Abstract
We have advanced the application of algorithms within a method of basic matrices, which are equipped with the technology of long arithmetic to improve the precision of performing the basic operations in the course of studying the ill-conditioned linear systems, specifically, the systems of linear algebraic equations (SLAE). Identification of the fact of ill-conditionality of a system is a rather time-consuming computational procedure. The possibility to control computations entering the state of incorrectness and the impossibility of accumulating calculation errors, which is a desirable property of the methods and algorithms for solving practical problems, were introduced. Modern computers typically use the standard types of integers whose size does not exceed 64 bytes. This hardware limitation was resolved using software, specifically, by developing a proprietary type of data in the form of a special Longnum library in the C++ language (using the STL (Standard Template Library)). Software implementation was aimed at carrying out computations for methods of basic matrices (MBM) and Gauss matrices, that is, long arithmetic for models with rational elements was used. We have proposed the algorithms and computer realization of the Gauss type methods and methods of artificial basic matrices (a variant of the method of basic matrices) in MatLAB environment and Visual C++ environment using precise computation of the methods' elements, first of all, for the ill-conditioned systems of varying dimensionality. The Longnum library with the types of long integers (longint3) and rational numbers (longrat3) with the numerator and denominator of the longint3 type was developed. Arithmetic operations on long integers were performed based on the modern methods, including the Strassen multiplication method. We give the results from the computational experiment employing the mentioned methods, in which test models of the systems were generated, specifically, based on the Gilbert matrices of different dimensionality
Highlights
Mathematical modeling of processes of different nature is known to lead to the need to explore non-linear equations and systems of varying complexity.In many cases, they are solved by the introduction of certain simplifications in statements, transition, in particular, to difference analogues and eventually to the systems of linear algebraic equations (SLAE) of different dimension, often with a square matrix of restrictions
Compared to the old version of the Longnum library of precise computations, the following was done in the new version: – implementations of arithmetic operations with long integers were optimized taking onto account the new standard programming language C++17; – outdated functions of standard C++ libraries were replaced by the new, more effective and safe analogues
The advantage of using the new version of Longnum compared with the previous version [22] is the higher computational effectiveness
Summary
Mathematical modeling of processes of different nature is known to lead to the need to explore non-linear equations and systems of varying complexity (mathematical models).In many cases, they are solved by the introduction of certain simplifications in statements, transition, in particular, to difference analogues (discrete variant) and eventually to the systems of linear algebraic equations (SLAE) of different dimension, often with a square matrix of restrictions.
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