Calculate a Quartic Equation

Calculator for quartic equations (fourth-degree equations). Calculates the four solutions of the equation $a{x}^4+b{x}^3+c{x}^2+dx+e=0$.
Such a fourth-degree equation does not necessarily have a real solution. All solutions lie in the complex plane, of which 0, 2 or 4 may be real. Please enter a value for each coefficient a, b, c, d and e. If a value is omitted, a is set to 1 and all other coefficients are set to 0. The equation is converted into normalized form so that the coefficient of x⁴ becomes 1. The resulting equation has the same zeros as the original.

The solutions are calculated using the Weierstrass (Durand-Kerner) method. This numerical method determines all four zeros simultaneously by first choosing four complex starting values. These values are then improved step by step. In each iteration, the calculated approximation is improved. The calculation is repeated until the results are sufficiently accurate. The maximum number of iterations can be changed. The default value of 1000 iterations is sufficient for most equations and is still calculated very quickly.

For each approximation, the value of the polynomial is recalculated. If this value is not equal to 0, it is not yet an exact zero. A correction is calculated from the polynomial value and the differences to the other approximations, moving the value closer to the actual zero. The correction is performed according to the formula

$$z_{i,\text{new}}=z_i-\frac{f\left(z_i\right)}{\prod _{j\ne i}(z_i-z_j)}$$
Here, f(z) is the polynomial and z_i is the current approximation of the i-th zero. z_j runs through the other zeros. The product in the denominator contains the differences between the current approximation and all remaining approximations.

There are also exact solution formulas for fourth-degree equations. However, the required expressions are very extensive and numerically sensitive. Even small rounding errors can lead to inaccurate results. In practice, the W-(D-K) method used here is usually more robust.

The method is also suitable for equations of higher degree. For general equations of degree five and higher, there are no exact solution formulas using only the basic arithmetic operations and root extractions. Numerical methods such as the W-(D-K) method can nevertheless solve such equations approximately.