Solución de una ecuación cúbica
https://stackoverflow.com/questions/1829330
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Como parte de un programa que estoy escribiendo, necesito resolver una ecuación cúbica exactamente (en lugar de utilizar un buscador de raíces numérica):
a*x**3 + b*x**2 + c*x + d = 0.
Estoy tratando de utilizar las ecuaciones de aquí . Sin embargo, considere el siguiente código (esto es Python, pero es bastante genérico código):
a = 1.0
b = 0.0
c = 0.2 - 1.0
d = -0.7 * 0.2
q = (3*a*c - b**2) / (9 * a**2)
r = (9*a*b*c - 27*a**2*d - 2*b**3) / (54*a**3)
print "q = ",q
print "r = ",r
delta = q**3 + r**2
print "delta = ",delta
# here delta is less than zero so we use the second set of equations from the article:
rho = (-q**3)**0.5
# For x1 the imaginary part is unimportant since it cancels out
s_real = rho**(1./3.)
t_real = rho**(1./3.)
print "s [real] = ",s_real
print "t [real] = ",t_real
x1 = s_real + t_real - b / (3. * a)
print "x1 = ", x1
print "should be zero: ",a*x1**3+b*x1**2+c*x1+d
Sin embargo, la salida es:
q = -0.266666666667
r = 0.07
delta = -0.014062962963
s [real] = 0.516397779494
t [real] = 0.516397779494
x1 = 1.03279555899
should be zero: 0.135412149064
lo que la salida no es cero, y así x1 no es realmente una solución. ¿Hay algún error en el artículo de Wikipedia?
PD: Sé que numpy.roots va a resolver este tipo de ecuaciones, pero tengo que hacer esto para millones de ecuaciones y así que necesito para implementar esta trabajando en las matrices de coeficientes
.Solución
(rho^(1/3), theta/3) la notación de Wikipedia no significa que rho^(1/3) es la parte real y theta/3 es la parte imaginaria. Más bien, esto es, en coordenadas polares. Por lo tanto, si desea que la parte real, que tomaría rho^(1/3) * cos(theta/3).
Hice estos cambios en su código y funcionó para mí:
theta = arccos(r/rho)
s_real = rho**(1./3.) * cos( theta/3)
t_real = rho**(1./3.) * cos(-theta/3)
(Por supuesto, s_real = t_real aquí porque cos es par.)
Otros consejos
He mirado en el artículo de Wikipedia y su programa.
También resolvió la ecuación usando Wolfram Alpha y los resultados allí no coinciden con lo que se obtiene.
Me acaba de ir a través de su programa en cada paso, uso una gran cantidad de declaraciones de impresión y obtener cada resultado intermedio. A continuación, seguir adelante con una calculadora y hacerlo usted mismo.
No puedo encontrar lo que está pasando, pero en sus cálculos de la mano y el programa divergen es un buen lugar para buscar.
Aquí está la solución A. Rex en JavaScript:
a = 1.0;
b = 0.0;
c = 0.2 - 1.0;
d = -0.7 * 0.2;
q = (3*a*c - Math.pow(b, 2)) / (9 * Math.pow(a, 2));
r = (9*a*b*c - 27*Math.pow(a, 2)*d - 2*Math.pow(b, 3)) / (54*Math.pow(a, 3));
console.log("q = "+q);
console.log("r = "+r);
delta = Math.pow(q, 3) + Math.pow(r, 2);
console.log("delta = "+delta);
// here delta is less than zero so we use the second set of equations from the article:
rho = Math.pow((-Math.pow(q, 3)), 0.5);
theta = Math.acos(r/rho);
// For x1 the imaginary part is unimportant since it cancels out
s_real = Math.pow(rho, (1./3.)) * Math.cos( theta/3);
t_real = Math.pow(rho, (1./3.)) * Math.cos(-theta/3);
console.log("s [real] = "+s_real);
console.log("t [real] = "+t_real);
x1 = s_real + t_real - b / (3. * a);
console.log("x1 = "+x1);
console.log("should be zero: "+(a*Math.pow(x1, 3)+b*Math.pow(x1, 2)+c*x1+d));
Aquí, puse una ecuación cúbica (con coeficientes complejos) solver.
#include <string>
#include <fstream>
#include <iostream>
#include <cstdlib>
using namespace std;
#define PI 3.141592
long double complex_multiply_r(long double xr, long double xi, long double yr, long double yi) {
return (xr * yr - xi * yi);
}
long double complex_multiply_i(long double xr, long double xi, long double yr, long double yi) {
return (xr * yi + xi * yr);
}
long double complex_triple_multiply_r(long double xr, long double xi, long double yr, long double yi, long double zr, long double zi) {
return (xr * yr * zr - xi * yi * zr - xr * yi * zi - xi * yr * zi);
}
long double complex_triple_multiply_i(long double xr, long double xi, long double yr, long double yi, long double zr, long double zi) {
return (xr * yr * zi - xi * yi * zi + xr * yi * zr + xi * yr * zr);
}
long double complex_quadraple_multiply_r(long double xr, long double xi, long double yr, long double yi, long double zr, long double zi, long double wr, long double wi) {
long double z1r, z1i, z2r, z2i;
z1r = complex_multiply_r(xr, xi, yr, yi);
z1i = complex_multiply_i(xr, xi, yr, yi);
z2r = complex_multiply_r(zr, zi, wr, wi);
z2i = complex_multiply_i(zr, zi, wr, wi);
return (complex_multiply_r(z1r, z1i, z2r, z2i));
}
long double complex_quadraple_multiply_i(long double xr, long double xi, long double yr, long double yi, long double zr, long double zi, long double wr, long double wi) {
long double z1r, z1i, z2r, z2i;
z1r = complex_multiply_r(xr, xi, yr, yi);
z1i = complex_multiply_i(xr, xi, yr, yi);
z2r = complex_multiply_r(zr, zi, wr, wi);
z2i = complex_multiply_i(zr, zi, wr, wi);
return (complex_multiply_i(z1r, z1i, z2r, z2i));
}
long double complex_divide_r(long double xr, long double xi, long double yr, long double yi) {
return ((xr * yr + xi * yi) / (yr * yr + yi * yi));
}
long double complex_divide_i(long double xr, long double xi, long double yr, long double yi) {
return ((-xr * yi + xi * yr) / (yr * yr + yi * yi));
}
long double complex_root_r(long double xr, long double xi) {
long double r, theta;
r = sqrt(xr*xr + xi*xi);
if (r != 0.0) {
if (xr >= 0 && xi >= 0) {
theta = atan(xi / xr);
}
else if (xr < 0 && xi >= 0) {
theta = PI - abs(atan(xi / xr));
}
else if (xr < 0 && xi < 0) {
theta = PI + abs(atan(xi / xr));
}
else {
theta = 2.0 * PI + atan(xi / xr);
}
return (sqrt(r) * cos(theta / 2.0));
}
else {
return 0.0;
}
}
long double complex_root_i(long double xr, long double xi) {
long double r, theta;
r = sqrt(xr*xr + xi*xi);
if (r != 0.0) {
if (xr >= 0 && xi >= 0) {
theta = atan(xi / xr);
}
else if (xr < 0 && xi >= 0) {
theta = PI - abs(atan(xi / xr));
}
else if (xr < 0 && xi < 0) {
theta = PI + abs(atan(xi / xr));
}
else {
theta = 2.0 * PI + atan(xi / xr);
}
return (sqrt(r) * sin(theta / 2.0));
}
else {
return 0.0;
}
}
long double complex_cuberoot_r(long double xr, long double xi) {
long double r, theta;
r = sqrt(xr*xr + xi*xi);
if (r != 0.0) {
if (xr >= 0 && xi >= 0) {
theta = atan(xi / xr);
}
else if (xr < 0 && xi >= 0) {
theta = PI - abs(atan(xi / xr));
}
else if (xr < 0 && xi < 0) {
theta = PI + abs(atan(xi / xr));
}
else {
theta = 2.0 * PI + atan(xi / xr);
}
return (pow(r, 1.0 / 3.0) * cos(theta / 3.0));
}
else {
return 0.0;
}
}
long double complex_cuberoot_i(long double xr, long double xi) {
long double r, theta;
r = sqrt(xr*xr + xi*xi);
if (r != 0.0) {
if (xr >= 0 && xi >= 0) {
theta = atan(xi / xr);
}
else if (xr < 0 && xi >= 0) {
theta = PI - abs(atan(xi / xr));
}
else if (xr < 0 && xi < 0) {
theta = PI + abs(atan(xi / xr));
}
else {
theta = 2.0 * PI + atan(xi / xr);
}
return (pow(r, 1.0 / 3.0) * sin(theta / 3.0));
}
else {
return 0.0;
}
}
void main() {
long double a[2], b[2], c[2], d[2], minusd[2];
long double r, theta;
cout << "ar?";
cin >> a[0];
cout << "ai?";
cin >> a[1];
cout << "br?";
cin >> b[0];
cout << "bi?";
cin >> b[1];
cout << "cr?";
cin >> c[0];
cout << "ci?";
cin >> c[1];
cout << "dr?";
cin >> d[0];
cout << "di?";
cin >> d[1];
if (b[0] == 0.0 && b[1] == 0.0 && c[0] == 0.0 && c[1] == 0.0) {
if (d[0] == 0.0 && d[1] == 0.0) {
cout << "x1r: 0.0 \n";
cout << "x1i: 0.0 \n";
cout << "x2r: 0.0 \n";
cout << "x2i: 0.0 \n";
cout << "x3r: 0.0 \n";
cout << "x3i: 0.0 \n";
}
else {
minusd[0] = -d[0];
minusd[1] = -d[1];
r = sqrt(minusd[0]*minusd[0] + minusd[1]*minusd[1]);
if (minusd[0] >= 0 && minusd[1] >= 0) {
theta = atan(minusd[1] / minusd[0]);
}
else if (minusd[0] < 0 && minusd[1] >= 0) {
theta = PI - abs(atan(minusd[1] / minusd[0]));
}
else if (minusd[0] < 0 && minusd[1] < 0) {
theta = PI + abs(atan(minusd[1] / minusd[0]));
}
else {
theta = 2.0 * PI + atan(minusd[1] / minusd[0]);
}
cout << "x1r: " << pow(r, 1.0 / 3.0) * cos(theta / 3.0) << "\n";
cout << "x1i: " << pow(r, 1.0 / 3.0) * sin(theta / 3.0) << "\n";
cout << "x2r: " << pow(r, 1.0 / 3.0) * cos((theta + 2.0 * PI) / 3.0) << "\n";
cout << "x2i: " << pow(r, 1.0 / 3.0) * sin((theta + 2.0 * PI) / 3.0) << "\n";
cout << "x3r: " << pow(r, 1.0 / 3.0) * cos((theta + 4.0 * PI) / 3.0) << "\n";
cout << "x3i: " << pow(r, 1.0 / 3.0) * sin((theta + 4.0 * PI) / 3.0) << "\n";
}
}
else {
// find eigenvalues
long double term0[2], term1[2], term2[2], term3[2], term3buf[2];
long double first[2], second[2], second2[2], third[2];
term0[0] = -4.0 * complex_quadraple_multiply_r(a[0], a[1], c[0], c[1], c[0], c[1], c[0], c[1]);
term0[1] = -4.0 * complex_quadraple_multiply_i(a[0], a[1], c[0], c[1], c[0], c[1], c[0], c[1]);
term0[0] += complex_quadraple_multiply_r(b[0], b[1], b[0], b[1], c[0], c[1], c[0], c[1]);
term0[1] += complex_quadraple_multiply_i(b[0], b[1], b[0], b[1], c[0], c[1], c[0], c[1]);
term0[0] += -4.0 * complex_quadraple_multiply_r(b[0], b[1], b[0], b[1], b[0], b[1], d[0], d[1]);
term0[1] += -4.0 * complex_quadraple_multiply_i(b[0], b[1], b[0], b[1], b[0], b[1], d[0], d[1]);
term0[0] += 18.0 * complex_quadraple_multiply_r(a[0], a[1], b[0], b[1], c[0], c[1], d[0], d[1]);
term0[1] += 18.0 * complex_quadraple_multiply_i(a[0], a[1], b[0], b[1], c[0], c[1], d[0], d[1]);
term0[0] += -27.0 * complex_quadraple_multiply_r(a[0], a[1], a[0], a[1], d[0], d[1], d[0], d[1]);
term0[1] += -27.0 * complex_quadraple_multiply_i(a[0], a[1], a[0], a[1], d[0], d[1], d[0], d[1]);
term1[0] = -27.0 * complex_triple_multiply_r(a[0], a[1], a[0], a[1], d[0], d[1]);
term1[1] = -27.0 * complex_triple_multiply_i(a[0], a[1], a[0], a[1], d[0], d[1]);
term1[0] += 9.0 * complex_triple_multiply_r(a[0], a[1], b[0], b[1], c[0], c[1]);
term1[1] += 9.0 * complex_triple_multiply_i(a[0], a[1], b[0], b[1], c[0], c[1]);
term1[0] -= 2.0 * complex_triple_multiply_r(b[0], b[1], b[0], b[1], b[0], b[1]);
term1[1] -= 2.0 * complex_triple_multiply_i(b[0], b[1], b[0], b[1], b[0], b[1]);
term2[0] = 3.0 * complex_multiply_r(a[0], a[1], c[0], c[1]);
term2[1] = 3.0 * complex_multiply_i(a[0], a[1], c[0], c[1]);
term2[0] -= complex_multiply_r(b[0], b[1], b[0], b[1]);
term2[1] -= complex_multiply_i(b[0], b[1], b[0], b[1]);
term3[0] = complex_multiply_r(term1[0], term1[1], term1[0], term1[1]);
term3[1] = complex_multiply_i(term1[0], term1[1], term1[0], term1[1]);
term3[0] += 4.0 * complex_triple_multiply_r(term2[0], term2[1], term2[0], term2[1], term2[0], term2[1]);
term3[1] += 4.0 * complex_triple_multiply_i(term2[0], term2[1], term2[0], term2[1], term2[0], term2[1]);
term3buf[0] = term3[0];
term3buf[1] = term3[1];
term3[0] = complex_root_r(term3buf[0], term3buf[1]);
term3[1] = complex_root_i(term3buf[0], term3buf[1]);
if (term0[0] == 0.0 && term0[1] == 0.0 && term1[0] == 0.0 && term1[1] == 0.0) {
cout << "x1r: " << -pow(d[0], 1.0 / 3.0) << "\n";
cout << "x1i: " << 0.0 << "\n";
cout << "x2r: " << -pow(d[0], 1.0 / 3.0) << "\n";
cout << "x2i: " << 0.0 << "\n";
cout << "x3r: " << -pow(d[0], 1.0 / 3.0) << "\n";
cout << "x3i: " << 0.0 << "\n";
}
else {
// eigenvalue1
first[0] = complex_divide_r(complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1]), 3.0 * pow(2.0, 1.0 / 3.0) * a[0], 3.0 * pow(2.0, 1.0 / 3.0) * a[1]);
first[1] = complex_divide_i(complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1]), 3.0 * pow(2.0, 1.0 / 3.0) * a[0], 3.0 * pow(2.0, 1.0 / 3.0) * a[1]);
second[0] = complex_divide_r(pow(2.0, 1.0 / 3.0) * term2[0], pow(2.0, 1.0 / 3.0) * term2[1], 3.0 * complex_multiply_r(a[0], a[1], complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1])), 3.0 * complex_multiply_i(a[0], a[1], complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1])));
second[1] = complex_divide_i(pow(2.0, 1.0 / 3.0) * term2[0], pow(2.0, 1.0 / 3.0) * term2[1], 3.0 * complex_multiply_r(a[0], a[1], complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1])), 3.0 * complex_multiply_i(a[0], a[1], complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1])));
third[0] = complex_divide_r(b[0], b[1], 3.0 * a[0], 3.0 * a[1]);
third[1] = complex_divide_i(b[0], b[1], 3.0 * a[0], 3.0 * a[1]);
cout << "x1r: " << first[0] - second[0] - third[0] << "\n";
cout << "x1i: " << first[1] - second[1] - third[1] << "\n";
// eigenvalue2
first[0] = complex_divide_r(complex_multiply_r(1.0, -sqrt(3.0), complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1])), complex_multiply_i(1.0, -sqrt(3.0), complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1])), 6.0 * pow(2.0, 1.0 / 3.0) * a[0], 6.0 * pow(2.0, 1.0 / 3.0) * a[1]);
first[1] = complex_divide_i(complex_multiply_r(1.0, -sqrt(3.0), complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1])), complex_multiply_i(1.0, -sqrt(3.0), complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1])), 6.0 * pow(2.0, 1.0 / 3.0) * a[0], 6.0 * pow(2.0, 1.0 / 3.0) * a[1]);
second[0] = complex_divide_r(complex_multiply_r(1.0, sqrt(3.0), term2[0], term2[1]), complex_multiply_i(1.0, sqrt(3.0), term2[0], term2[1]), 3.0 * pow(2.0, 2.0 / 3.0) * complex_multiply_r(a[0], a[1], complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1])), 3.0 * pow(2.0, 2.0 / 3.0) * complex_multiply_i(a[0], a[1], complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1])));
second[1] = complex_divide_i(complex_multiply_r(1.0, sqrt(3.0), term2[0], term2[1]), complex_multiply_i(1.0, sqrt(3.0), term2[0], term2[1]), 3.0 * pow(2.0, 2.0 / 3.0) * complex_multiply_r(a[0], a[1], complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1])), 3.0 * pow(2.0, 2.0 / 3.0) * complex_multiply_i(a[0], a[1], complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1])));
third[0] = complex_divide_r(b[0], b[1], 3.0 * a[0], 3.0 * a[1]);
third[1] = complex_divide_i(b[0], b[1], 3.0 * a[0], 3.0 * a[1]);
cout << "x2r: " << -first[0] + second[0] - third[0] << "\n";
cout << "x2i: " << -first[1] + second[1] - third[1] << "\n";
// eigenvalue3
first[0] = complex_divide_r(complex_multiply_r(1.0, sqrt(3.0), complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1])), complex_multiply_i(1.0, sqrt(3.0), complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1])), 6.0 * pow(2.0, 1.0 / 3.0) * a[0], 6.0 * pow(2.0, 1.0 / 3.0) * a[1]);
first[1] = complex_divide_i(complex_multiply_r(1.0, sqrt(3.0), complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1])), complex_multiply_i(1.0, sqrt(3.0), complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1])), 6.0 * pow(2.0, 1.0 / 3.0) * a[0], 6.0 * pow(2.0, 1.0 / 3.0) * a[1]);
second[0] = complex_divide_r(complex_multiply_r(1.0, -sqrt(3.0), term2[0], term2[1]), complex_multiply_i(1.0, -sqrt(3.0), term2[0], term2[1]), 3.0 * pow(2.0, 2.0 / 3.0) * complex_multiply_r(a[0], a[1], complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1])), 3.0 * pow(2.0, 2.0 / 3.0) * complex_multiply_i(a[0], a[1], complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1])));
second[1] = complex_divide_i(complex_multiply_r(1.0, -sqrt(3.0), term2[0], term2[1]), complex_multiply_i(1.0, -sqrt(3.0), term2[0], term2[1]), 3.0 * pow(2.0, 2.0 / 3.0) * complex_multiply_r(a[0], a[1], complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1])), 3.0 * pow(2.0, 2.0 / 3.0) * complex_multiply_i(a[0], a[1], complex_cuberoot_r(term3[0] + term1[0], term3[1] + term1[1]), complex_cuberoot_i(term3[0] + term1[0], term3[1] + term1[1])));
third[0] = complex_divide_r(b[0], b[1], 3.0 * a[0], 3.0 * a[1]);
third[1] = complex_divide_i(b[0], b[1], 3.0 * a[0], 3.0 * a[1]);
cout << "x3r: " << -first[0] + second[0] - third[0] << "\n";
cout << "x3i: " << -first[1] + second[1] - third[1] << "\n";
}
}
int end;
cin >> end;
}
En caso de que alguien necesita el código C ++, puede utilizar esta pieza de OpenCV:
https://github.com/opencv /opencv/blob/master/modules/calib3d/src/polynom_solver.cpp
El programa siguiente perfectamente resuelve la ecuación cúbica de la forma Ax ^ 3 + Bx ^ 2 + Cx + D.
El código impecable está escrito en C # .Net.
Las raíces reales e imaginarios están separados por colores negro y rojo, respectivamente.
Simplemente actualizar el valor de las variables A, B, C y D para ver las respuestas es decir, la solución de la ecuación.
Para encontrar las raíces 2, la solución de una ecuación cuadrática por favor visite aquí.
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