解決三次方程

Question

作爲一個程序我寫的一部分，我需要確切地解決了三次方程（而不是使用數值求根）：

a*x**3 + b*x**2 + c*x + d = 0. 

我試圖用公式從here。然而，考慮下面的代碼（這是Python的，但它是非常通用的代碼）：

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 

但輸出是：

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 

所以輸出不爲零，所以X1是不實際一個辦法。維基百科文章中是否有錯誤？

ps：我知道numpy.roots會解決這種方程式，但我需要爲數百萬個方程組做這個工作，所以我需要實現這個來處理係數數組。

Answer 1

維基百科的符號(rho^(1/3), theta/3)並不意味着rho^(1/3)是真正的部分，theta/3是虛構的部分。相反，這是在極座標中。因此，如果你想要真正的部分，你會採取rho^(1/3) * cos(theta/3)。

我做了這些改變你的代碼，它的工作對我來說：（當然，s_real = t_real這裏，因爲cos爲偶數）

theta = arccos(r/rho) 
s_real = rho**(1./3.) * cos(theta/3) 
t_real = rho**(1./3.) * cos(-theta/3) 

Answer 2

我看過維基百科的文章和你的程序。

我也用Wolfram Alpha解決了方程式，結果與你得到的結果不符。

我只是通過你的程序在每一步，使用大量的打印語句，並獲得每個中間結果。然後通過一個計算器並自己做。

我找不到發生了什麼，但是你的手算和程序分歧的地方是一個好看的地方。

Answer 3

這裏是A.雷克斯在JavaScript解決方案：

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)); 

Answer 4

下面的程序完美地解決了形式Ax^3 + Bx^2 + Cx + D的三次方程。

無瑕的代碼是用.Net C＃編寫的。

真實和想象的根分別由黑色和紅色分開。

只需更新變量A，B，C和D的值以查看答案即方程的解。

要找到2個根，解二次方程請訪問這裏。

Source

Answer 5

在這裏，我把三次方程（復係數）求解。

#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; 
}