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The problem of restoration of an element f of Euclidean functional space  L²(X) based on the results of measurements of a finite set of its linear functionals, distorted by (random) error is solved. A priori data aren't assumed. Family of linear subspaces of the maximum (effective) dimension for which the projections of element f to them allow estimates with a given accuracy, is received. The effective rank ρ(δ) of the estimation problem is defined as the function equal to the maximum dimension of an orthogonal component Pf of the element f which can be estimated with a error, which is not surpassed the value δ. The example of restoration of a spectrum of radiation based on a finite set of experimental data is given.

The basic subset of two-stage Rosenbrock schemes with complex coefficients for numerical solution of autonomous systems of ordinary differential equations (ODE) has been considered. Numerical realization of such schemes requires one LU-decomposition, two computations of right side function and one computation of Jacoby matrix of the system per one step. The full theoretical investigation of accuracy and stability of such schemes have been done. New A-stable methods of the 3-rd order of accuracy with different properties have been constructed. There are high order L-decremented schemes as well as schemes with simple estimation of the main term of truncation error which is necessary for automatic evaluation of time step. Testing of new methods has been performed.

Difference of software development for parallel computing technology from sequential programming is dicussed. Arguements for introduction of new phases into technology of software engineering are given. These phases are: decomposition of algorithms, assignment of jobs to performers, conducting and mapping of logical to physical performers. Issues of performance evaluation of algorithms are briefly discussed. Decomposition of algorithms and programs into parts that can be executed in parallel is dicussed.

We developed an algorithm for fast simulation of VLSI CMOS (Very Large Scale Integration with Complementary Metal-Oxide-Semiconductors) with an accuracy control. The algorithm provides an ability of parallel numerical experiments in multiprocessor computational environment. There is computation speed up by means of block-matrix and structural (DCCC) decompositions application. A feature of the approach is both in a choice of moments and ways of parameters synchronization and application of multi-rate integration methods. Due to this fact we have ability to estimate and control error of given characteristics.

We consider a numerically stable direct multiplicative algorithm of solving linear equations systems, which takes into account the sparseness of matrices presented in a packed form. The advantage of the algorithm is the ability to minimize the filling of the main rows of multipliers without losing the accuracy of the results. Moreover, changes in the position of the next processed row of the matrix are not made, what allows using static data storage formats. Linear system solving by a direct multiplicative algorithm is, like the solving with $LU$-decomposition, just another scheme of the Gaussian elimination method implementation.

In this paper, this algorithm is the basis for solving the following problems:

Problem 1. Setting the descent direction in Newtonian methods of unconditional optimization by integrating one of the known techniques of constructing an essentially positive definite matrix. This approach allows us to weaken or remove additional specific difficulties caused by the need to solve large equation systems with sparse matrices presented in a packed form.

Problem 2. Construction of a new mathematical formulation of the problem of quadratic programming and a new form of specifying necessary and sufficient optimality conditions. They are quite simple and can be used to construct mathematical programming methods, for example, to find the minimum of a quadratic function on a polyhedral set of constraints, based on solving linear equations systems, which dimension is not higher than the number of variables of the objective function.

Problem 3. Construction of a continuous analogue of the problem of minimizing a real quadratic polynomial in Boolean variables and a new form of defining necessary and sufficient conditions of optimality for the development of methods for solving them in polynomial time. As a result, the original problem is reduced to the problem of finding the minimum distance between the origin and the angular point of a convex polyhedron, which is a perturbation of the $n$-dimensional cube and is described by a system of double linear inequalities with an upper triangular matrix of coefficients with units on the main diagonal. Only two faces are subject to investigation, one of which or both contains the vertices closest to the origin. To calculate them, it is sufficient to solve $4n – 4$ linear equations systems and choose among them all the nearest equidistant vertices in polynomial time. The problem of minimizing a quadratic polynomial is $NP$-hard, since an $NP$-hard problem about a vertex covering for an arbitrary graph comes down to it. It follows therefrom that $P = NP$, which is based on the development beyond the limits of integer optimization methods.

A numerically stable direct multiplicative method for solving systems of linear equations that takes into account the sparseness of matrices presented in a packed form is considered. The advantage of the method is the calculation of the Cholesky factors for a positive definite matrix of the system of equations and its solution within the framework of one procedure. And also in the possibility of minimizing the filling of the main rows of multipliers without losing the accuracy of the results, and no changes are made to the position of the next processed row of the matrix, which allows using static data storage formats. The solution of the system of linear equations by a direct multiplicative algorithm is, like the solution with LU-decomposition, just another scheme for implementing the Gaussian elimination method.

The calculation of the Cholesky factors for a positive definite matrix of the system and its solution underlies the construction of a new mathematical formulation of the unconditional problem of quadratic programming and a new form of specifying necessary and sufficient conditions for optimality that are quite simple and are used in this paper to construct a new mathematical formulation for the problem of quadratic programming on a polyhedral set of constraints, which is the problem of finding the minimum distance between the origin ordinate and polyhedral boundary by means of a set of constraints and linear algebra dimensional geometry.

To determine the distance, it is proposed to apply the known exact method based on solving systems of linear equations whose dimension is not higher than the number of variables of the objective function. The distances are determined by the construction of perpendiculars to the faces of a polyhedron of different dimensions. To reduce the number of faces examined, the proposed method involves a special order of sorting the faces. Only the faces containing the vertex closest to the point of the unconditional extremum and visible from this point are subject to investigation. In the case of the presence of several nearest equidistant vertices, we investigate a face containing all these vertices and faces of smaller dimension that have at least two common nearest vertices with the first face.

The using of determining the measure of interaction between channels when choosing the configuration structure of a control system for complex dynamic objects is considered in the work. The main methods for determining the measure of interaction between subsystems of complex control systems based on the methods RGA (Relative Gain Array), Dynamic RGA, HIIA (Hankel Interaction Index Array), PM (Participation matrix) are presented. When choosing a control configuration, simple configurations are preferable, as they are simple in design, maintenance and more resistant to failures. However, complex configurations provide higher performance control systems. Processes in large dynamic objects are characterized by a high degree of interaction between process variables. For the design of the control structure interaction measures are used, namely, the selection of the control structure and the decision on the configuration of the controller. The choice of control structure is to determine which dynamic connections should be used to design the controller. When a structure is selected, connections can be used to configure the controller. For large systems, it is proposed to pre-group the components of the vectors of input and output signals of the actuators and sensitive elements into sets in which the number of variables decreases significantly in order to select a control structure. A quantitative estimation of the decentralization of the control system based on minimizing the sum of the off-diagonal elements of the PM matrix is given. An example of estimation the measure of interaction between components of strong coupled subsystems and the measure of interaction between components of weak coupled subsystems is given. A quantitative estimation is given of neglecting the interaction of components of weak coupled subsystems. The construction of a weighted graph for visualizing the interaction of the subsystems of a complex system is considered. A method for the formation of the controllability gramian on the vector of output signals that is invariant to state vector transformations is proposed in the paper. An example of the decomposition of the stabilization system of the components of the flying vehicle angular velocity vector is given. The estimation of measures of the mutual influence of processes in the channels of control systems makes it possible to increase the reliability of the systems when accounting for the use of analytical redundancy of information from various devices, which reduces the mass and energy consumption. Methods for assessing measures of the interaction of processes in subsystems of control systems can be used in the design of complex systems, for example, motion control systems, orientation and stabilization systems of vehicles.

An approach to the decrease of norm of the correction in Newton’s methods of optimization, based on the Cholesky’s factorization is presented, which is based on the integration with the technique of the choice of leading element of algorithm of linear programming as a method of solving the system of equations. We investigate the issues of increasing of the numerical stability of the Cholesky’s decomposition and the Gauss’ method of exception.

The work is devoted to the structural analysis of complex technical systems. Mechanical structures are considered, the properties of which affect the behavior of products during assembly, repair and operation. The main source of data on parts and mechanical connections between them is a hypergraph. This model formalizes the multidimensional basing relation. The hypergraph correctly describes the connectivity and mutual coordination of parts, which is achieved during the assembly of the product. When developing complex products in CAD systems, an engineer often makes serious design mistakes: overbasing of parts and non-sequential assembly operations. Effective ways of identifying these structural defects have been proposed. It is shown that the property of independent assembly can be represented as a closure operator whose domain is the boolean of the set of product parts. The images of this operator are connected and coordinated subsets of parts that can be assembled independently. A lattice model is described, which is the state space of the product during assembly, disassembly and decomposition into assembly units. The lattice model serves as a source of various structural information about the project. Numerical estimates of the cardinality of the set of admissible alternatives in the problems of choosing an assembly sequence and decomposition into assembly units are proposed. For many technical operations (for example, control, testing, etc.), it is necessary to mount all the operand parts in one assembly unit. A simple formalization of the technical conditions requiring the inclusion (exclusion) of parts in the assembly unit (from the assembly unit) has been developed. A theorem that gives an mathematical description of product decomposition into assembly units in exact lattice terms is given. A method for numerical evaluation of the robustness of the mechanical structure of a complex technical system is proposed.