Semenov V. Solution of systems of nonlinear algebraic equations and functions’ minimization tasks: elements of theory and applications

In the thesis the methods of localization of all the roots of nonlinear algebraic equations systems consisting of twice continuously differentiable functions were developed. Based on the method of searching all the roots of systems of nonlinear algebraic equations, the method of global minimization of functions of many variables is developed. Also, based on the developed methods for finding the roots of nonlinear equations, new efficient high-speed methods of calculating the immittance spectral frequencies of speech signals have been developed. Conditions for orthogonality and minimization of the Riesz ratio for wavelets based on Jacobi polynomials were established. Basing on maximizing a posterior probability function, the method of demodulation and estimation of channel parameters on the basis of particle filtering, the method of demodulating signals with a continuous phase, which is based on the use of signal phase values only and the method of blind separation of amplitude-phase modulated signals were developed. The method of semi-automatic data classification based on the minimization of Tikhonov functional using discrete Laplacian and the quasi-optimal choice of the regularization parameter has been developed and verified for different data sources. Also a new computationally efficient method for automatic identification of the speaker’s gender based on the Gaussian mixture models was developed based on the maximization of likelihood function.