Sergei Khashin

Solution of cubic and quartic equations C++

Cubic equation

Linear and quadratic equations with real coefficients are easy to solve. For the solution of the cubic equation we take a trigonometric Viete method, C++ code takes about two dozen lines. The roots of equation

int   SolveP3(double *x,double a,double b,double c);

In the case of three real roots function returns the number 3, the roots themselves back in x[0],x[1],x[2].

Remark 1. Roots are not necessarily ordered!
If two roots are match, the function returns 2 and in the array x still there are a three numbers.

If the function returns 1, then x[0] is a real root and x[1]張*x[2] is a pair of coplex-conjugated roots.

Remark 2. Due to rounding errors pair of complex conjugate roots with a very small imaginary part can sometimes be a real root of multiplicity 2. For example, for equation x³ - 5x² + 8x - 4 = 0 with roots 1,2,2 we obtain roots     1.0, 2.0張*9.6e-17. If the absolute value of the imaginary part of the root not greater than 1e-14, the SolveP3 itself replaces a pair on one valid double root, but the user must still be aware of the possibility of such a situation.

Quartic equation

For the solution of a quartic equation we take a Descartes-Euler method. Roots of the equation

int   SolveP4(double *x,double a,double b,double c,double d);

In the case of 4 real roots function returns the number 4, the roots themselves back in x[0],x[1],x[2],x[3].

In the case of 2 real and a pair of complex conjugate roots function returns the number 2, x[0],x[1] are the real roots and x[2]張*x[3] is a complex.

If the equation has two pairs of pairs of complex conjugate roots, the function returns 0, the root are x[0]張*x[1] and x[2]張*x[3].
Remark 3. Numerical experiments show that in some cases the resulting tolerance is quite large, up to 10^-12. So at the end each found real root are specified with a single step of Newton's method.

Quintic equation

All roots of the quintic equation

Quintic equation have at least one real root. To find it, starting with the interval [-brd, brd] make 6 "bisections". Then check the root using the Newton method.

Finding one real root x₀, divide it original polynomial f(x) and find the roots of the resulting polynomial of degree 4.

The solution of a quintic equation may be computed by the function

int   SolveP5(double *x,double a,double b,double c,double d, double e);

In the case of 5 real roots function returns the number 5, the roots themselves back in x[0],x[1],x[2],x[3], x[4].

In the case of 3 real and a pair of complex conjugate roots function returns the number 3, x[0],x[1],x[2] are the real roots and x[3]張*x[4] is a complex.

If the equation has one real root and two pairs of pairs of complex conjugate roots, the function returns 1, x[0] is a real root and x[1]張*x[2] , x[3]張*x[4] are a complex roots.

C++-realization

math.h. Dynamic memory allocation is not used.

Solution of cubic equations is performed in a single function SolveP3. To solve the equations of degree 4 are three auxiliary functions:

void  CSqrt( double x, double y, double &a, double &b);  // returns as a+i*s,  sqrt(x+i*y)
int   SolveP4Bi(double *x, double b, double d);	         // solve equation x^4 + b*x^2 + d = 0
int   SolveP4De(double *x, double b, double c, double d);// solve equation x^4 + b*x^2 + c*x + d = 0

The first is to extract the square root of a complex number: a+i*s = sqrt(x+i*y).

Second to solve the biquadratic equation, the third for the solutions of the depressed equation.
Remark 4. As in the case of cubic equations the root of multiplicity 2 or a pair of very close real roots can be shown as a pair of complex conjugate roots with small imaginary part.

Solution of quintic equations is performed in a function SolveP5:

int   SolveP5(double *x,double a,double b,double c,double d,double e);// solve equation x^5 + a*x^4 + b*x^3 + c*x^2 + d*x + e = 0