + SQRT(CMPLX(discriminant))) / (2.0_dp*a) croot2 = CONJG(croot1) WRITE(*,*) "The roots are complex:" WRITE(*,"(2(A,2E23.15,A))") "Root1 = ", croot1, "j ", " Root2 = ", croot2, "j" END IF</lang>
- Output:
Coefficients are: a = 0.300000000000000E+01 b = 0.400000000000000E+01 c = 0.133333333330000E+01 The roots are real and equal: Root = -0.666666666666667E+00 Coefficients are: a = 0.300000000000000E+01 b = 0.200000000000000E+01 c = -0.100000000000000E+01 The roots are real: Root1 = -0.100000000000000E+01 Root2 = 0.333333333333333E+00 Coefficients are: a = 0.300000000000000E+01 b = 0.200000000000000E+01 c = 0.100000000000000E+01 The roots are complex: Root1 = -0.333333333333333E+00 0.471404512723287E+00j Root2 = -0.333333333333333E+00 -0.471404512723287E+00j Coefficients are: a = 0.100000000000000E+01 b = -0.100000000000000E+07 c = 0.100000000000000E+01 The roots are real: Root1 = 0.999999999999000E+06 Root2 = 0.100000000000100E-05
Fortran I
Source code written in FORTRAN I (october 1956) for the IBM 704. <lang fortran> COMPUTE ROOTS OF A QUADRATIC FUNCTION - 1956
READ 100,A,B,C
100 FORMAT(3F8.3)
PRINT 100,A,B,C
DISC=B**2-4.*A*C
IF(DISC),1,2,3
1 XR=-B/(2.*A)
XI=SQRT(-DISC)/(2.*A)
XJ=-XI
PRINT 311
PRINT 312,XR,XI,XR,XJ
311 FORMAT(13HCOMPLEX ROOTS)
312 FORMAT(4HX1=(,2E12.4,6H),X2=(,2E12.4,1H))
GO TO 999
2 X1=-B/(2.*A)
X2=X1
PRINT 321
PRINT 332,X1,X2
321 FORMAT(16HEQUAL REAL ROOTS)
GO TO 999
3 X1= (-B+SQRT(DISC)) / (2.*A)
X2= (-B-SQRT(DISC)) / (2.*A)
PRINT 331
PRINT 332,X1,X2
331 FORMAT(10HREAL ROOTS)
332 FORMAT(3HX1=,E12.5,4H,X2=,E12.5)
999 STOP
</lang>
GAP
<lang gap>QuadraticRoots := function(a, b, c)
local d; d := Sqrt(b*b - 4*a*c); return [ (-b+d)/(2*a), (-b-d)/(2*a) ];
end;
- Hint : E(12) is a 12th primitive root of 1
QuadraticRoots(2, 2, -1);
- [ 1/2*E(12)^4-1/2*E(12)^7+1/2*E(12)^8+1/2*E(12)^11,
- 1/2*E(12)^4+1/2*E(12)^7+1/2*E(12)^8-1/2*E(12)^11 ]
- This works also with floating-point numbers
QuadraticRoots(2.0, 2.0, -1.0);
- [ 0.366025, -1.36603 ]</lang>
Go
<lang go>package main
import (
"fmt" "math"
)
func qr(a, b, c float64) ([]float64, []complex128) {
d := b*b-4*a*c
switch {
case d == 0:
// single root
return []float64{-b/(2*a)}, nil
case d > 0:
// two real roots
if b < 0 {
d = math.Sqrt(d)-b
} else {
d = -math.Sqrt(d)-b
}
return []float64{d/(2*a), (2*c)/d}, nil
case d < 0:
// two complex roots
den := 1/(2*a)
t1 := complex(-b*den, 0)
t2 := complex(0, math.Sqrt(-d)*den)
return nil, []complex128{t1+t2, t1-t2}
}
// otherwise d overflowed or a coefficient was NAN
return []float64{d}, nil
}
func test(a, b, c float64) {
fmt.Print("coefficients: ", a, b, c, " -> ")
r, i := qr(a, b, c)
switch len(r) {
case 1:
fmt.Println("one real root:", r[0])
case 2:
fmt.Println("two real roots:", r[0], r[1])
default:
fmt.Println("two complex roots:", i[0], i[1])
}
}
func main() {
for _, c := range [][3]float64{
{1, -2, 1},
{1, 0, 1},
{1, -10, 1},
{1, -1000, 1},
{1, -1e9, 1},
} {
test(c[0], c[1], c[2])
}
}</lang>
- Output:
coefficients: 1 -2 1 -> one real root: 1 coefficients: 1 0 1 -> two complex roots: (0+1i) (-0-1i) coefficients: 1 -10 1 -> two real roots: 9.898979485566356 0.10102051443364381 coefficients: 1 -1000 1 -> two real roots: 999.998999999 0.001000001000002 coefficients: 1 -1e+09 1 -> two real roots: 1e+09 1e-09
Haskell
<lang haskell>import Data.Complex (Complex, realPart)
type CD = Complex Double
quadraticRoots :: (CD, CD, CD) -> (CD, CD) quadraticRoots (a, b, c) =
if realPart b > 0 then ((2 * c) / (-b - d), (-b - d) / (2 * a)) else ((-b + d) / (2 * a), (2 * c) / (-b + d)) where d = sqrt $ b ^ 2 - 4 * a * c
main :: IO () main =
mapM_ (print . quadraticRoots) [(3, 4, 4 / 3), (3, 2, -1), (3, 2, 1), (1, -10e5, 1), (1, -10e9, 1)]</lang>
- Output:
((-0.6666666666666666) :+ 0.0,(-0.6666666666666666) :+ 0.0) (0.3333333333333333 :+ 0.0,(-1.0) :+ 0.0) ((-0.33333333333333326) :+ 0.4714045207910316,(-0.3333333333333333) :+ (-0.47140452079103173)) (999999.999999 :+ 0.0,1.000000000001e-6 :+ 0.0) (1.0e10 :+ 0.0,1.0e-10 :+ 0.0)
Icon and Unicon
Works in both languages. <lang unicon>procedure main()
solve(1.0, -10.0e5, 1.0)
end
procedure solve(a,b,c)
d := sqrt(b*b - 4.0*a*c)
roots := if b < 0 then [r1 := (-b+d)/(2.0*a), c/(a*r1)]
else [r1 := (-b-d)/(2.0*a), c/(a*r1)]
write(a,"*x^2 + ",b,"*x + ",c," has roots ",roots[1]," and ",roots[2])
end</lang>
- Output:
->rqf 1.0 -0.000000001 1.0 1.0*x^2 + -1000000.0*x + 1.0 has roots 999999.999999 and 1.000000000001e-06 ->
IDL
<lang idl>compile_OPT IDL2
print, "input a, press enter, input b, press enter, input c, press enter" read,a,b,c Promt='Enter values of a,b,c and hit enter'
a0=0.0 b0=0.0 c0=0.0 ;make them floating point variables
x=-b+sqrt((b^2)-4*a*c) y=-b-sqrt((b^2)-4*a*c) z=2*a d= x/z e= y/z
print, d,e</lang>
IS-BASIC
<lang IS-BASIC> 100 PROGRAM "Quadratic.bas" 110 PRINT "Enter coefficients a, b and c:":INPUT PROMPT "a= ,b= ,c= ":A,B,C 120 IF A=0 THEN 130 PRINT "The coefficient of x^2 can not be 0." 140 ELSE 150 LET D=B^2-4*A*C 160 SELECT CASE SGN(D) 170 CASE 0 180 PRINT "The single root is ";-B/2/A 190 CASE 1 200 PRINT "The real roots are ";(-B+SQR(D))/(2*A);"and ";(-B-SQR(D))/(2*A) 210 CASE -1 220 PRINT "The complex roots are ";-B/2/A;"+/- ";STR$(SQR(-D)/2/A);"*i" 230 END SELECT 240 END IF</lang>
J
Solution use J's built-in polynomial solver:
p.
Example using inputs from other solutions and the unstable example from the task description:
<lang j> coeff =. _3 |.\ 3 4 4r3 3 2 _1 3 2 1 1 _1e6 1 1 _1e9 1
> {:"1 p. coeff
_0.666667 _0.666667
_1 0.333333
_0.333333j0.471405 _0.333333j_0.471405
1e6 1e_6
1e9 1e_9</lang>
Of course p. generalizes to polynomials of arbitrary order (which isn't as great as that might sound, because of practical limitations). Given the coefficients p. returns the multiplier and roots of the polynomial. Given the multiplier and roots it returns the coefficients. For example using the cubic :
<lang j> p. 0 16 _12 2 NB. return multiplier ; roots
+-+-----+
|2|4 2 0|
+-+-----+
p. 2 ; 4 2 0 NB. return coefficients
0 16 _12 2</lang>
Exploring the limits of precision:
<lang j> 1{::p. 1 _1e5 1 NB. display roots 100000 1e_5
1 _1e5 1 p. 1{::p. 1 _1e5 1 NB. test roots
_3.38436e_7 0
1 _1e5 1 p. 1e5 1e_5 NB. test displayed roots
1 9.99999e_11
1e5 1e_5 - 1{::p. 1 _1e5 1 NB. find difference
1e_5 _1e_15
1 _1e5 1 p. 1e5 1e_5-1e_5 _1e_15 NB. test displayed roots with adjustment
_3.38436e_7 0</lang>
When these "roots" are plugged back into the original polynomial, the results are nowhere near zero. However, double precision floating point does not have enough bits to represent the (extremely close) answers that would give a zero result.
Middlebrook formula implemented in J
<lang j>q_r=: verb define
'a b c' =. y q=. b %~ %: a * c f=. 0.5 + 0.5 * %:(1-4*q*q) (-b*f%a),(-c%b*f)
)
q_r 1 _1e6 1
1e6 1e_6</lang>
Java
<lang java>public class QuadraticRoots {
private static class Complex {
double re, im;
public Complex(double re, double im) {
this.re = re;
this.im = im;
}
@Override
public boolean equals(Object obj) {
if (obj == this) {return true;}
if (!(obj instanceof Complex)) {return false;}
Complex other = (Complex) obj;
return (re == other.re) && (im == other.im);
}
@Override
public String toString() {
if (im == 0.0) {return String.format("%g", re);}
if (re == 0.0) {return String.format("%gi", im);}
return String.format("%g %c %gi", re,
(im < 0.0 ? '-' : '+'), Math.abs(im));
}
}
private static Complex[] quadraticRoots(double a, double b, double c) {
Complex[] roots = new Complex[2];
double d = b * b - 4.0 * a * c; // discriminant
double aa = a + a;
if (d < 0.0) {
double re = -b / aa;
double im = Math.sqrt(-d) / aa;
roots[0] = new Complex(re, im);
roots[1] = new Complex(re, -im);
} else if (b < 0.0) {
// Avoid calculating -b - Math.sqrt(d), to avoid any
// subtractive cancellation when it is near zero.
double re = (-b + Math.sqrt(d)) / aa;
roots[0] = new Complex(re, 0.0);
roots[1] = new Complex(c / (a * re), 0.0);
} else {
// Avoid calculating -b + Math.sqrt(d).
double re = (-b - Math.sqrt(d)) / aa;
roots[1] = new Complex(re, 0.0);
roots[0] = new Complex(c / (a * re), 0.0);
}
return roots;
}
public static void main(String[] args) {
double[][] equations = {
{1.0, 22.0, -1323.0}, // two distinct real roots
{6.0, -23.0, 20.0}, // with a != 1.0
{1.0, -1.0e9, 1.0}, // with one root near zero
{1.0, 2.0, 1.0}, // one real root (double root)
{1.0, 0.0, 1.0}, // two imaginary roots
{1.0, 1.0, 1.0} // two complex roots
};
for (int i = 0; i < equations.length; i++) {
Complex[] roots = quadraticRoots(
equations[i][0], equations[i][1], equations[i][2]);
System.out.format("%na = %g b = %g c = %g%n",
equations[i][0], equations[i][1], equations[i][2]);
if (roots[0].equals(roots[1])) {
System.out.format("X1,2 = %s%n", roots[0]);
} else {
System.out.format("X1 = %s%n", roots[0]);
System.out.format("X2 = %s%n", roots[1]);
}
}
}
}</lang>
- Output:
a = 1.00000 b = 22.0000 c = -1323.00 X1 = 27.0000 X2 = -49.0000 a = 6.00000 b = -23.0000 c = 20.0000 X1 = 2.50000 X2 = 1.33333 a = 1.00000 b = -1.00000e+09 c = 1.00000 X1 = 1.00000e+09 X2 = 1.00000e-09 a = 1.00000 b = 2.00000 c = 1.00000 X1,2 = -1.00000 a = 1.00000 b = 0.00000 c = 1.00000 X1 = 1.00000i X2 = -1.00000i a = 1.00000 b = 1.00000 c = 1.00000 X1 = -0.500000 + 0.866025i X2 = -0.500000 - 0.866025i
jq
Currently jq does not include support for complex number operations, so a small library is included in the first section.
The second section defines quadratic_roots(a;b;c), which emits a stream of 0 or two solutions, or the value true if a==b==c==0.
The third section defines a function for producing a table showing (i, error, solution) for solutions to x^2 - 10^i + 1 = 0 for various values of i.
Section 1: Complex numbers (scrolling window)
<lang jq># Complex numbers as points [x,y] in the Cartesian plane def real(z): if (z|type) == "number" then z else z[0] end;
def imag(z): if (z|type) == "number" then 0 else z[1] end;
def plus(x; y):
if (x|type) == "number" then
if (y|type) == "number" then [ x+y, 0 ]
else [ x + y[0], y[1]]
end
elif (y|type) == "number" then plus(y;x)
else [ x[0] + y[0], x[1] + y[1] ]
end;
def multiply(x; y):
if (x|type) == "number" then
if (y|type) == "number" then [ x*y, 0 ]
else [x * y[0], x * y[1]]
end
elif (y|type) == "number" then multiply(y;x)
else [ x[0] * y[0] - x[1] * y[1], x[0] * y[1] + x[1] * y[0]]
end;
def negate(x): multiply(-1; x);
def minus(x; y): plus(x; multiply(-1; y));
def conjugate(z):
if (z|type) == "number" then [z, 0] else [z[0], -(z[1]) ] end;
def invert(z):
if (z|type) == "number" then [1/z, 0] else ( (z[0] * z[0]) + (z[1] * z[1]) ) as $d # use "0 + ." to convert -0 back to 0 | [ z[0]/$d, (0 + -(z[1]) / $d)] end;
def divide(x;y): multiply(x; invert(y));
def magnitude(z):
real( multiply(z; conjugate(z))) | sqrt;
- exp^z
def complex_exp(z):
def expi(x): [ (x|cos), (x|sin) ]; if (z|type) == "number" then z|exp elif z[0] == 0 then expi(z[1]) # for efficiency else multiply( (z[0]|exp); expi(z[1]) ) end ;
def complex_sqrt(z):
if imag(z) == 0 and real(z) >= 0 then [(real(z)|sqrt), 0]
else
magnitude(z) as $r
| if $r == 0 then [0,0]
else
(real(z)/$r) as $a
| (imag(z)/$r) as $b
| $r | sqrt as $r
| (($a | acos) / 2)
| [ ($r * cos), ($r * sin)]
end
end ;</lang>Section 2: quadratic_roots(a;b;c) <lang jq># If there are infinitely many solutions, emit true;
- if none, emit empty;
- otherwise always emit two.
- For numerical accuracy, Middlebrook's approach is adopted:
def quadratic_roots(a; b; c):
if a == 0 and b == 0 then
if c == 0 then true # infinitely many
else empty # none
end
elif a == 0 then [-c/b, 0]
elif b == 0 then (complex_sqrt(1/a) | (., negate(.)))
else
divide( plus(1.0; complex_sqrt( minus(1.0; (4 * a * c / (b*b))))); 2) as $f
| negate(divide(multiply(b; $f); a)),
negate(divide(c; multiply(b; $f)))
end
- </lang>
Section 3: Produce a table showing [i, error, solution] for solutions to x^2 - 10^i + 1 = 0 <lang jq>def example:
def pow(i): . as $in | reduce range(0;i) as $i (1; . * $in); def poly(a;b;c): plus( plus( multiply(a; multiply(.;.)); multiply(b;.)); c); def abs: if . < 0 then -. else . end; def zero(z): if z == 0 then 0 else (magnitude(z)|abs) as $zero | if $zero < 1e-10 then "+0" else $zero end end; def lpad(n): tostring | (n - length) * " " + .;
range(0; 13) as $i | (- (10|pow($i))) as $b | quadratic_roots(1; $b; 1) as $x | $x | poly(1; $b; 1) as $zero | "\($i|lpad(4)): error = \(zero($zero)|lpad(18)) x=\($x)"
example</lang>
- Output:
(scrolling window)
<lang sh> $ jq -M -r -n -f Roots_of_a_quadratic_function.jq
0: error = +0 x=[0.5,0.8660254037844386] 0: error = +0 x=[0.5000000000000001,-0.8660254037844387] 1: error = +0 x=[9.898979485566356,0] 1: error = +0 x=[0.10102051443364382,-0] 2: error = +0 x=[99.98999899979995,0] 2: error = +0 x=[0.010001000200050014,-0] 3: error = 1.1641532182693481e-10 x=[999.998999999,0] 3: error = +0 x=[0.0010000010000019998,-0] 4: error = +0 x=[9999.999899999999,0] 4: error = +0 x=[0.00010000000100000003,-0] 5: error = +0 x=[99999.99999,0] 5: error = +0 x=[1.0000000001e-05,-0] 6: error = 0.0001220703125 x=[999999.9999989999,0] 6: error = +0 x=[1.000000000001e-06,-0] 7: error = 0.015625 x=[9999999.9999999,0] 7: error = +0 x=[1.0000000000000101e-07,-0] 8: error = 1 x=[99999999.99999999,0] 8: error = +0 x=[1e-08,-0] 9: error = 1 x=[1000000000,0] 9: error = +0 x=[1e-09,-0] 10: error = 1 x=[10000000000,0] 10: error = +0 x=[1e-10,-0] 11: error = 1 x=[100000000000,0] 11: error = +0 x=[1e-11,-0] 12: error = 1 x=[1000000000000,0]12: error = +0 x=[1e-12,-0]</lang>
Julia
This solution is an implementation of algorithm from the Goldberg paper cited in the task description. It does check for a=0 and returns the linear solution in that case. Julia's sqrt throws a domain error for negative real inputs, so negative discriminants are converted to complex by adding 0im prior to taking the square root.
Alternative solutions might make use of Julia's Polynomials or Roots packages.
<lang julia>using Printf
function quadroots(x::Real, y::Real, z::Real)
a, b, c = promote(float(x), y, z)
if a ≈ 0.0 return [-c / b] end
Δ = b ^ 2 - 4a * c
if Δ ≈ 0.0 return [-sqrt(c / a)] end
if Δ < 0.0 Δ = complex(Δ) end
d = sqrt(Δ)
if b < 0.0
d -= b
return [d / 2a, 2c / d]
else
d = -d - b
return [2c / d, d / 2a]
end
end
a = [1, 1, 1.0, 10] b = [10, 2, -10.0 ^ 9, 1] c = [1, 1, 1, 1]
for (x, y, z) in zip(a, b, c)
@printf "The roots of %.2fx² + %.2fx + %.2f\n\tx₀ = (%s)\n" x y z join(round.(quadroots(x, y, z), 2), ", ")
end</lang>
- Output:
The roots of 1.00x² + 10.00x + 1.00 x₀ = (-0.1, -9.9) The roots of 1.00x² + 2.00x + 1.00 x₀ = (-1.0) The roots of 1.00x² + -1000000000.00x + 1.00 x₀ = (1.0e9, 0.0) The roots of 10.00x² + 1.00x + 1.00 x₀ = (-0.05 + 0.31im, -0.05 - 0.31im)
Kotlin
<lang scala>import java.lang.Math.*
data class Equation(val a: Double, val b: Double, val c: Double) {
data class Complex(val r: Double, val i: Double) {
override fun toString() = when {
i == 0.0 -> r.toString()
r == 0.0 -> "${i}i"
else -> "$r + ${i}i"
}
}
data class Solution(val x1: Any, val x2: Any) {
override fun toString() = when(x1) {
x2 -> "X1,2 = $x1"
else -> "X1 = $x1, X2 = $x2"
}
}
val quadraticRoots by lazy {
val _2a = a + a
val d = b * b - 4.0 * a * c // discriminant
if (d < 0.0) {
val r = -b / _2a
val i = sqrt(-d) / _2a
Solution(Complex(r, i), Complex(r, -i))
} else {
// avoid calculating -b +/- sqrt(d), to avoid any
// subtractive cancellation when it is near zero.
val r = if (b < 0.0) (-b + sqrt(d)) / _2a else (-b - sqrt(d)) / _2a
Solution(r, c / (a * r))
}
}
}
fun main(args: Array<String>) {
val equations = listOf(Equation(1.0, 22.0, -1323.0), // two distinct real roots
Equation(6.0, -23.0, 20.0), // with a != 1.0
Equation(1.0, -1.0e9, 1.0), // with one root near zero
Equation(1.0, 2.0, 1.0), // one real root (double root)
Equation(1.0, 0.0, 1.0), // two imaginary roots
Equation(1.0, 1.0, 1.0)) // two complex roots
equations.forEach { println("$it\n" + it.quadraticRoots) }
}</lang>
- Output:
Equation(a=1.0, b=22.0, c=-1323.0) X1 = -49.0, X2 = 27.0 Equation(a=6.0, b=-23.0, c=20.0) X1 = 2.5, X2 = 1.3333333333333333 Equation(a=1.0, b=-1.0E9, c=1.0) X1 = 1.0E9, X2 = 1.0E-9 Equation(a=1.0, b=2.0, c=1.0) X1,2 = -1.0 Equation(a=1.0, b=0.0, c=1.0) X1 = 1.0i, X2 = -1.0i Equation(a=1.0, b=1.0, c=1.0) X1 = -0.5 + 0.8660254037844386i, X2 = -0.5 + -0.8660254037844386i
lambdatalk
<lang scheme> 1) using lambdas:
{def equation
{lambda {:a :b :c}
{b equation :a*x{sup 2}+:b*x+:c=0}
{{lambda {:a' :b' :d}
{if {> :d 0}
then {{lambda {:b' :d'}
{equation.disp {+ :b' :d'} {- :b' :d'} 2 real roots}
} :b' {/ {sqrt :d} :a'}}
else {if {< :d 0}
then {{lambda {:b' :d'}
{equation.disp [:b',:d'] [:b',-:d'] 2 complex roots}
} :b' {/ {sqrt {- :d}} :a'} }
else {equation.disp :b' :b' one real double root}
}}
} {* 2 :a} {/ {- :b} {* 2 :a}} {- {* :b :b} {* 4 :a :c}} } }}
2) using let:
{def equation
{lambda {:a :b :c}
{b equation :a*x{sup 2}+:b*x+:c=0}
{let { {:a' {* 2 :a}}
{:b' {/ {- :b} {* 2 :a}}}
{:d {- {* :b :b} {* 4 :a :c}}} }
{if {> :d 0}
then {let { {:b' :b'}
{:d' {/ {sqrt :d} :a'}} }
{equation.disp {+ :b' :d'} {- :b' :d'} 2 real roots} }
else {if {< :d 0}
then {let { {:b' :b'}
{:d' {/ {sqrt {- :d}} :a'}} }
{equation.disp [:b',:d'] [:b',-:d'] 2 complex roots} }
else {equation.disp :b' :b' one real double root} }} }}}
3) a function to display results in an HTML table format
{def equation.disp
{lambda {:x1 :x2 :txt}
{table {@ style="background:#ffa"}
{tr {td :txt: }}
{tr {td x1 = :x1 }}
{tr {td x2 = :x2 }} } }}
4) testing:
equation 1*x2+1*x+-1=0 2 real roots:
x1 = 0.6180339887498949 x2 = -1.618033988749895
equation 1*x2+1*x+1=0 2 complex roots:
x1 = [-0.5,0.8660254037844386] x2 = [-0.5,-0.8660254037844386]
equation 1*x2+-2*x+1=0 one real double root:
x1 = 1 x2 = 1
</lang>
Liberty BASIC
<lang lb>a=1:b=2:c=3
'assume a<>0 print quad$(a,b,c) end
function quad$(a,b,c)
D=b^2-4*a*c
x=-1*b
if D<0 then
quad$=str$(x/(2*a));" +i";str$(sqr(abs(D))/(2*a));" , ";str$(x/(2*a));" -i";str$(sqr(abs(D))/abs(2*a))
else
quad$=str$(x/(2*a)+sqr(D)/(2*a));" , ";str$(x/(2*a)-sqr(D)/(2*a))
end if
end function</lang>
Logo
<lang logo>to quadratic :a :b :c
localmake "d sqrt (:b*:b - 4*:a*:c) if :b < 0 [make "d minus :d] output list (:d-:b)/(2*:a) (2*:c)/(:d-:b)
end</lang>
Lua
In order to correctly handle complex roots, qsolve must be given objects from a suitable complex number library, like that from the Complex Numbers article. However, this should be enough to demonstrate its accuracy:
<lang lua>function qsolve(a, b, c)
if b < 0 then return qsolve(-a, -b, -c) end val = b + (b^2 - 4*a*c)^(1/2) --this never exhibits instability if b > 0 return -val / (2 * a), -2 * c / val --2c / val is the same as the "unstable" second root
end
for i = 1, 12 do
print(qsolve(1, 0-10^i, 1))
end</lang> The "trick" lies in avoiding subtracting large values that differ by a small amount, which is the source of instability in the "normal" formula. It is trivial to prove that 2c/(b + sqrt(b^2-4ac)) = (b - sqrt(b^2-4ac))/2a.
Maple
<lang Maple>solve(a*x^2+b*x+c,x);
solve(1.0*x^2-10.0^9*x+1.0,x,explicit,allsolutions);
fsolve(x^2-10^9*x+1,x,complex);</lang>
- Output:
(1/2) (1/2)
/ 2\ / 2\
-b + \-4 a c + b / b + \-4 a c + b /
-----------------------, - ----------------------
2 a 2 a
9 -9
1.000000000 10 , 1.000000000 10
-9 9
1.000000000 10 , 1.000000000 10
Mathematica
Possible ways to do this are (symbolic and numeric examples): <lang Mathematica>Solve[a x^2 + b x + c == 0, x] Solve[x^2 - 10^5 x + 1 == 0, x] Root[#1^2 - 10^5 #1 + 1 &, 1] Root[#1^2 - 10^5 #1 + 1 &, 2] Reduce[a x^2 + b x + c == 0, x] Reduce[x^2 - 10^5 x + 1 == 0, x] FindInstance[x^2 - 10^5 x + 1 == 0, x, Reals, 2] FindRoot[x^2 - 10^5 x + 1 == 0, {x, 0}] FindRoot[x^2 - 10^5 x + 1 == 0, {x, 10^6}]</lang> gives back:
Note that some functions do not really give the answer (like reduce) rather it gives another way of writing it (boolean expression). However note that reduce gives the explicit cases for a zero and nonzero, b zero and nonzero, et cetera. Some functions are numeric by nature, other can handle both symbolic and numeric. In generals the solution will be exact if the input is exact. Any exact result can be approximated to arbitrary precision using the function N[expression,number of digits]. Further notice that some functions give back exact answers in a different form then others, however the answers are both correct, the answers are just written differently.
MATLAB / Octave
<lang Matlab>roots([1 -3 2]) % coefficients in decreasing order of power e.g. [x^n ... x^2 x^1 x^0]</lang>
Maxima
<lang maxima>solve(a*x^2 + b*x + c = 0, x);
/* 2 2
sqrt(b - 4 a c) + b sqrt(b - 4 a c) - b
[x = - --------------------, x = --------------------]
2 a 2 a */
fpprec: 40$
solve(x^2 - 10^9*x + 1 = 0, x); /* [x = 500000000 - sqrt(249999999999999999),
x = sqrt(249999999999999999) + 500000000] */
bfloat(%); /* [x = 1.0000000000000000009999920675269450501b-9,
x = 9.99999999999999998999999999999999999b8] */</lang>
МК-61/52
<lang>П2 С/П /-/ <-> / 2 / П3 x^2 С/П ИП2 / - Вx <-> КвКор НОП x>=0 28 ИП3 x<0 24 <-> /-/ + / Вx С/П /-/ КвКор ИП3 С/П</lang>
Input: a С/П b С/П c С/П
- Output:
x1 - РX; x2 - РY (or error message, if D < 0).
Modula-3
<lang modula3>MODULE Quad EXPORTS Main;
IMPORT IO, Fmt, Math;
TYPE Roots = ARRAY [1..2] OF LONGREAL;
VAR r: Roots;
PROCEDURE Solve(a, b, c: LONGREAL): Roots =
VAR sd: LONGREAL := Math.sqrt(b * b - 4.0D0 * a * c);
x: LONGREAL;
BEGIN
IF b < 0.0D0 THEN
x := (-b + sd) / 2.0D0 * a;
RETURN Roots{x, c / (a * x)};
ELSE
x := (-b - sd) / 2.0D0 * a;
RETURN Roots{c / (a * x), x};
END;
END Solve;
BEGIN
r := Solve(1.0D0, -10.0D5, 1.0D0);
IO.Put("X1 = " & Fmt.LongReal(r[1]) & " X2 = " & Fmt.LongReal(r[2]) & "\n");
END Quad.</lang>
OCaml
<lang ocaml>type quadroots =
| RealRoots of float * float | ComplexRoots of Complex.t * Complex.t ;;
let quadsolve a b c =
let d = (b *. b) -. (4.0 *. a *. c) in
if d < 0.0
then
let r = -. b /. (2.0 *. a)
and i = sqrt(-. d) /. (2.0 *. a) in
ComplexRoots ({ Complex.re = r; Complex.im = i },
{ Complex.re = r; Complex.im = (-.i) })
else
let r =
if b < 0.0
then ((sqrt d) -. b) /. (2.0 *. a)
else ((sqrt d) +. b) /. (-2.0 *. a)
in
RealRoots (r, c /. (r *. a))
- </lang>
- Output:
<lang ocaml># quadsolve 1.0 0.0 (-2.0) ;; - : quadroots = RealRoots (-1.4142135623730951, 1.4142135623730949)
- quadsolve 1.0 0.0 2.0 ;;
- : quadroots = ComplexRoots ({Complex.re = 0.; Complex.im = 1.4142135623730951},
{Complex.re = 0.; Complex.im = -1.4142135623730951})
- quadsolve 1.0 (-1.0e5) 1.0 ;;
- : quadroots = RealRoots (99999.99999, 1.0000000001000001e-005)</lang>
Octave
See MATLAB.
PARI/GP
<lang parigp>roots(a,b,c)=polrootsreal(Pol([a,b,c]))</lang>
Otherwise, coding directly: <lang parigp>roots(a,b,c)={
b /= a; c /= a; my (delta = b^2 - 4*c, root=sqrt(delta)); if (delta < 0, [root-b,-root-b]/2 , my(sol=if(b>0, -b - root,-b + root)/2); [sol,c/sol] )
};</lang>
Either way, <lang parigp>roots(1,-1e9,1)</lang> gives one root around 0.000000001000000000000000001 and one root around 999999999.999999999.
Pascal
some parts translated from Modula2 <lang pascal>Program QuadraticRoots;
var
a, b, c, q, f: double;
begin
a := 1; b := -10e9; c := 1; q := sqrt(a * c) / b; f := (1 + sqrt(1 - 4 * q * q)) / 2;
writeln ('Version 1:');
writeln ('x1: ', (-b * f / a):16, ', x2: ', (-c / (b * f)):16);
writeln ('Version 2:');
q := sqrt(b * b - 4 * a * c);
if b < 0 then
begin
f := (-b + q) / 2 * a;
writeln ('x1: ', f:16, ', x2: ', (c / (a * f)):16);
end
else
begin
f := (-b - q) / 2 * a;
writeln ('x1: ', (c / (a * f)):16, ', x2: ', f:16);
end;
end. </lang>
- Output:
Version 1: x1: 1.00000000E+010, x2: 1.00000000E-010 Version 2: x1: 1.00000000E+010, x2: 1.00000000E-010
Perl
When using Math::Complex perl automatically convert numbers when necessary. <lang perl>use Math::Complex;
($x1,$x2) = solveQuad(1,2,3);
print "x1 = $x1, x2 = $x2\n";
sub solveQuad { my ($a,$b,$c) = @_; my $root = sqrt($b**2 - 4*$a*$c); return ( -$b + $root )/(2*$a), ( -$b - $root )/(2*$a); }</lang>
Phix
<lang Phix>procedure solve_quadratic(sequence t3) atom {a,b,c} = t3 atom d = b*b-4*a*c, f string s = sprintf("for a=%g,b=%g,c=%g",t3), t sequence u
if abs(d)<1e-6 then d=0 end if
switch sign(d) do
case 0: t = "single root is %g"
u = {-b/2/a}
case 1: t = "real roots are %g and %g"
f = (1+sqrt(1-4*a*c/(b*b)))/2
u = {-f*b/a,-c/b/f}
case-1: t = "complex roots are %g +/- %g*i"
u = {-b/2/a,sqrt(-d)/2/a}
end switch
printf(1,"%-25s the %s\n",{s,sprintf(t,u)})
end procedure
constant tests = {{1,-1E9,1},
{1,0,1},
{2,-1,-6},
{1,2,-2},
{0.5,1.4142135,1},
{1,3,2},
{3,4,5}}
for i=1 to length(tests) do
solve_quadratic(tests[i])
end for</lang>
for a=1,b=-1e+9,c=1 the real roots are 1e+9 and 1e-9 for a=1,b=0,c=1 the complex roots are 0 +/- 1*i for a=2,b=-1,c=-6 the real roots are 2 and -1.5 for a=1,b=2,c=-2 the real roots are -2.73205 and 0.732051 for a=0.5,b=1.41421,c=1 the single root is -1.41421 for a=1,b=3,c=2 the real roots are -2 and -1 for a=3,b=4,c=5 the complex roots are -0.666667 +/- 1.10554*i
PicoLisp
<lang PicoLisp>(scl 40)
(de solveQuad (A B C)
(let SD (sqrt (- (* B B) (* 4 A C)))
(if (lt0 B)
(list
(*/ (- SD B) A 2.0)
(*/ C 2.0 (*/ A A (- SD B) `(* 1.0 1.0))) )
(list
(*/ C 2.0 (*/ A A (- 0 B SD) `(* 1.0 1.0)))
(*/ (- 0 B SD) A 2.0) ) ) ) )
(mapcar round
(solveQuad 1.0 -1000000.0 1.0) (6 .) )</lang>
- Output:
-> ("999,999.999999" "0.000001")
PL/I
<lang PL/I>
declare (c1, c2) float complex,
(a, b, c, x1, x2) float;
get list (a, b, c);
if b**2 < 4*a*c then
do;
c1 = (-b + sqrt(b**2 - 4+0i*a*c)) / (2*a);
c2 = (-b - sqrt(b**2 - 4+0i*a*c)) / (2*a);
put data (c1, c2);
end;
else
do;
x1 = (-b + sqrt(b**2 - 4*a*c)) / (2*a);
x2 = (-b - sqrt(b**2 - 4*a*c)) / (2*a);
put data (x1, x2);
end;
</lang>
Python
This solution compares the naïve method with three "better" methods. <lang python>#!/usr/bin/env python3
import math import cmath import numpy
def quad_discriminating_roots(a,b,c, entier = 1e-5):
"""For reference, the naive algorithm which shows complete loss of
precision on the quadratic in question. (This function also returns a
characterization of the roots.)"""
discriminant = b*b - 4*a*c
a,b,c,d =complex(a), complex(b), complex(c), complex(discriminant)
root1 = (-b + cmath.sqrt(d))/2./a
root2 = (-b - cmath.sqrt(d))/2./a
if abs(discriminant) < entier:
return "real and equal", abs(root1), abs(root1)
if discriminant > 0:
return "real", root1.real, root2.real
return "complex", root1, root2
def middlebrook(a, b, c):
try:
q = math.sqrt(a*c)/b
f = .5+ math.sqrt(1-4*q*q)/2
except ValueError:
q = cmath.sqrt(a*c)/b
f = .5+ cmath.sqrt(1-4*q*q)/2
return (-b/a)*f, -c/(b*f)
def whatevery(a, b, c):
try:
d = math.sqrt(b*b-4*a*c)
except ValueError:
d = cmath.sqrt(b*b-4*a*c)
if b > 0:
return div(2*c, (-b-d)), div((-b-d), 2*a)
else:
return div((-b+d), 2*a), div(2*c, (-b+d))
def div(n, d):
"""Divide, with a useful interpretation of division by zero."""
try:
return n/d
except ZeroDivisionError:
if n:
return n*float('inf')
return float('nan')
testcases = [
(3, 4, 4/3), # real, equal (3, 2, -1), # real, unequal (3, 2, 1), # complex (1, -1e9, 1), # ill-conditioned "quadratic in question" required by task. (1, -1e100, 1), (1, -1e200, 1), (1, -1e300, 1),
]
print('Naive:') for c in testcases:
print("{} {:.5} {:.5}".format(*quad_discriminating_roots(*c)))
print('\nMiddlebrook:') for c in testcases:
print(("{:.5} "*2).format(*middlebrook(*c)))
print('\nWhat Every...') for c in testcases:
print(("{:.5} "*2).format(*whatevery(*c)))
print('\nNumpy:') for c in testcases:
print(("{:.5} "*2).format(*numpy.roots(c)))</lang>
- Output:
Naive: real and equal 0.66667 0.66667 real 0.33333 -1.0 complex (-0.33333+0.4714j) (-0.33333-0.4714j) real 1e+09 0.0 real 1e+100 0.0 real nan nan real nan nan Middlebrook: -0.66667 -0.66667 (-1+0j) (0.33333+0j) (-0.33333-0.4714j) (-0.33333+0.4714j) 1e+09 1e-09 1e+100 1e-100 1e+200 1e-200 1e+300 1e-300 What Every... -0.66667 -0.66667 0.33333 -1.0 (-0.33333+0.4714j) (-0.33333-0.4714j) 1e+09 1e-09 1e+100 1e-100 inf 0.0 inf 0.0 Numpy: -0.66667 -0.66667 -1.0 0.33333 (-0.33333+0.4714j) (-0.33333-0.4714j) 1e+09 1e-09 1e+100 1e-100 1e+200 1e-200 1e+300 0.0
R
<lang R>qroots <- function(a, b, c) {
r <- sqrt(b * b - 4 * a * c + 0i)
if (abs(b - r) > abs(b + r)) {
z <- (-b + r) / (2 * a)
} else {
z <- (-b - r) / (2 * a)
}
c(z, c / (z * a))
}
qroots(1, 0, 2i) [1] -1+1i 1-1i
qroots(1, -1e9, 1) [1] 1e+09+0i 1e-09+0i</lang>
Using the builtin polyroot function (note the order of coefficients is reversed):
<lang R>polyroot(c(2i, 0, 1)) [1] -1+1i 1-1i
polyroot(c(1, -1e9, 1)) [1] 1e-09+0i 1e+09+0i</lang>
Racket
<lang Racket>#lang racket (define (quadratic a b c)
(let* ((-b (- b))
(delta (- (expt b 2) (* 4 a c)))
(denominator (* 2 a)))
(list
(/ (+ -b (sqrt delta)) denominator)
(/ (- -b (sqrt delta)) denominator))))
- (quadratic 1 0.0000000000001 -1)
- '(0.99999999999995 -1.00000000000005)
- (quadratic 1 0.0000000000001 1)
- '(-5e-014+1.0i -5e-014-1.0i)</lang>
Raku
(formerly Perl 6)
Raku has complex number handling built in.
<lang perl6>for [1, 2, 1], [1, 2, 3], [1, -2, 1], [1, 0, -4], [1, -10**6, 1] -> @coefficients {
printf "Roots for %d, %d, %d\t=> (%s, %s)\n", |@coefficients, |quadroots(@coefficients);
}
sub quadroots (*[$a, $b, $c]) {
( -$b + $_ ) / (2 * $a), ( -$b - $_ ) / (2 * $a) given ($b ** 2 - 4 * $a * $c ).Complex.sqrt.narrow
}</lang>
- Output:
Roots for 1, 2, 1 => (-1, -1) Roots for 1, 2, 3 => (-1+1.4142135623731i, -1-1.4142135623731i) Roots for 1, -2, 1 => (1, 1) Roots for 1, 0, -4 => (2, -2) Roots for 1, -1000000, 1 => (999999.999999, 1.00000761449337e-06)
REXX
version 1
The REXX language doesn't have a sqrt function, nor does it support complex numbers natively.
Since "unlimited" decimal precision is part of the REXX language, the numeric digits was increased
(from a default of 9) to 200 to accommodate when a root is closer to zero than the other root.
Note that only ten decimal digits (precision) are shown in the displaying of the output.
This REXX version supports complex numbers for the result. <lang rexx>/*REXX program finds the roots (which may be complex) of a quadratic function.*/ numeric digits 200 /*use enough digits to handle extremes.*/ parse arg a b c . /*obtain the specified arguments: A B C*/ call quadratic a,b,c /*solve quadratic function via the sub.*/ numeric digits 10 /*reduce (output) digits for human eyes*/ r1=r1/1; r2=r2/1; a=a/1; b=b/1; c=c/1 /*normalize numbers to the new digits. */ if r1j\=0 then r1=r1 || left('+',r1j>0)(r1j/1)"i" /*handle complex number.*/ if r2j\=0 then r2=r2 || left('+',r2j>0)(r2j/1)"i" /* " " " */
say ' a =' a /*display the normalized value of A. */
say ' b =' b /* " " " " " B. */
say ' c =' c /* " " " " " C. */
say; say 'root1 =' r1 /* " " " " 1st root*/
say 'root2 =' r2 /* " " " " 2nd root*/
exit /*stick a fork in it, we're all done. */ /*────────────────────────────────────────────────────────────────────────────*/ quadratic: parse arg aa,bb,cc /*obtain the specified three arguments.*/
$=sqrt(bb**2-4*aa*cc); L=length($) /*compute SQRT (which may be complex).*/
r=1/(aa+aa); ?=right($,1)=='i' /*compute reciprocal of 2*aa; Complex?*/
if ? then do; r1= -bb *r; r2=r1; r1j=left($,L-1)*r; r2j=-r1j; end
else do; r1=(-bb+$)*r; r2=(-bb-$)*r; r1j=0; r2j= 0; end
return
/*────────────────────────────────────────────────────────────────────────────*/ sqrt: procedure; parse arg x 1 ox; if x=0 then return 0; d=digits(); m.=9
numeric digits 9; numeric form; h=d+6; x=abs(x)
parse value format(x,2,1,,0) 'E0' with g 'E' _ .; g=g *.5'e'_ %2
do j=0 while h>9; m.j=h; h=h%2+1; end /*j*/
do k=j+5 to 0 by -1; numeric digits m.k; g=(g+x/g)*.5; end /*k*/
numeric digits d; return (g/1)left('i',ox<0) /*make complex if OX<0.*/</lang>
output when using the input of: 1 -10e5 1
a = 1
b = -1000000
c = 1
root1 = 1000000
root2 = 0.000001
The following output is when Regina 3.9.1 REXX is used.
output when using the input of: 1 -10e9 1
a = 1
b = -10000000000
c = 1
root1 = 1.000000000E+10
root2 = 1E-10
The following output is when R4 REXX is used.
output when using the input of: 1 -10e9 1
a = 1
b = -1E+10
c = 1
root1 = 1E+10
root2 = 0.0000000001
output when using the input of: 3 2 1
a = 3
b = 2
c = 1
root1 = -0.3333333333+0.4714045208i
root2 = -0.3333333333-0.4714045208i
output when using the input of: 1 0 1
a = 1
b = 0
c = 1
root1 = 0+1i
root2 = 0-1i
Version 2
<lang rexx>/* REXX ***************************************************************
- 26.07.2913 Walter Pachl
- /
Numeric Digits 30
Parse Arg a b c 1 alist
Select
When a= | a='?' Then
Call exit 'rexx qgl a b c solves a*x**2+b*x+c'
When words(alist)<>3 Then
Call exit 'three numbers are required'
Otherwise
Nop
End
gl=a'*x**2'
Select
When b<0 Then gl=gl||b'*x'
When b>0 Then gl=gl||'+'||b'*x'
Otherwise Nop
End
Select
When c<0 Then gl=gl||c
When c>0 Then gl=gl||'+'||c
Otherwise Nop
End
Say gl '= 0'
d=b**2-4*a*c If d<0 Then Do dd=sqrt(-d) r=-b/(2*a) i=dd/(2*a) x1=r'+'i'i' x2=r'-'i'i' End Else Do dd=sqrt(d) x1=(-b+dd)/(2*a) x2=(-b-dd)/(2*a) End Say 'x1='||x1 Say 'x2='||x2 Exit
sqrt: /* REXX ***************************************************************
- EXEC to calculate the square root of x with high precision
- /
Parse Arg x prec=digits() prec1=2*prec eps=10**(-prec1) k = 1 Numeric Digits prec1 r0= x r = 1 Do i=1 By 1 Until r=r0 | (abs(r*r-x)<eps) r0 = r r = (r + x/r) / 2 k = min(prec1,2*k) Numeric Digits (k + 5) End Numeric Digits prec Return (r+0)
exit: Say arg(1) </lang>
- Output:
Version 1:
a = 1
b = -1
c = 0
root1 = 1
root2 = 0
Version 2:
1*x**2-1.0000000001*x+1.e-9 = 0
x1=0.9999999991000000000025
x2=0.0000000009999999999975
Ring
<lang> x1 = 0 x2 = 0 quadratic(3, 4, 4/3.0) # [-2/3] see "x1 = " + x1 + " x2 = " + x2 + nl quadratic(3, 2, -1) # [1/3, -1] see "x1 = " + x1 + " x2 = " + x2 + nl quadratic(-2, 7, 15) # [-3/2, 5] see "x1 = " + x1 + " x2 = " + x2 + nl quadratic(1, -2, 1) # [1] see "x1 = " + x1 + " x2 = " + x2 + nl
func quadratic a, b, c
sqrtDiscriminant = sqrt(pow(b,2) - 4*a*c)
x1 = (-b + sqrtDiscriminant) / (2.0*a)
x2 = (-b - sqrtDiscriminant) / (2.0*a)
return [x1, x2]
</lang>
Ruby
The CMath#sqrt method will return a Complex instance if necessary. <lang ruby>require 'cmath'
def quadratic(a, b, c)
sqrt_discriminant = CMath.sqrt(b**2 - 4*a*c) [(-b + sqrt_discriminant) / (2.0*a), (-b - sqrt_discriminant) / (2.0*a)]
end
p quadratic(3, 4, 4/3.0) # [-2/3] p quadratic(3, 2, -1) # [1/3, -1] p quadratic(3, 2, 1) # [(-1/3 + sqrt(2/9)i), (-1/3 - sqrt(2/9)i)] p quadratic(1, 0, 1) # [(0+i), (0-i)] p quadratic(1, -1e6, 1) # [1e6, 1e-6] p quadratic(-2, 7, 15) # [-3/2, 5] p quadratic(1, -2, 1) # [1] p quadratic(1, 3, 3) # [(-3 + sqrt(3)i)/2), (-3 - sqrt(3)i)/2)]</lang>
- Output:
[-0.6666666666666666, -0.6666666666666666] [0.3333333333333333, -1.0] [(-0.3333333333333333+0.47140452079103173i), (-0.3333333333333333-0.47140452079103173i)] [(0.0+1.0i), (0.0-1.0i)] [999999.999999, 1.00000761449337e-06] [-1.5, 5.0] [1.0, 1.0] [(-1.5+0.8660254037844386i), (-1.5-0.8660254037844386i)]
Run BASIC
<lang runbasic>print "FOR 1,2,3 => ";quad$(1,2,3) print "FOR 4,5,6 => ";quad$(4,5,6)
FUNCTION quad$(a,b,c)
d = b^2-4 * a*c
x = -1*b
if d<0 then
quad$ = str$(x/(2*a));" +i";str$(sqr(abs(d))/(2*a))+" , "+str$(x/(2*a));" -i";str$(sqr(abs(d))/abs(2*a))
else
quad$ = str$(x/(2*a)+sqr(d)/(2*a))+" , "+str$(x/(2*a)-sqr(d)/(2*a))
end if
END FUNCTION</lang>
FOR 1,2,3 => -1 +i1.41421356 , -1 -i1.41421356 FOR 4,5,6 => -0.625 +i1.05326872 , -0.625 -i1.05326872
Scala
Using Complex class from task Arithmetic/Complex. <lang scala>import ArithmeticComplex._ object QuadraticRoots {
def solve(a:Double, b:Double, c:Double)={
val d = b*b-4.0*a*c
val aa = a+a
if (d < 0.0) { // complex roots
val re= -b/aa;
val im = math.sqrt(-d)/aa;
(Complex(re, im), Complex(re, -im))
}
else { // real roots
val re=if (b < 0.0) (-b+math.sqrt(d))/aa else (-b -math.sqrt(d))/aa
(re, (c/(a*re)))
}
}
}</lang> Usage: <lang scala>val equations=Array(
(1.0, 22.0, -1323.0), // two distinct real roots (6.0, -23.0, 20.0), // with a != 1.0 (1.0, -1.0e9, 1.0), // with one root near zero (1.0, 2.0, 1.0), // one real root (double root) (1.0, 0.0, 1.0), // two imaginary roots (1.0, 1.0, 1.0) // two complex roots
);
equations.foreach{v =>
val (a,b,c)=v
println("a=%g b=%g c=%g".format(a,b,c))
val roots=solve(a, b, c)
println("x1="+roots._1)
if(roots._1 != roots._2) println("x2="+roots._2)
println
}</lang>
- Output:
a=1.00000 b=22.0000 c=-1323.00 x1=-49.0 x2=27.0 a=6.00000 b=-23.0000 c=20.0000 x1=2.5 x2=1.3333333333333333 a=1.00000 b=-1.00000e+09 c=1.00000 x1=1.0E9 x2=1.0E-9 a=1.00000 b=2.00000 c=1.00000 x1=-1.0 a=1.00000 b=0.00000 c=1.00000 x1=-0.0 + 1.0i x2=-0.0 + -1.0i a=1.00000 b=1.00000 c=1.00000 x1=-0.5 + 0.8660254037844386i x2=-0.5 + -0.8660254037844386i
Scheme
<lang scheme>(define (quadratic a b c) (if (= a 0) (if (= b 0) 'fail (- (/ c b))) (let ((delta (- (* b b) (* 4 a c)))) (if (and (real? delta) (> delta 0)) (let ((u (+ b (* (if (>= b 0) 1 -1) (sqrt delta))))) (list (/ u -2 a) (/ (* -2 c) u))) (list (/ (- (sqrt delta) b) 2 a) (/ (+ (sqrt delta) b) -2 a))))))
- examples
(quadratic 1 -1 -1)
- (1.618033988749895 -0.6180339887498948)
(quadratic 1 0 -2)
- (-1.4142135623730951 1.414213562373095)
(quadratic 1 0 2)
- (0+1.4142135623730951i 0-1.4142135623730951i)
(quadratic 1+1i 2 5)
- (-1.0922677260818898-1.1884256155834088i 0.09226772608188982+2.1884256155834088i)
(quadratic 0 4 3)
- -3/4
(quadratic 0 0 1)
- fail
(quadratic 1 2 0)
- (-2 0)
(quadratic 1 2 1)
- (-1 -1)
(quadratic 1 -1e5 1)
- (99999.99999 1.0000000001000001e-05)</lang>
Seed7
<lang seed7>$ include "seed7_05.s7i";
include "float.s7i"; include "math.s7i";
const type: roots is new struct
var float: x1 is 0.0; var float: x2 is 0.0; end struct;
const func roots: solve (in float: a, in float: b, in float: c) is func
result
var roots: solution is roots.value;
local
var float: sd is 0.0;
var float: x is 0.0;
begin
sd := sqrt(b**2 - 4.0 * a * c);
if b < 0.0 then
x := (-b + sd) / 2.0 * a;
solution.x1 := x;
solution.x2 := c / (a * x);
else
x := (-b - sd) / 2.0 * a;
solution.x1 := c / (a * x);
solution.x2 := x;
end if;
end func;
const proc: main is func
local
var roots: r is roots.value;
begin
r := solve(1.0, -10.0E5, 1.0);
writeln("X1 = " <& r.x1 digits 6 <& " X2 = " <& r.x2 digits 6);
end func;</lang>
- Output:
X1 = 1000000.000000 X2 = 0.000001
Sidef
<lang ruby>var sets = [
[1, 2, 1],
[1, 2, 3],
[1, -2, 1],
[1, 0, -4],
[1, -1e6, 1],
]
func quadroots(a, b, c) {
var root = sqrt(b**2 - 4*a*c)
[(-b + root) / (2 * a),
(-b - root) / (2 * a)]
}
sets.each { |coefficients|
say ("Roots for #{coefficients}",
"=> (#{quadroots(coefficients...).join(', ')})")
}</lang>
- Output:
Roots for [1, 2, 1]=> (-1, -1) Roots for [1, 2, 3]=> (-1+1.41421356237309504880168872420969807856967187538i, -1-1.41421356237309504880168872420969807856967187538i) Roots for [1, -2, 1]=> (1, 1) Roots for [1, 0, -4]=> (2, -2) Roots for [1, -1000000, 1]=> (999999.999998999999999998999999999997999999999995, 0.00000100000000000100000000000200000000000500000000002)
Stata
<lang stata>mata
- polyroots((-2,0,1))
1 2 +-----------------------------+ 1 | 1.41421356 -1.41421356 | +-----------------------------+
- polyroots((2,0,1))
1 2 +-------------------------------+ 1 | -1.41421356i 1.41421356i | +-------------------------------+</lang>
Tcl
<lang tcl>package require math::complexnumbers namespace import math::complexnumbers::complex math::complexnumbers::tostring
proc quadratic {a b c} {
set discrim [expr {$b**2 - 4*$a*$c}]
set roots [list]
if {$discrim < 0} {
set term1 [expr {(-1.0*$b)/(2*$a)}]
set term2 [expr {sqrt(abs($discrim))/(2*$a)}]
lappend roots [tostring [complex $term1 $term2]] \
[tostring [complex $term1 [expr {-1 * $term2}]]]
} elseif {$discrim == 0} {
lappend roots [expr {-1.0*$b / (2*$a)}]
} else {
lappend roots [expr {(-1*$b + sqrt($discrim)) / (2 * $a)}] \
[expr {(-1*$b - sqrt($discrim)) / (2 * $a)}]
}
return $roots
}
proc report_quad {a b c} {
puts [format "%sx**2 + %sx + %s = 0" $a $b $c]
foreach root [quadratic $a $b $c] {
puts " x = $root"
}
}
- examples on this page
report_quad 3 4 [expr {4/3.0}] ;# {-2/3} report_quad 3 2 -1 ;# {1/3, -1} report_quad 3 2 1 ;# {(-1/3 + sqrt(2/9)i), (-1/3 - sqrt(2/9)i)} report_quad 1 0 1 ;# {(0+i), (0-i)} report_quad 1 -1e6 1 ;# {1e6, 1e-6}
- examples from http://en.wikipedia.org/wiki/Quadratic_equation
report_quad -2 7 15 ;# {5, -3/2} report_quad 1 -2 1 ;# {1} report_quad 1 3 3 ;# {(-3 - sqrt(3)i)/2), (-3 + sqrt(3)i)/2)}</lang>
- Output:
3x**2 + 4x + 1.3333333333333333 = 0
x = -0.6666666666666666
3x**2 + 2x + -1 = 0
x = 0.3333333333333333
x = -1.0
3x**2 + 2x + 1 = 0
x = -0.3333333333333333+0.47140452079103173i
x = -0.3333333333333333-0.47140452079103173i
1x**2 + 0x + 1 = 0
x = i
x = -i
1x**2 + -1e6x + 1 = 0
x = 999999.999999
x = 1.00000761449337e-6
-2x**2 + 7x + 15 = 0
x = -1.5
x = 5.0
1x**2 + -2x + 1 = 0
x = 1.0
1x**2 + 3x + 3 = 0
x = -1.5+0.8660254037844386i
x = -1.5-0.8660254037844386i
TI-89 BASIC
TI-89 BASIC has built-in numeric and algebraic solvers. <lang>solve(x^2-1E9x+1.0)</lang> returns
x=1.E-9 or x=1.E9
zkl
zkl doesn't have a complex number package.
<lang zkl>fcn quadratic(a,b,c){ b=b.toFloat();
println("Roots of a quadratic function %s, %s, %s".fmt(a,b,c));
d,a2:=(b*b - 4*a*c), a+a;
if(d>0){
sd:=d.sqrt();
println(" the real roots are %s and %s".fmt((-b + sd)/a2,(-b - sd)/a2));
}
else if(d==0) println(" the single root is ",-b/a2);
else{
sd:=(-d).sqrt();
println(" the complex roots are %s and \U00B1;%si".fmt(-b/a2,sd/a2));
}
}</lang> <lang zkl>foreach a,b,c in (T( T(1,-2,1), T(1,-3,2), T(1,0,1), T(1,-1.0e10,1), T(1,2,3), T(2,-1,-6)) ){
quadratic(a,b,c)
}</lang>
- Output:
Roots of a quadratic function 1, -2, 1 the single root is 1 Roots of a quadratic function 1, -3, 2 the real roots are 2 and 1 Roots of a quadratic function 1, 0, 1 the complex roots are 0 and ±1i Roots of a quadratic function 1, -1e+10, 1 the real roots are 1e+10 and 0 Roots of a quadratic function 1, 2, 3 the complex roots are -1 and ±1.41421i Roots of a quadratic function 2, -1, -6 the real roots are 2 and -1.5
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