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A set of implicit difference schemes on the five-pointwise stensil is under construction. The analysis of properties of difference schemes is carried out in a space of undetermined coefficients. The spaces were introduced for the first time by A. S. Kholodov. Usually for properties of difference schemes investigation the problem of the linear programming was constructed. The coefficient at the main term of a discrepancy was considered as the target function. The optimization task with inequalities type restrictions was considered for construction of the monotonic difference schemes. The limitation of such an approach becomes clear taking into account that approximation of the difference scheme is defined only on the classical (smooth) solutions of partial differential equations.

The functional which minimum will be found put in compliance to the difference scheme. The functional must be the linear on the difference schemes coefficients. It is possible that the functional depends on net function – the solution of a difference task or a grid projection of the differential problem solution. If the initial terms of the functional expansion in a Taylor series on grid parameters are equal to conditions of classical approximation, we will call that the functional will be the generalized condition of approximation. It is shown that such functionals exist. For the simple linear partial differential equation with constant coefficients construction of the functional is possible also for the generalized (non-smooth) solution of a differential problem.

Families of functionals both for smooth solutions of an initial differential problem and for the generalized solution are constructed. The new difference schemes based on the analysis of the functionals by linear programming methods are constructed. At the same time the research of couple of self-dual problems of the linear programming is used. The optimum monotonic difference scheme possessing the first order of approximation on the smooth solution of differential problem is found. The possibility of application of the new schemes for creation of hybrid difference methods of the raised approximation order on smooth solutions is discussed.

The example of numerical implementation of the simplest difference scheme with the generalized approximation is given.

In this paper, we introduce a new version of Gauss–Newton method for solving a system of nonlinear equations based on ideas of the residual upper bound for a system of nonlinear equations and a quadratic regularization term. The introduced Gauss–Newton method in practice virtually forms the whole parameterized family of the methods solving systems of nonlinear equations and regression problems. The developed family of Gauss–Newton methods completely consists of iterative methods with generalization for cases of non-euclidean normed spaces, including special forms of Levenberg–Marquardt algorithms. The developed methods use the local model based on a parameterized proximal mapping allowing us to use an inexact oracle of «black–box» form with restrictions for the computational precision and computational complexity. We perform an efficiency analysis including global and local convergence for the developed family of methods with an arbitrary oracle in terms of iteration complexity, precision and complexity of both local model and oracle, problem dimensionality. We present global sublinear convergence rates for methods of the proposed family for solving a system of nonlinear equations, consisting of Lipschitz smooth functions. We prove local superlinear convergence under extra natural non-degeneracy assumptions for system of nonlinear functions. We prove both local and global linear convergence for a system of nonlinear equations under Polyak–Lojasiewicz condition for proposed Gauss– Newton methods. Besides theoretical justifications of methods we also consider practical implementation issues. In particular, for conducted experiments we present effective computational schemes for the exact oracle regarding to the dimensionality of a problem. The proposed family of methods unites several existing and frequent in practice Gauss–Newton method modifications, allowing us to construct a flexible and convenient method implementable using standard convex optimization and computational linear algebra techniques.

A vector-matrix approach to the theoretical design of metabolic pathways converting chemical compounds, viz., preset substrates, into desirable products is described. It is a mathematical basis for computer–aided generation of alternative biochemical reaction sets executing the given substrate–product conversion. The pathways are retrieved from the used database of biochemical reactions and utilize the reaction stoichiometry and restrictions based on the irreversibility of a part of them. Particular attention is paid to the analysis of restriction interrelations. It is shown that the number of restrictions can be notably reduced due to the existence of families of parallel restricting planes in the space of reaction flows. Coinciding planes of contradirectional restrictions result in the existence of fixed reaction flow values. The problem of exclusion of so called futile cycles is also considered. Utilization of these factors allows essential lowering of the problem complexity and necessary computational resources. An example of alternative biochemical pathway computation for conversion of glucose and glycerol into succinic acid is given. It is found that for a preset “substrate–product” pair many pathways have the same high-energy bond balance.