A transformation that reduces coefficient count to one greatly simplifies the task of finding roots.
Edwin P. Russo, P.E.
National Center for Advanced Mfg.
Carsie A. Hall
Duane J. Jardine
Mechanical Engineering Dept.
University of New Orleans
New Orleans, La.
William W. St. Cyr, P.E.
NASA Stennis Space Center
The general cubic equation given by:(1)
has roots that are solvable by classical techniques involving the computation of inverse cosines, cosines of multiple angles, and so forth. But equation 1 may be transformed into a simpler equation with only one coefficient. Such a transformation resembles those given elsewhere, though the other transformations only reduce the number of coefficients from three to two, rather than to one.
Jahnke and Emde in 1945 derived a formulation giving a single coefficient with a short table of roots. But it needed a highly complicated graphical procedure to find the roots. The transformation presented here as well reduces the coefficient count to one but gives a convenient means by which to tabularize the roots of the cubic equation, eliminating various approximate or tedious methods of finding the roots.
Equation 1 may be transformed into:
The above transformation first defines a new variable, X, by the relation:
where D and K are arbitrary constants to be determined later. Substituting equation 6 into equation 1 gives:
The X 2 term is eliminated by requiring that K = a/3. D is now defined by requiring that the coefficients of X 3 and X be the same. Therefore:
or, after substituting K:
Dividing equation 7 by D3 and substituting for K gives an equation with only one coefficient, that is, equation 2. This technique also works to reduce the number of coefficients for higher-order equations (quartic, for example).
An abbreviated Table of roots contains real values of P in rather large increments. Interested readers may wish to expand the Table to include finer increments of P. Note that the real part of the complex roots, X2 and X3, is simply X1/2, and that X2 and X3 are complex conjugates. Examination of equation 5 shows that if a, b, and c are real, then the coefficient P can only be real (positive or negative) or imaginary if b < a 2 /3.
When P is imaginary (b < a 2 /3) the cubic equation 2 may be rewritten in a more convenient form, namely:(10)
where P has been replaced by iQ and X replaced by iY (Q is real and i = √–1). That is:
Table 2 lists the roots of equation 10 for various values of the coefficient Q. Only when Q is in the interval, –4/27 < Q < √4/27, will the roots of equation 10 all be real. These roots are listed in Table 3.
After selecting the roots of equation 2 or 10 from the appropriate Table, it is a simple matter to obtain the roots for the general cubic equation 1 from equation 4 or 12.
Note when P is large:
And when Q is large:
The Tables may be used to solve principal stresses, pump curves, eigenvalues, hydraulic jumps, control systems, spillway flow, moving wave/bores, setting initial conditions for Newton iterations, and so on.