Wilkinson's polynomial illustrates that high precision may be necessary when computing the roots of a polynomial given its coefficients : the problem of finding the roots from the coefficients is in general ill-conditioned.
32.
Some methods result in a \ tau which is a closed-form continuous function while others need to be decomposed into a series of computational steps involving, for example, SVD or finding the roots of a polynomial.
33.
In theory, the coefficients of the characteristic polynomial can be computed exactly, since they are sums of products of matrix elements; and there are algorithms that can find all the roots of a polynomial of arbitrary degree to any required accuracy.
34.
The rule of signs can be thought of as stating that the number of real roots of a polynomial is dependent on the polynomial's complexity, and that this complexity is proportional to the number of monomials it has, not its degree.
35.
Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting.
36.
In 1830, �variste Galois, studying the permutations of the roots of a polynomial, extended the Abel Ruffini theorem by showing that, given a polynomial equation, one may decide whether it is solvable by radicals, and, if it is, solve it.
37.
In 1824, Niels Henrik Abel proved the striking result that there can be no general ( finite ) formula, involving only arithmetic operations and radicals, that expresses the roots of a polynomial of degree 5 or greater in terms of its coefficients ( see Abel Ruffini theorem ).
38.
In that book Rolle firmly established the notation for the " n " th root of a polynomial, and proved a polynomial version of the theorem that today bears his name . ( " Rolle s Theorem " was named by Giusto Bellavitis in 1846 .)
39.
The method is much more difficult though for " m " > 2 than it is for " m " = 1 or " m " = 2 because it is much harder to determine the roots of a polynomial of degree 3 or higher.
40.
We have a reduction to the Bring & ndash; Jerrard form in terms of solvable polynomial equations, and we used transformations involving polynomial expressions in the roots only up to the fourth degree, which means inverting the transformation may be done by finding the roots of a polynomial solvable in radicals.
