Решение систем линейных алгебраичес

Решение систем линейных алгебраических уравнений – одна из основных задач вычислительной линейной алгебры. Хотя задача решения системы линейных уравнений сравнительно редко представляет самостоятельный интерес для приложений, от умения эффективно решать такие системы часто зависит сама возможность математического моделирования самых разнообразных процессов с применением ЭВМ. Значительная часть численных методов решения различных (в особенности – нелинейных) задач включает в себя решение систем линейных уравнений как элементарный шаг соответствующего алгоритма.
Одна из трудностей практического решения систем большой размерности связанна с ограниченностью оперативной памяти ЭВМ. Хотя обьем оперативной памяти вновь создаваемых вычислительных машин растет очень быстро, тем не менее, еще быстрее возрастают потребности практики в решении задач все большей размерности. В значительной степени ограничения на размерность решаемых систем можно снять, если использовать для хранения матрицы внешние запоминающие устройства. Однако в этом случае многократно возрастают как затраты машинного времени, так и сложность соответствующих алгоритмов. Поэтому при создании вычислительных алгоритмов линейной алгебры большое внимание уделяют способам компактного размещения элементов матриц в памяти ЭВМ.
К счастью, приложения очень часто приводят к матрицам, в которых число ненулевых элементов много меньше общего чила элементов матрицы. Такие матрицы принято называть разреженными. Одним из основных источников разреженных матриц являются математические модели технических устройств, состоящих из большого числа элементов, связи между которыми локальны. Простейшие примеры таких устройств – сложные строительные конструкции и большие электрические цепи.
Известны примеры решенных в последние годы задач, где число неизвестных достигало сотен тысяч. Естественно, это было бы невозможно, если бы соответствующие матрицы не являлись разреженными (матрица системы из 100 тыс. уравнений в формате двойной точности заняла бы около 75 Гбайт).

Результаты (английский) 1:

Solving systems of linear algebraic equations-one of the main tasks of the computational linear algebra. While the task of solving the system of linear equations is relatively rarely represents an independent interest for applications, the ability to deal effectively with such systems often depend on the very possibility of mathematical modeling of a wide variety of processes with the use of computers. A large part of solving various numerical methods (in particular, nonlinear) includes solving systems of linear equations as an elementary step appropriate algorithm. One of the difficulties of a practical solution of systems of large dimension is associated with limited RAM. Although the volume of RAM newly created computing machines is growing very fast, however, even faster increasing needs practice in addressing the challenges of increasing dimension. To a large extent restrictions on dimension of current systems can be removed if used to store matrix external storage devices. However, in this case magnified as the cost of computer time, and the complexity of the relevant algorithms. Therefore, when you create linear algebra computational algorithms much attention to methods of a compact layout of matrices in COMPUTER memory. Fortunately, applications very often lead to matrices with nonzero elements a lot less than the total numbers of elements of a matrix. Such matrices commonly called sparse. One of the main sources of sparse matrix are mathematical models of technical devices, consisting of a large number of items that are local. The simplest examples of such devices are complex constructions and large electrical circuits. Examples of outstanding tasks in recent years, where the number of unknown reached hundreds of thousands. Naturally, this would be impossible if the matrices were not sparse (Matrix system from 100 thousand equations in double-precision format would take about 75 GB).

Результаты (английский) 2:

Solution of systems of linear algebraic equations - one of the main problems of computational linear algebra. Although the problem of solving a system of linear equations is relatively rare independent interest for applications, the ability to deal effectively with such systems often depends on the very possibility of mathematical modeling of a wide variety of processes using a computer. A significant part of numerical methods for solving different. (Especially - non-linear) tasks involves solving systems of linear equations as an elementary step of the corresponding algorithm
One of the difficulties of practical solutions of high dimension systems connected with the limited RAM computer. Although the volume of newly created memory computers is growing very rapidly, however, even faster increase in the practical needs solving problems of increasing dimension. To a large extent on the dimension of the system constraints can be removed, if used to store the matrix external storage devices. However, in this case, many times increasing as the cost of computer time and the complexity of the algorithms. Therefore, when creating computational algorithms of linear algebra pay great attention to the methods of a compact placement of the matrix elements in the computer memory.
Fortunately, the app very often leads to a matrix in which the number of non-zero elements is much smaller than the total Chila elements of the matrix. Such matrices are called sparse. One of the main sources of sparse matrices are mathematical models of technical devices, consisting of a large number of items, connections between them are local. The simplest examples of such devices - the complex engineering structures and large circuits.
There are examples of solved problems in recent years, where unknown numbers reach hundreds of thousands. Naturally, it would not be possible if the corresponding matrix is not sparse (matrix of the system of 100 thousand. Equations in double-precision format would take about 75 GB).

Результаты (английский) 3:

solution of systems of linear equations is one of the main tasks of computational linear algebra. although the solution system of linear equations is a relatively rare interest for applications, the ability to deal effectively with such systems often depends on the possibility of mathematical modelling of various processes with the use of computers. a large part of the numerical methods for solving various (especially nonlinear) tasks include systems of linear equations as an elementary step of the algorithm.one of the difficulties of practical solution of high dimensional systems is linked to the limited memory of computer. although the amount of ram newly emerging computing machines is growing very fast, however, faster growing needs of practice in addressing the challenges of increasing dimensionality. greatly limits the dimensionality of the systems can be removed, if used for the storage of the external storage device. in this case, however, increases as the cost of machine time, and complexity of algorithms. however, with the establishment of computational linear algebra algorithms, attach importance to the ways of the compact of element matrices in memory of computer.fortunately, applications often lead to матрицам, in which the number of ненулевых much less common elements of the matrix elements. such a matrix is called a разреженными. one of the main sources of the diluted matrices are mathematical models of technical devices, consisting of a large number of elements, the relationship between the local. the simplest examples of such devices are complicated construction and large electrical circuits.in recent years, outstanding examples of tasks where the number of unknown reached hundreds of thousands. of course, this would be impossible if the matrix were not разреженными (matrix system of 100 thousand. the equations in the format of double precision would be about 75 gb).