Search results
Results From The WOW.Com Content Network
Chapter 8 (The solution of equations of the fifth degree at the Wayback Machine (archived 31 March 2010)) gives a description of the solution of solvable quintics x 5 + cx + d. Victor S. Adamchik and David J. Jeffrey, "Polynomial transformations of Tschirnhaus, Bring and Jerrard," ACM SIGSAM Bulletin , Vol. 37, No. 3, September 2003, pp. 90–94.
For these problems, it is very easy to tell whether solutions exist, but thought to be very hard to tell how many. Many of these problems are #P-complete, and hence among the hardest problems in #P, since a polynomial time solution to any of them would allow a polynomial time solution to all other #P problems.
A solution of a polynomial system is a tuple of values of (x 1, ..., x m) that satisfies all equations of the polynomial system. The solutions are sought in the complex numbers, or more generally in an algebraically closed field containing the coefficients. In particular, in characteristic zero, all complex solutions are sought. Searching for ...
Bairstow's approach is to use Newton's method to adjust the coefficients u and v in the quadratic + + until its roots are also roots of the polynomial being solved. The roots of the quadratic may then be determined, and the polynomial may be divided by the quadratic to eliminate those roots.
This problem is commonly resolved by the use of spline interpolation. Here, the interpolant is not a polynomial but a spline: a chain of several polynomials of a lower degree. Interpolation of periodic functions by harmonic functions is accomplished by Fourier transform.
if s 1 dominates s 2, then g(s 1) ≤ g(s 2) (in minimization problems); g(s 1) ≥ g(s 2) (in maximization problems). Technical conditions (in addition to the above): The quasi-dominance relation can be decided in polynomial time. Conditions on the filter functions: For any r>1, for any filter function h in H, for any input-vector x, and for ...
The fact that every polynomial equation of positive degree has solutions, possibly non-real, was asserted during the 17th century, but completely proved only at the beginning of the 19th century. This is the fundamental theorem of algebra , which does not provide any tool for computing exactly the solutions, although Newton's method allows ...
The Galois group of f(x) modulo 2 is cyclic of order 6, because f(x) modulo 2 factors into polynomials of orders 2 and 3, (x 2 + x + 1)(x 3 + x 2 + 1). f(x) modulo 3 has no linear or quadratic factor, and hence is irreducible. Thus its modulo 3 Galois group contains an element of order 5.