A method of solving non-linear equations, using a priori probability estimates of the roots

Algorithms are described for evaluating the roots of non-linear equations, using a priori probability estimates of the required root in the form of a distribution density. By indicating the maximum likelihood estimate for the point of the required root, and the likelihood ratios for the estimates of locally maximum likelihood, subintervals of the domain of specification of the left-hand side of the equation are distinguished as being the most likely to contain the root. The construction of the algorithms is based on an approach described earlier, whereby the left-hand side is regarded to the realization of a random process, and enabling a conditional probability density (with respect to the computed values of the discrepancy) of the position of the root to be constructed, and a decision rule to be justified for choosing the points of the iterations. Sufficient conditions are obtained for the algorithms to converge.