According to your experience, what number is most frequently obtained as a root using any known method? You can indicate not just one number but an interval.
Nonlinear algebraic equation means roots of a polynomial? It depends on the method employed and initial guess used. If you apply a method to find a real root, then if it exists and simple then the value of root depend on the interval selected.
If the equation is a polynomial, You should find the solutions in complex numbers. Numerically, Newton-Rafson method is still available, since a complex number consists of two real numbers, the initial guess is difficult. This is from my old knowledge. I think there are many good method to find complex solutions numerically.
a generic nonlinear algebraic equation can have solutions anywhere in R (or C) ! there is no such thing as a "most frequent number" (in other words your question is not well posed, unless you are talking about "digits" and not "numbers". In that case maybe Benford's Law applies, nut I am not sure this is true for equations)
If we look for the roots of a simple equation, z^3+1=0, the solution can be found easily even by juniar high school student. if we put z=x+iy, the roots are (x,y)=(-1,0), (1/sqrt(2),sqrt(3/2)), and (1/sqrt(2),-sqrt(3/2)). but numerical methods are not easy. You should use a numerical algebra, such as Mathematica or Maple.
I am interested not in methods, but in numbers that often appear to be the roots of nonlinear equations. I am thinking about stochastic algorithm, that could guess the root. Thank you, Simone Orcioni. Without sarcasm, your answer is the best, in this sense.
Checking a few dozen equations, I found that the most frequent solutions were 0, 1, -1, i and -i in that order. Well, I guess the equations were constructed that way since they were mostly from schoolbooks...
That aside, if the equations come from applications, I would assume that the solutions actually mean something, and that the equations are therefore properly scaled. So I guess that probability that the norm of the solution exceeds y decays exponentially with y. For a probabilistic approach I would try some exponential distribution...
I am rather interested in nonlinear partial differential equations not algebraic equations.
If we are considering the numerous equations posed as artificial test examples in the literature then zero and one will probably occur pretty frequently!
Giancarlo Gajani is right. A generic algebraic equation can have roots anywhere in C. There is simply no number that appears more often as root than any other number.
Do you think there is dependence between the magnitude of the root and the magnitude of the coefficients of a generic nonlinear equation? I mean if the magnitude of the coefficients increase then the magnitude of the roots also may increase.
Asking often occur is nonsense. We usurely meet equations whose orders are higher than 4, which cannot be solved by algebraic method by Galois theory but have roots in complex numbers by the fundamental theorem of algebra.
Dzmitry,
There is definitly no dependence, as you can multiply an algebraic equation by an arbitrary number, making the coefficients as large or as small as you like, without changing the roots.
Dear Shogo Inagaki and Hermann Schichl, thank you for your comments. I think I am fully satisfied with all answers. Thanks all for discussion.
In light of Hermann's answer, would this question be better posed if we restricted the set of algebraic equations to some standard scaling (for example, the coefficient of the highest order term is unity)?
Yes, Ivan. It is always possible to write the equation in a monic way. This is always possible if the coefficients of the equation are in a field.
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