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Real-root isolation is useful because usual root-finding algorithms for computing the real roots of a polynomial may produce some real roots, but, cannot generally certify having found all real roots. In particular, if such an algorithm does not find any root, one does not know whether it is because there is no real root.
The non-real factors come in pairs which when multiplied give quadratic polynomials with real coefficients. Since every polynomial with complex coefficients can be factored into 1st-degree factors (that is one way of stating the fundamental theorem of algebra), it follows that every polynomial with real coefficients can be factored into factors ...
For polynomials with real or complex coefficients, it is not possible to express a lower bound of the root separation in terms of the degree and the absolute values of the coefficients only, because a small change on a single coefficient transforms a polynomial with multiple roots into a square-free polynomial with a small root separation, and ...
This is illustrated by Wilkinson's polynomial: the roots of this polynomial of degree 20 are the 20 first positive integers; changing the last bit of the 32-bit representation of one of its coefficient (equal to –210) produces a polynomial with only 10 real roots and 10 complex roots with imaginary parts larger than 0.6.
If <, the cubic has one real root and two non-real complex conjugate roots. This can be proved as follows. First, if r is a root of a polynomial with real coefficients, then its complex conjugate is also a root. So the non-real roots, if any, occur as pairs of complex conjugate roots.
The fundamental theorem of algebra, also called d'Alembert's theorem [1] or the d'Alembert–Gauss theorem, [2] states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part ...
The resulting (Sturm's) method for computing the real roots of polynomials has been the only one widely known and used ever since—up to about 1980, when it was replaced (in almost all computer algebra systems) by methods derived from Vincent's theorem, the fastest one being the Vincent–Akritas–StrzeboĊski (VAS) method.
The subtraction of only multiples of 2 from the maximal number of positive roots occurs because the polynomial may have nonreal roots, which always come in pairs since the rule applies to polynomials whose coefficients are real. Thus if the polynomial is known to have all real roots, this rule allows one to find the exact number of positive and ...