In 1941 Niven pioneered root-finding for a quaternion polynomial P(x), proving the fundamental theorem of algebra (FTA) and proposing an algorithm, practical if the norm and trace of a solution are known. We pr
Our analysis and experiments show efficiency of the resulting algorithms.doi:10.48550/arXiv.1311.6077Pan, Victor YZheng, Ai-LongMathematicsV. Y. Pan, A. Zheng, New Structured Matrix Methods for Real and Complex Polyno- mial Root-finding" (by Victor Y. Pan, Ai-Long Zheng), Nov. 23, ...
Finding roots of a polynomial is therefore equivalent to polynomial factorization into factors of degree 1. Any polynomial can be numerically factored, although different algorithms have different strengths and weaknesses. The roots of a polynomial equation may be found exactly in the Wolfram ...
An updated QPmR algorithm implementation for computation and analysis of the spectrum of quasi-polynomials is presented. The objective is to compute all the zeros of a quasi-polynomial located in a given region of the complex plane. The root-finding task
This algorithm can be used for counting and exclusion tests in a subdivision algorithms for polynomial root-fnding, and would be especially useful in application scenarios where high-precision polynomial cofficients are hard to obtain but we succeed with counting already by using polynomial evaluation...
Polynomial algorithms are at the core of classical "computer algebra". Incorporating methods that span from antiquity to the latest cutting-edge research at Wolfram Research, the Wolfram Language has the world's broadest and deepest integrated web of pol
Some classes of iterative methods for finding zeros of polynomials and analytic functions 3859 Fine and coarse grained parallel implementations of polynomial root finding algorithms 3871 Divide and conquer techniques for the polynomial root-finding problem 3885 A unified approach to some classes of ...
We dramatically accelerate the known algorithms in this case by exploiting the correlation between the computations with matrices and polynomi- als, extending the techniques of the matrix sign iteration, and exploiting the structure of the companion matrix of the input polynomial. We ex- tend some ...
Then, the (worst) running times of real root finding algorithms are functions of m, L(d) and are given in [14]. On the other hand, bisection methods for finding all roots of f, real and complex with similar running times, can be found in [78],[97]. As it can be seen from the...
In all polynomial zerofinding algorithms, a good convergence requires a very good initial approximation of the exact roots. The objective of the work is to study the conditions for determining the initial approximations for an iterative matrix zerofinding method. The investigation is based on the ...