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Selection of criteria to reduce the unknowns in Newton–Raphson method in the problem of finding an equilibrium composition of iron ore and fuel

P.A. Sechenov

Vestnik IGEU, 2024 issue 5, pp. 82—90

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Abstract in English: 

Background. Finding the equilibrium composition of a multicomponent system is required in various industries associated with chemical reactions. Reduction of the unknowns in the original nonlinear equation is possible if most of the unknowns receive values equal to zero as a result of the solution. This situation is typical to solve the problem of finding an equilibrium composition of iron ore and fuel, where the unknowns are the number of moles of all possible reaction products from given simple substances. The dependence of algorithmic complexity on the number of unknowns to solve a SLAEs is expressed through the function O(N3). Therefore, reduction of the unknowns will speed up the computational process. Thus, the solution to the problem of selecting criteria to reduce the number of the unknowns at each iteration of the Newton-Raphson method as applied to the problem of finding the equilibrium composition of iron ore and fuel is relevant.

Materials and methods. The matrix dimensionality reduction has been achieved using four strategies. The first three strategies involve finding a minimum threshold (removing values that most quickly tend to zero), after which the number of unknowns will decrease. The fourth strategy is associated with reducing the dimension of the original matrix.

Results. The results of numerical experiments to reduce the unknowns and speed up calculations are presented. Criteria have been found under which the calculation speed increases by 2–4 times and variables whose value is not equal to zero are not reduced.

Conclusions. The criteria to reduce the unknowns implemented in the T-Energу software package make it possible to solve the problem of finding the equilibrium composition of iron ore and fuel 2–4,2 times faster. In this case, the obtained solution for the components of the equilibrium composition with the reduction of the unknowns corresponds to the complete solution with an accuracy of 10–3. Due to acceleration at the same time, it is possible to construct equilibrium compositions over a temperature range with a temperature step that is 4 times smaller.

References in English: 

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Key words in Russian: 
редукция матрицы, итерационный процесс, метод Ньютона–Рафсона, задача нахождения равновесного состава
Key words in English: 
matrix reduction, iterative process, Newton–Raphson method, equilibrium composition problem
The DOI index: 
10.17588/2072-2672.2024.5.082-090
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