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270 | with Ada.Numerics;
with Ada.Numerics.Generic_Elementary_Functions;
with Ada.Numerics.Generic_Complex_Elementary_Functions;
-- with Ada.Text_IO;
package body Complex_Polynomial_Roots is
package GEF is new
Ada.Numerics.Generic_Elementary_Functions(Complex_Types.Real'Base);
package GCEF is new
Ada.Numerics.Generic_Complex_Elementary_Functions(Complex_Types);
use Ada.Numerics, GEF, GCEF;
epsilon : constant Real:= Real'Model_Epsilon;
epsilon2 : constant Real:= epsilon**2;
procedure Solve (a,b,c: Real; r1,r2: out Complex) is
dis_sqrt: Complex; denom: Real;
begin
if abs a <= epsilon then
raise dominant_coefficient_a_is_zero with "there is at most 1 root";
end if;
dis_sqrt := Sqrt((b*b - a * 4.0 * c, 0.0));
denom := 1.0 / (2.0 * a);
r1 := (-b + dis_sqrt) * denom;
r2 := (-b - dis_sqrt) * denom;
end Solve;
procedure Solve (a,b,c: Complex; r1,r2: out Complex) is
dis_sqrt: Complex; denom: Complex;
begin
if abs a <= epsilon then
raise dominant_coefficient_a_is_zero with "there is at most 1 root";
end if;
dis_sqrt := Sqrt(b*b - a * 4.0 * c);
denom := 1.0 / (2.0 * a);
r1 := (-b + dis_sqrt) * denom;
r2 := (-b - dis_sqrt) * denom;
end Solve;
third : constant := 1.0/3.0;
sqrt3 : constant Real := Sqrt(3.0);
procedure Solve (a,b,c,d: Real; r1,r2,r3: out Complex) is
p,q, r, dr, qsr3, ymx, phi: Real;
begin
if abs a <= epsilon then
raise dominant_coefficient_a_is_zero with "there are at most 2 roots";
end if;
if abs d < epsilon then
-- "d = 0" case.
-- Then we have to solve: a x^3 + b x^2 + c x = x * (a x^2 + b x + c).
r1 := (0.0, 0.0);
Solve (a,b,c, r2,r3);
else
-- Equivalent equation: y^3 + 3p y + 2q = 0 where y = x + b/(3*a)
ymx:= b/(3.0*a); -- = y-x. Then x = y - ymx
p:= (3.0 * a*c - b**2) / (9.0 * a**2);
q:= (b**3) / (27.0 * a**3) - b*c / (6.0*a**2) + d / (2.0 * a);
if abs p < epsilon then -- y^3 = -2q
if q < 0.0 then -- y^3 = 2|q|
r1 := ( 2.0 * (abs q)**third - ymx, 0.0);
else -- y^3 = -2|q|
r1 := (-2.0 * (abs q)**third - ymx, 0.0);
end if;
r2 := r1;
r3 := r1;
else
r := Sqrt(abs p);
if q < 0.0 then
r := -r;
end if;
qsr3 := q / (r**3);
dr := r+r;
if p < 0.0 then -- also: p < -epsilon < 0
if p**3 + q**2 <= 0.0 then
phi := Arccos (qsr3);
r1 := ( -dr * Cos (phi * third) - ymx, 0.0 );
r2 := ( dr * Cos ((Pi - phi) * third) - ymx, 0.0);
r3 := ( dr * Cos ((Pi + phi) * third) - ymx, 0.0);
else
phi := Arccosh (qsr3);
p := Cosh (phi* third);
r1 := ( -dr * p - ymx, 0.0 );
r2 := ( r * p - ymx, sqrt3 * r * Sinh (phi* third));
r3 := Conjugate (r2);
end if;
else -- p > epsilon > 0
phi := Arcsinh (qsr3);
p := Sinh (phi* third);
r1 := ( -dr * p - ymx, 0.0 );
r2 := ( r * p - ymx, sqrt3 * r * Cosh(phi* third));
r3 := Conjugate (r2);
end if;
end if;
end if;
end Solve;
procedure Solve_A (a,b,c,d,e: Real; r1,r2,r3,r4: out Complex) is
-- Bronstein-Semendjajev
bb,cc,dd,ee, y, L2,L, f: Real;
yc, d1,d2: Complex;
begin
if abs a <= epsilon then
raise dominant_coefficient_a_is_zero with "there are at most 3 roots";
end if;
if abs e < epsilon then
r1:= (0.0,0.0);
Solve (a,b,c,d, r2,r3,r4);
else
bb := b / a;
cc := c / a;
dd := d / a;
ee := e / a;
-- Solve the resolvant cubic
Solve ( 8.0, -4.0*cc, 2.0*bb*dd - 8.0*ee, ee*(4.0*cc-bb**2)-dd**2, yc,d1,d2 );
y := Re(yc);
-- y is the cubic's root in \IR
L2 := 8.0*y + bb**2 - 4.0*cc;
if abs L2 < epsilon2 then
r1 := (0.0, 0.0);
elsif L2 > 0.0 then
L := Sqrt(L2);
f := (bb*y - dd) / L;
Solve ( 1.0, 0.5 * (bb + L), y + f, r1,r2 );
Solve ( 1.0, 0.5 * (bb - L), y - f, r3,r4 );
else
L := Sqrt(-L2);
f := (bb*y - dd) / L;
Solve ( (1.0, 0.0), 0.5 * (bb, -L), (y, - f), r1,r2 );
Solve ( (1.0, 0.0), 0.5 * (bb, L), (y, f), r3,r4 );
end if;
end if;
end Solve_A;
procedure Solve_B (a,b,c,d,e: Real; r1,r2,r3,r4: out Complex) is
-- Ferrari (John M. Gamble)
b4,bb,cc,dd,ee,f,g,h: Real;
alpha, beta, gamma, rho: Complex;
p, q, z: Complex;
begin
if abs a <= epsilon then
raise dominant_coefficient_a_is_zero with "there are at most 3 roots";
end if;
if abs e < epsilon then
r1:= (0.0,0.0);
Solve(a,b,c,d, r2,r3,r4);
else
--
-- First step: Divide by the leading coefficient.
--
bb:= b / a;
cc:= c / a;
dd:= d / a;
ee:= e / a;
--
-- Second step: simplify the equation to the
-- "resolvant cubic" y**4 + fy**2 + gy + h.
--
-- (This is done by setting x:= y - b/4).
--
b4:= bb * 0.25;
--
-- The f, g, and h values are:
--
f:= cc - 6.0 * b4 * b4;
g:= dd + 2.0 * b4 * (-cc + 4.0 * b4 * b4);
h:= ee + b4 * (-dd + b4 * (cc - 3.0 * b4 * b4));
if abs h < epsilon then
--
-- Special case: h = 0. We have a cubic times y.
--
r1 := (0.0,0.0);
Solve (1.0, 0.0, f, g, r2,r3,r4);
elsif abs g < epsilon then
--
-- Another special case: g = 0. We have a quadratic
-- with y-squared.
--
Solve (1.0, f, h, p, q);
p := Sqrt (p);
q := Sqrt (q);
r1 := p;
r2 := -p;
r3 := q;
r4 := -q;
else
--
-- Special cases don't apply, so continue on with Ferrari's
-- method. This involves setting up the resolvant cubic
-- as the product of two quadratics.
--
-- After setting up conditions that guarantee that the
-- coefficients come out right (including the zero value
-- for the third-power term), we wind up with a 6th
-- degree polynomial with, fortunately, only even-powered
-- terms. In other words, a cubic with z:= y**2.
--
-- Take a root of that equation, and get the
-- quadratics from it.
--
Solve(1.0, 2.0*f, f*f - 4.0*h, -g*g, z, p, q); -- p, q: dummy
alpha := Sqrt(z);
rho := g/alpha;
beta := (f + z - rho)*0.5;
gamma := (f + z + rho)*0.5;
--
Solve ((1.0, 0.0), alpha, beta, r1,r2);
Solve ((1.0, 0.0), -alpha, gamma, r3,r4);
end if;
r1:= r1 - b4;
r2:= r2 - b4;
r3:= r3 - b4;
r4:= r4 - b4;
end if;
end Solve_B;
procedure Solve_compare (a,b,c,d,e: Real; r1,r2,r3,r4: out Complex) is
type Method is (mA, mB);
r : array (Method, 1..4) of Complex;
n : array (Method) of Real;
mm : Method;
p, x : Complex;
begin
if abs a <= epsilon then
raise dominant_coefficient_a_is_zero with "there are at most 3 roots";
end if;
if abs e < epsilon then
-- "e = 0" case. Then we solve: a x^4 + b x^3 + c x^2 + d x = 0,
-- and x = 0 is an obvious root.
-- Then we have to solve: a x^3 + b x^2 + c x + d = 0
-- to find the other roots.
r1 := (0.0, 0.0);
Solve (a,b,c,d, r2,r3,r4);
else
Solve_A (a,b,c,d,e, r(mA, 1), r(mA, 2), r(mA, 3), r(mA, 4));
Solve_B (a,b,c,d,e, r(mB, 1), r(mB, 2), r(mB, 3), r(mB, 4));
for m in Method loop
n (m) := 0.0;
-- We evaluate the polynomial for root j.
for j in 1 .. 4 loop
x := r(m, j);
p := (((a * x + b) * x + c) * x + d) * x + e;
-- Theoretically, p = 0.
n(m) := n(m) + abs p;
end loop;
end loop;
if n(mA) < n(mB) then -- Choosing the method with the smallest error.
mm:= mA; -- Ada.Text_IO.Put('A');
else
mm:= mB; -- Ada.Text_IO.Put('B');
end if;
r1 := r (mm, 1);
r2 := r (mm, 2);
r3 := r (mm, 3);
r4 := r (mm, 4);
end if;
end Solve_compare;
procedure Solve (a,b,c,d,e: Real; r1,r2,r3,r4: out Complex)
renames Solve_compare;
end Complex_Polynomial_Roots;
|