cubic:
Function that can resolve the cubic equation numerical stable and any lower dimensional case except unsolvable cases.
Description
A*x^3+B*x^2+C*x+D=0. All coefficients had to be numeric or integers.
This function calculates from 4 coefficient all possible and senfully solutions. D=0 returns no values at all. This would be a impossibel case. It returns upto 3 potential complex solutions.
Less solutions are copied to get the tripple solution format.
Usage
cubic(A, B, C, D)
Arguments
A
The coefficient of x^3.
B
The coefficient of x^2.
C
The coefficient of X.
D
The constant.
Value
Returns cubic(A,B,C,D)[c(1:3)] roots of the at most cubic equation.
Details
This function is called by haplotypeit. The results are returned as vector of the three possible solutions: output[1],output[2],output[3].
Further data for checks of the roots. p,q and the discriminat. 10 and 11 are only usable for symmetry checks.
References
Cardans formular as in e.g. The Mathematical Gazette (1993); 77 (Nov, No 480), 354-359 (jstor)
http://www.nickalls.org/dick/papers/maths/cubic1993.pdf
or any other book for algebraic solutions.
See also : http://de.wikipedia.org/wiki/Cardanische_Formeln
and http://en.wikipedia.org/wiki/Cubic_equation