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We consider the approaches to the construction of methods for solving four-dimensional programming problems for calculating directions for multiple minimizations of smooth functions on a set of a given set of linear equalities. The approach consists of two stages.

At the first stage, the problem of quadratic programming is transformed by a numerically stable direct multiplicative algorithm into an equivalent problem of designing the origin of coordinates on a linear manifold, which defines a new mathematical formulation of the dual quadratic problem. For this, a numerically stable direct multiplicative method for solving systems of linear equations is proposed, taking into account the sparsity of matrices presented in packaged form. The advantage of this approach is to calculate the modified Cholesky factors to construct a substantially positive definite matrix of the system of equations and its solution in 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 in the position of the next processed row of the matrix, which allows the use of static data storage formats.

At the second stage, the necessary and sufficient optimality conditions in the form of Kuhn–Tucker determine the calculation of the direction of descent — the solution of the dual quadratic problem is reduced to solving a system of linear equations with symmetric positive definite matrix for calculating of Lagrange's coefficients multipliers and to substituting the solution into the formula for calculating the direction of descent.

It is proved that the proposed approach to the calculation of the direction of descent by numerically stable direct multiplicative methods at one iteration requires a cubic law less computation than one iteration compared to the well-known dual method of Gill and Murray. Besides, the proposed method allows the organization of the computational process from any starting point that the user chooses as the initial approximation of the solution.

Variants of the problem of designing the origin of coordinates on a linear manifold, a convex polyhedron and a vertex of a convex polyhedron are presented. Also the relationship and implementation of methods for solving these problems are described.

The paper describes a free software for research in the field of holomorphic dynamics based on the computational capabilities of the MATLAB environment. The software allows constructing not only single complex-valued mappings, but also their collectives as linearly connected, on a square or hexagonal lattice. In the first case, analogs of the Julia set (in the form of escaping points with color indication of the escape velocity), Fatou (with chaotic dynamics highlighting), and the Mandelbrot set generated by one of two free parameters are constructed. In the second case, only the dynamics of a cellular automaton with a complex-valued state of the cells and of all the coefficients in the local transition function is considered. The abstract nature of object-oriented programming makes it possible to combine both types of calculations within a single program that describes the iterated dynamics of one object.

The presented software provides a set of options for the field shape, initial conditions, neighborhood template, and boundary cells neighborhood features. The mapping display type can be specified by a regular expression for the MATLAB interpreter. This paper provides some UML diagrams, a short introduction to the user interface, and some examples.

The following cases are considered as example illustrations containing new scientific knowledge:

1) a linear fractional mapping in the form $Az^{n} +B/z^{n} $, for which the cases $n=2$, $4$, $n>1$, are known. In the portrait of the Fatou set, attention is drawn to the characteristic (for the classical quadratic mapping) figures of <>, showing short-period regimes, components of conventionally chaotic dynamics in the sea;

2) for the Mandelbrot set with a non-standard position of the parameter in the exponent $z(t+1)\Leftarrow z(t)^{\mu } $ sketch calculations reveal some jagged structures and point clouds resembling Cantor's dust, which are not Cantor's bouquets that are characteristic for exponential mapping. Further detailing of these objects with complex topology is required.

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.

This work presents the model of heat wall functions FlowVision (WFFV), which allows simulation of nonisothermal flows of fluid and gas near solid surfaces on relatively coarse grids with use of turbulence models. The work follows the research on the development of wall functions applicable in wide range of the values of quantity y+. Model WFFV assumes smooth profiles of the tangential component of velocity, turbulent viscosity, temperature, and turbulent heat conductivity near a solid surface. Possibility of using a simple algebraic model for calculation of variable turbulent Prandtl number is investigated in this study (the turbulent Prandtl number enters model WFFV as parameter). The results are satisfactory. The details of implementation of model WFFV in the FlowVision software are explained. In particular, the boundary condition for the energy equation used in high-Reynolds number calculations of non-isothermal flows is considered. The boundary condition is deduced for the energy equation written via thermodynamic enthalpy and via full enthalpy. The capability of the model is demonstrated on two test problems: flow of incompressible fluid past a plate and supersonic flow of gas past a plate (M = 3).

Analysis of literature shows that there exists essential ambiguity in experimental data and, as a consequence, in empirical correlations for the Stanton number (that being a dimensionless heat flux). The calculations suggest that the default values of the model parameters, automatically specified in the program, allow calculations of heat fluxes at extended solid surfaces with engineering accuracy. At the same time, it is obvious that one cannot invent universal wall functions. For this reason, the controls of model WFFV are made accessible from the FlowVision interface. When it is necessary, a user can tune the model for simulation of the required type of flow.

The proposed model of wall functions is compatible with all the turbulence models implemented in the FlowVision software: the algebraic model of Smagorinsky, the Spalart-Allmaras model, the SST $k-\omega$ model, the standard $k-\varepsilon$ model, the $k-\varepsilon$ model of Abe, Kondoh, Nagano, the quadratic $k-\varepsilon$ model, and $k-\varepsilon$ model FlowVision.