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This project aims to employ the aptitude treatment interaction model in the teaching system of linear equations as an alternative learning approach for high school students pursuing vocational careers. Instructional design was utilized to develop the learning model. An essential aspect of applying the aptitude treatment interaction model to learning design is the analysis of student requirements. Learning outcomes and performance goals are thoughtfully considered and aligned with instructional objectives. The findings suggest that educators should utilize the aptitude treatment interaction model when teaching systems of linear equations. The effectiveness of the learning model depends on how well it is tailored to the student's abilities., Tujuan penelitian ini adalah untuk memberikan alternatif pilihan desain pembelajaran, dengan menggunakan model aptitude treatment interaction pada materi sistem persamaan linear untuk siswa SMK. Penulisan ini menggunakan desain instruksional untuk mengembangkan model pembelajaran. Pengembangan desain pembelajaran dilakukan dengan menganalisis kebutuhan peserta didik. Tujuan kinerja dan hasil belajar dibahas secara rinci dan disesuaikan dengan tujuan instruksional. Hasil penelitian ini menunjukkan bahwa desain pembelajaran pada materi sistem persamaan linear dengan menerapkan model aptitude treatment interaction direkomendasikan bagi pendidik dalam melaksanakan pembelajaran sistem persamaan linear. Hal ini terjadi karena strategi pembelajaran (treatment) yang efektif diterapkan kepada siswa adalah berdasarkan kemampuannya masing-masing, sehingga dapat membantu siswa dalam memahami sistem persamaan linear.

Solving linear systems of equations is a common situation in many scientific and technological problems. Many methods either analytical or numerical, have been developed to solve them so, in this paper, I will identify the best method among all the methods which are used to solve the linear system of equations for this we need to define some concepts. The most commonly used general method linear algebra is Gaussian elimination, Gauss Jordan, Crammer's rule method, Gauss Jacobies, and Gauss Seidel methods.. In this paper ,I ,will explain these by taking an example. Also in this paper, I will explain the Researcher’s works that how they explain – different methods by taking a different example. And I worked on using these different methods in solving a single example; finally, I will enquire about the best method among them. In this paper, I will explain the real-life application that how the System of Linear Equation is used in our daily life

The focus in this paper is interior-point methods for bound-constrained nonlinear optimization, where the system of nonlinear equations that arise are solved with Newton’s method. There is a trade-off between solving Newton systems directly, which give high quality solutions, and solving many approximate Newton systems which are computationally less expensive but give lower quality solutions. We propose partial and full approximate solutions to the Newton systems. The specific approximate solution depends on estimates of the active and inactive constraints at the solution. These sets are at each iteration estimated by basic heuristics. The partial approximate solutions are computationally inexpensive, whereas a system of linear equations needs to be solved for the full approximate solution. The size of the system is determined by the estimate of the inactive constraints at the solution. In addition, we motivate and suggest two Newton-like approaches which are based on an intermediate step that consists of the partial approximate solutions. The theoretical setting is introduced and asymptotic error bounds are given. We also give numerical results to investigate the performance of the approximate solutions within and beyond the theoretical framework., QC 20211130

Graphical Processing Units (GPUs) have become more accessible peripheral devices with great computing capacity. Moreover, GPUs can be used not only to accelerate the graphics produced by a computer but also for general purpose computing. Many researchers use this technique on their personal workstations to accelerate the execution of their programs and have often encountered that the amount of memory available on GPU cards is typically smaller than the amount of memory available on the host computer. We are interested in exploring approaches to solve problems with this restriction. Our main contribution is to devise ways in which portions of the problem can be moved to the memory of the GPU to be solved using its multiprocessing capabilities. We implemented on a GPU the Jacobi iterative method to solve systems of linear equations and report the details from the results obtained, analyzing its performance and accuracy. Our code solves a system of linear equations large enough to exceed the card’s memory, but not the host memory. Significant speedups were observed, as the execution time taken to solve each system is faster than those obtained with Intel® MKL and Eigen, libraries designed to work on CPUs. © Springer Nature Switzerland AG 2018.

Bakalářská práce se zabývá tvorbou výukové aplikace pro vybrané oblasti lineární algebry v jazyce Java. V první části této práce jsou představena vybraná témata a typy úloh z lineární algebry. Také jsou popsány možnosti řešení těchto úloh. V další části práce je proveden návrh výukové aplikace. Aplikace umožňuje generování úloh z vybraných oblastí lineární algebry, které jsou vhodné pro procvičování. Tyto úlohy jsou následně vypočteny a je možné zobrazit jednotlivé kroky výpočtu simulující postup lidského řešitele., This bachelor thesis deals with the creation of an educational application in Java for selected areas of linear algebra. The first part of this work summarizes selected topics and types of exercises from linear algebra. There are also described possible solutions of these exercises. The second part of the thesis explains design of the application. The application enables to generate exercises from the selected areas of linear algebra which are suitable for practice. These exercises are calculated by the application and its user can view particular steps of the solution, simulating calculation procedure of a human solver., Fakulta elektrotechniky a informatiky, Cílem bakalářské práce je vytvořit aplikaci pro generování příkladů z vybraných oblastí lineární algebry. Aplikace bude schopna animovat postup řešení příkladů. Aplikace bude vytvořena v programovacím jazyce Java. Existuje spousta aplikací, které dokážou vyřešit úlohy z lineární algebry. Student však ve své práci vytvořil aplikaci, která kromě samotného řešení úlohy také zobrazí elementární kroky výpočtu tak, aby byl co nejpřesněji simulován postup řešení lidským řešitelem. Každý krok v průběhu výpočtu je ještě slovně okomentován. V aplikaci je obsaženo široké spektrum témat z lineární algebry. Soustavy rovnic, determinanty, inverzní matice, lineární kombinace a lineární závislost vektorů, vlastní čísla a vlastní vektory matice. Také lze zvolit různé úrovně obtížnosti příkladů. Textová část práce pak popisuje vybraná témata na teoretické rovině. Vzniklá práce tak může být výborným pomocníkem při výuce lineární algebry. Vydařená bakalářská práce, hodnocena klasifikací výborně. Rovněž odpovědi na otázky odpovídají celkovému hodnocení výborně. Celkově prospěl s vyznamenáním. Samotný text práce vykazuje shodu méně než 5 procent a není plagiátem. Ve zdrojových kódech byla nalezena vyšší míra shody, to však souvisí se standardními konstrukcemi, které se používají v jazyce Java. Kromě těchto konstrukcí nebyla ve zdrojových kódech identifikována žádná shoda. Práce byla zkontrolována v systému IS/STAG, dle hodnocení komise nejeví známky plagiátorství., Dokončená práce s úspěšnou obhajobou

The Induced Dimension Reduction method (IDR(s)) (Sonneveld and van Gijzen, 2008) is a short-recurrences Krylov method to solve systems of linear equations. In this work, we accelerate this method using spectral information. We construct a Hessenberg relation from the IDR(s) residual recurrences formulas, from which we approximate the eigenvalues and eigenvectors. Using the Ritz values, we propose a self-contained variant of the Ritz-IDR(s) method (Simoncini and Szyld, 2010) for solving a system of linear equations. In addition, the Ritz vectors are used to speed-up IDR(s) for the solution of sequence of systems of linear equations., Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public., Numerical Analysis

Tato bakalářská práce se zabývá teorií maticového počtu a matematickými modely používanými při řešení rozhodovacích úloh. První část práce je věnována základním pojmům teorie matic. V další části jsou presentovány matematické modely a uvedeny oblasti jejich vyžití. Cílem poslední části je ukázat na příkladech, jak se využívají bilanční modely a modely lineárního programování v praxi, This Bachelor's thesis studies the theory of matrix and mathematical models used to solve decision-making problems. The first part of the work is devoted to the fundamental concepts of matrix theory. In the next part are presented mathematical models and areas of application of these models. The aim of the last part is to show examples of how balance models and linear programming models are used in practice, Fakulta ekonomicko-správní, Studentka představila komisi bakalářskou práci s názvem Některé aplikace maticového počtu v ekonomických výpočtech v rámci provozu podniku. Cílem práce bylo vytvoření přehledu základů maticového počtu, charakteristika vybraných matematických modelů a některé aspekty stability tržní rovnováhy. Studentka zodpověděla následující otázky: Popište alespoň jeden další problém z ekonomické praxe, při jehož řešení se účelně využije práce s maticemi. Jaké doporučení na základě Vašich výsledků odvozujete? Jaké změny v podniku na základě Vašich výsledků by měly být provedeny? Dá se Váš výpočet využít v praxi? Cíl práce byl splněn., Dokončená práce s úspěšnou obhajobou

The Induced Dimension Reduction method (IDR(s)) (Sonneveld and van Gijzen, 2008) is a short-recurrences Krylov method to solve systems of linear equations. In this work, we accelerate this method using spectral information. We construct a Hessenberg relation from the IDR(s) residual recurrences formulas, from which we approximate the eigenvalues and eigenvectors. Using the Ritz values, we propose a self-contained variant of the Ritz-IDR(s) method (Simoncini and Szyld, 2010) for solving a system of linear equations. In addition, the Ritz vectors are used to speed-up IDR(s) for the solution of sequence of systems of linear equations., Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public., Numerical Analysis

In several applications in science and engineering, different types of matrix problems emerge from the discretization of partial differential equations. This thesis is devoted to the development of new algorithms to solve this kind of problems. In particular, when the matrices involved are sparse and non-symmetric. The new algorithms are based on the Induced Dimension Reduction method [IDR(s)]. IDR(s) is a Krylov subspace method originally proposed in 2008 to solve systems of linear equations. IDR(s) has received considerable attention due to its stable and fast convergence. It is, therefore, natural to ask if it is possible to extend IDR(s) to solve other matrix problems, and if so, to compare those extensions with other well-established methods. This work aims to answer these questions. The main matrix problems considered in this dissertation are: the standard eigenvalue problem, the quadratic eigenvalue problem, the solution of systems of linear equations, the solution of sequences of systems of linear equations, and linear matrix equations. We focus on examples that arise from the discretization of partial differential equations., Numerical Analysis

In several applications in science and engineering, different types of matrix problems emerge from the discretization of partial differential equations. This thesis is devoted to the development of new algorithms to solve this kind of problems. In particular, when the matrices involved are sparse and non-symmetric. The new algorithms are based on the Induced Dimension Reduction method [IDR(s)]. IDR(s) is a Krylov subspace method originally proposed in 2008 to solve systems of linear equations. IDR(s) has received considerable attention due to its stable and fast convergence. It is, therefore, natural to ask if it is possible to extend IDR(s) to solve other matrix problems, and if so, to compare those extensions with other well-established methods. This work aims to answer these questions. The main matrix problems considered in this dissertation are: the standard eigenvalue problem, the quadratic eigenvalue problem, the solution of systems of linear equations, the solution of sequences of systems of linear equations, and linear matrix equations. We focus on examples that arise from the discretization of partial differential equations., Numerical Analysis

The present work evaluates a recently developed particle method (DFPM). The basic idea behind this method is to utilize a Newtonian system of interacting particles that through dissipation solves mathematical problems. We find that this second order dynamical system results in an algorithm that is among the best methods known. The present work studies large systems of linear equations. Of special interest is the wide eigenvalue spectrum. This case is common as the discretization of the continuous problem becomes dense. The convergence rate of DFPM is shown to be in parity with that of the conjugate gradient method, both analytically and through numerical examples. However, an advantage with DFPM is that it is cheaper per iteration. Another advantage is that it is not restricted to symmetric matrices only, as is the case for the conjugate gradient method. The convergence properties of DFPM are shown to be superior to the closely related approach utilizing only a first order dynamical system, and also to several other iterative methods in numerical linear algebra. The performance properties are understood and optimized by taking advantage of critically damped oscillators in classical mechanics. Just as in the case of the conjugate gradient method, a limitation is that all eigenvalues (spring constants) are required to be of the same sign. DFPM has no other limitation such as matrix structure or a spectral radius as is common among iterative methods. Examples are provided to test the particle algorithm’s merits and also various performance comparisons with existent numerical algorithms are provided.

The present work evaluates a recently developed particle method (DFPM). The basic idea behind this method is to utilize a Newtonian system of interacting particles that through dissipation solves mathematical problems. We find that this second order dynamical system results in an algorithm that is among the best methods known. The present work studies large systems of linear equations. Of special interest is the wide eigenvalue spectrum. This case is common as the discretization of the continuous problem becomes dense. The convergence rate of DFPM is shown to be in parity with that of the conjugate gradient method, both analytically and through numerical examples. However, an advantage with DFPM is that it is cheaper per iteration. Another advantage is that it is not restricted to symmetric matrices only, as is the case for the conjugate gradient method. The convergence properties of DFPM are shown to be superior to the closely related approach utilizing only a first order dynamical system, and also to several other iterative methods in numerical linear algebra. The performance properties are understood and optimized by taking advantage of critically damped oscillators in classical mechanics. Just as in the case of the conjugate gradient method, a limitation is that all eigenvalues (spring constants) are required to be of the same sign. DFPM has no other limitation such as matrix structure or a spectral radius as is common among iterative methods. Examples are provided to test the particle algorithm’s merits and also various performance comparisons with existent numerical algorithms are provided.

Розглядається можливість розпаралелення методу простої ітерації — найпростішого серед ітераційних методів розв’язування систем лінійних алгебричних рівнянь. Побудовано однорідне обчислювальне середовище — систолічний масив реалізації цього алгоритму., Построена схема однородной вычислительной среды (специализированной многопроцессорной вычислительной машины) для нахождения решения систем линейных алгебраических уравнений методом простой итерации., The chart of homogeneous calculable environment (specialized multiprocessor calculable machine) is built for decision making of the systems of linear algebraic equalizations by the method of simple iteration.

У роботі розглядається аналітичний багато частотним спосіб вимірювання відстаней до багатьох об‘єктів. Шляхом вимірювання фазових зсувів та амплітуд гармонійних сигналів в заданому діапазоні частот із рівномірним кроком відбитих від трьох та чотирьох об‘єктів та розв‘язання відповідної системи рівнянь встановлено що даний підхід можна розповсюдити на довільну кількість об’єктів. Для вимірювання відстаней до деякої кількості об‘єктів необхідно застосування в два рази більшої кількості частот ніж об‘єктів., Introduction. Existing methods for measuring broadband over distances do not allow further raise the accuracy and distinguishing ability, because it leads to further unnecessary complication equipment. Phase methods have the distinguished ability. Developed multifrequency phase methods can both raise the accuracy and distinctive ability and significantly narrow the frequency range of measurement signals. Requirements for document. The many analytical frequency methods of measuring objects distances are examined in this paper. It is shown that the proposed approach can be extended to any number of objects by measuring the phase shift and amplitude harmonic signals in a given frequency range with uniform reflected from step three and four objects and solving the corresponding system of equations. It is necessary to use two times more frequency than objects measuring the distance to objects. Conclusions. This approach allows us to develop methods for Multiphase ranging many objects. Further improvement approach should be performed in feasibility study to use in radar, and also to measure the radial velocity measuring the Doppler frequency shift., В работе рассматривается аналитический многочастотный способ измерения расстояний до многих объектов. Путем измерения фазовых сдвигов и амплитуд гармонических сигналов в заданном диапазоне частот с равномерным шагом отраженных от трех и четырех объектов и решения соответствующей системы уравнений установлено, что данный подход можно распространить на любое количество объектов. Для измерения расстояний до некоторого количества объектов необходимо применение в два раза большего количества частот, чем объектов.

У роботі розглядається аналітичний багато частотним спосіб вимірювання відстаней до багатьох об‘єктів. Шляхом вимірювання фазових зсувів та амплітуд гармонійних сигналів в заданому діапазоні частот із рівномірним кроком відбитих від трьох та чотирьох об‘єктів та розв‘язання відповідної системи рівнянь встановлено що даний підхід можна розповсюдити на довільну кількість об’єктів. Для вимірювання відстаней до деякої кількості об‘єктів необхідно застосування в два рази більшої кількості частот ніж об‘єктів.

This course offers a rigorous treatment of linear algebra, including vector spaces, systems of linear equations, bases, linear independence, matrices, determinants, eigenvalues, inner products, quadratic forms, and canonical forms of matrices. Compared with Linear Algebra (18.06), more emphasis is placed on theory and proofs.

This course offers a rigorous treatment of linear algebra, including vector spaces, systems of linear equations, bases, linear independence, matrices, determinants, eigenvalues, inner products, quadratic forms, and canonical forms of matrices. Compared with Linear Algebra (18.06), more emphasis is placed on theory and proofs.

This course offers a rigorous treatment of linear algebra, including vector spaces, systems of linear equations, bases, linear independence, matrices, determinants, eigenvalues, inner products, quadratic forms, and canonical forms of matrices. Compared with Linear Algebra (18.06), more emphasis is placed on theory and proofs.

The stochastic fluid flow model (SFF) is one of the leading models in performance evaluation for tele- and datacommunication systems, especially in fast packet-switching networks and ATM. However, the numerical analysis of the SFF is widely considered to be unstable. In this paper, some investigations and results are presented concerning the numerical stability of the SFF analysis also for large systems with finite buffer. We identify the main source of the numerical problems and give hints how to circumvent them. The usefulness of different solution methods are compared and the most robust methods for systems with large numbers of sources and large buffer sizes are identifed., Pappret identifierar anledningar varför den stokastiska flödesmodellen analysen gärna kantrar numeriskt sett och berättar om möjliga åtgärder. Flödesmodell, QoS, numerisk analys, Eigensystem, linjär ekvationssystem

The stochastic fluid flow model (SFF) is one of the leading models in performance evaluation for tele- and datacommunication systems, especially in fast packet-switching networks and ATM. However, the numerical analysis of the SFF is widely considered to be unstable. In this paper, some investigations and results are presented concerning the numerical stability of the SFF analysis also for large systems with finite buffer. We identify the main source of the numerical problems and give hints how to circumvent them. The usefulness of different solution methods are compared and the most robust methods for systems with large numbers of sources and large buffer sizes are identifed., Pappret identifierar anledningar varför den stokastiska flödesmodellen analysen gärna kantrar numeriskt sett och berättar om möjliga åtgärder. Flödesmodell, QoS, numerisk analys, Eigensystem, linjär ekvationssystem

The stochastic fluid flow model (SFF) is one of the leading models in performance evaluation for tele- and datacommunication systems, especially in fast packet-switching networks and ATM. However, the numerical analysis of the SFF is widely considered to be unstable. In this paper, some investigations and results are presented concerning the numerical stability of the SFF analysis also for large systems with finite buffer. We identify the main source of the numerical problems and give hints how to circumvent them. The usefulness of different solution methods are compared and the most robust methods for systems with large numbers of sources and large buffer sizes are identifed., Pappret identifierar anledningar varför den stokastiska flödesmodellen analysen gärna kantrar numeriskt sett och berättar om möjliga åtgärder. Flödesmodell, QoS, numerisk analys, Eigensystem, linjär ekvationssystem

The stochastic fluid flow model (SFF) is one of the leading models in performance evaluation for tele- and datacommunication systems, especially in fast packet-switching networks and ATM. However, the numerical analysis of the SFF is widely considered to be unstable. In this paper, some investigations and results are presented concerning the numerical stability of the SFF analysis also for large systems with finite buffer. We identify the main source of the numerical problems and give hints how to circumvent them. The usefulness of different solution methods are compared and the most robust methods for systems with large numbers of sources and large buffer sizes are identifed., Pappret identifierar anledningar varför den stokastiska flödesmodellen analysen gärna kantrar numeriskt sett och berättar om möjliga åtgärder. Flödesmodell, QoS, numerisk analys, Eigensystem, linjär ekvationssystem

The stochastic fluid flow model (SFF) is one of the leading models in performance evaluation for tele- and datacommunication systems, especially in fast packet-switching networks and ATM. However, the numerical analysis of the SFF is widely considered to be unstable. In this paper, some investigations and results are presented concerning the numerical stability of the SFF analysis also for large systems with finite buffer. We identify the main source of the numerical problems and give hints how to circumvent them. The usefulness of different solution methods are compared and the most robust methods for systems with large numbers of sources and large buffer sizes are identifed., Pappret identifierar anledningar varför den stokastiska flödesmodellen analysen gärna kantrar numeriskt sett och berättar om möjliga åtgärder. Flödesmodell, QoS, numerisk analys, Eigensystem, linjär ekvationssystem

The stochastic fluid flow model (SFF) is one of the leading models in performance evaluation for tele- and datacommunication systems, especially in fast packet-switching networks and ATM. However, the numerical analysis of the SFF is widely considered to be unstable. In this paper, some investigations and results are presented concerning the numerical stability of the SFF analysis also for large systems with finite buffer. We identify the main source of the numerical problems and give hints how to circumvent them. The usefulness of different solution methods are compared and the most robust methods for systems with large numbers of sources and large buffer sizes are identifed., Pappret identifierar anledningar varför den stokastiska flödesmodellen analysen gärna kantrar numeriskt sett och berättar om möjliga åtgärder. Flödesmodell, QoS, numerisk analys, Eigensystem, linjär ekvationssystem

Tato bakalářská práce se zaobírá s restaurací audiosignálů, které na známých místech obsahují neznáme vzorky, s použitím dvou algoritmů. První z nich je Jannsenův algoritmus a druhá je metoda založená na faktorizaci nezáporných matic. Janssenův algoritmus je založen na principu autoregresního modelu. Restaurace vzorků se provádí tak, aby obnovený signál co nejlépe odpovídal předpokládanému modelu. Algoritmus založený na faktorizaci nezáporných matic se používá k rozkladu frekvenčního spektrogramu signálu jako součinu nezáporných matic., This bachelor's thesis deals with the restoration of audio signals containing unknown samples at known locations using two algorithms. The first is the Janssen algorithm and the second is a method based on non-negative matrix factorization. Janssen algorithm is built on the principle of the autoregressive model. The restoration of the samples is performed in such a way that the restored signal matches the predicted model as precisely as possible. The algorithm based on non-negative matrix factorization is used to decompose the frequency spectrogram of the signal as the product of non-negative matrices.

© 2018, Pleiades Publishing, Ltd. In this article we present a new algebraic approach to the greatest common divisor (GCD) computation of two polynomials based on Bezout’s identity. This approach is based on the solution of system of linear equations. Also we introduce the dmod operation for polynomials. This operation on polynomials f, g is used to reduce the degree of the larger polynomial f in a finite field Fp. This operation saves GCD(f, g). Also we present some ideas how to reduce spurious factors that arise at the procedure.

В данной статье рассматривается метод вывода формулы Крамера для системы двух и трех линейных уравнений и относительно 2 и 3 переменных, получаемых непосредственно методом подстановки и вычислением определителей второго и третьего порядка., This article discusses the method of derivation of Cramer's formula for the system of two and three linear equations and with respect to 2 and 3 variables obtained directly by the method of substitution and calculation of determinants of the second and third order.

В данной статье рассматривается методика продуктивного изучения теории определителей студентами нематематических специальностей с помощью метода Крамера. Формулы Крамера для системы двух линейных уравнений и трех линейных уравнений получаются непосредственно мето-дом подстановки. Определитель предлагается вычислять разложением по первой строке. Для доказательства свойств определителей широко используется метод математической индукции., This article discusses the technique of productive study of the theory of determinants of student non-mathematical specialties using the method of cramer solution of linear equations. Cramer's formula for a system of two linear equations and three linear equations are obtained directly by the method of substitution. Determinant serves to calculate the expansion in the first row. To prove the properties of determinants widely used method of induction.