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As R is a homogeneous polynomial in two indeterminates, the fundamental theorem of algebra implies that R is a product of pq linear polynomials. If one defines the multiplicity of a common zero of P and Q as the number of occurrences of the corresponding factor in the product, Bézout's theorem is thus proved.
Interpolating two values yields a line: a polynomial of degree one. This is the basis of the secant method . Regula falsi is also an interpolation method that interpolates two points at a time but it differs from the secant method by using two points that are not necessarily the last two computed points.
Both use the polynomial and its two first derivations for an iterative process that has a cubic convergence. Combining two consecutive steps of these methods into a single test, one gets a rate of convergence of 9, at the cost of 6 polynomial evaluations (with Horner's rule). On the other hand, combining three steps of Newtons method gives a ...
In algebra and algebraic geometry, the multi-homogeneous Bézout theorem is a generalization to multi-homogeneous polynomials of Bézout's theorem, which counts the number of isolated common zeros of a set of homogeneous polynomials. This generalization is due to Igor Shafarevich. [1]
On the other hand, if the multiplicity m of the root is not known, it is possible to estimate m after carrying out one or two iterations, and then use that value to increase the rate of convergence. If the multiplicity m of the root is finite then g ( x ) = f ( x ) / f ′ ( x ) will have a root at the same location with multiplicity 1.
Any nth degree polynomial has exactly n roots in the complex plane, if counted according to multiplicity. So if f(x) is a polynomial with real coefficients which does not have a root at 0 (that is a polynomial with a nonzero constant term) then the minimum number of nonreal roots is equal to (+),
The Jenkins–Traub algorithm for polynomial zeros is a fast globally convergent iterative polynomial root-finding method published in 1970 by Michael A. Jenkins and Joseph F. Traub. They gave two variants, one for general polynomials with complex coefficients, commonly known as the "CPOLY" algorithm, and a more complicated variant for the ...
The multiplicity is always finite if the solution is isolated, is perturbation invariant in the sense that a -fold solution becomes a cluster of solutions with a combined multiplicity under perturbation in complex spaces, and is identical to the intersection multiplicity on polynomial systems.