There exist equations the roots of which cannot be explicitly written as a function of the constants, for example $xe^x = a$, where the roots can be found using the product log, the value of which is found using Newtons method. As in the example before, however, there already exist algorithms for finding zeros of (to my knowledge) all imaginable explicit functions (take, for example, the Newton method). So if we already have an algorithm for finding roots, why do we seek an explicit function of the constants specifically? Why do we even need solutions for polynomials in radicals if we can very well just employ the Newton thing?
Asked
Active
Viewed 50 times
1
-
I don't have enough knowledge about abstract algebra, but note that this root finding algorithms give you an approximation , also a root finding algorithm like Newton's method is not always useful (see https://en.wikipedia.org/wiki/Newton%27s_method), another problem is that using an explicit formula we then have all the roots at the same time, thought using this kind of algorithm gives you just one root, also we sometimes have the problem of convergence or the problem with slow convergence. – Jan 12 '20 at 07:14
1 Answers
1
For all practical purposes as engineering or astronomy, e.g., these approximations are sufficient. But the mathematician heads to exactness.
Using an approximation for $\pi$ one may square the circle. But the nature of $\pi$ prohibits that squaring.
Michael Hoppe
- 18,103
- 3
- 32
- 49