Perturbation Effect Of Scalar Polynomial Coefficients And Real Root Finding Algorithms With Applications

This research has three very interrelated topics. Analytic proofs with experimental perturbation calculation method are used to determine the conditioning of polynomial roots occurring due to the perturbed coefficients and the conditioning of algorithms to solve the roots.Function construction, Taylor’s interpolation, variational method, Newton-correction and a theorem on higher derivative estimations in root finding (a new idea) are applied to develop new models which motivate efficient higher order iterative algorithms. Methods for simple roots and multiple roots for solving nonlinear equations are investigated.Root finding methods are applied to solve the nonlinear polynomials generated during the stability analyses of some numerical methods for ordinary differential equations. This enables us to obtain the absolute stability limits and critical points of the stability regions of the methods. Graphs with level curves and vector fields are shown for some stability functions. We also solve polynomial models of total water amount of Wuhan city (in China) obtained using matlab tools.Numerical results of applications and perturbation effects on different roots based on (a posteriori) error analyses are presented. The new iterative methods are compared with some existing methods using test equations implemented in C++for rate of convergence analysis, efficiency indices and function evaluations and results are tabulated.