Investigation of the Basic Notions in Numerical Analysis (original) (raw)
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Journal of Scientific Computing, 2006
In many numerical procedures one wishes to improve the basic approach either to improve efficiency or else to improve accuracy. Frequently this is based on an analysis of the properties of the discrete system being solved. Using a linear algebra approach one then improves the algorithm. We review methods that instead use a continuous analysis and properties of the differential equation rather than the algebraic system. We shall see that frequently one wishes to develop methods that destroy the physical significance of intermediate results. We present cases where this procedure works and others where it fails. Finally we present the opposite case where the physical intuition can be used to develop improved algorithms.
Numerical modeling for some fields related to engineering and obstacle problems.
Numerical analysis is a branch of mathematics devoted to the development of iterative matrix calculation techniques. We are searching for operations optimization as objective to calculate and solve systems of equations of order n with time and energy saving for computers that are conducted to calculate and analyze big data by solving matrix equations. Furthermore, this scientific discipline is producing results with a margin of error of approximation called rates. Thus, the results obtained from the numerical analysis techniques that are held on computer software such as MATLAB or Simulink offers a preliminary diagnosis of the situation of the environment or space targets. By this we can offer technical procedures needed for engineering or scientific studies exploitable by engineers and scientists. We will propose in this paper the following scientific applications: The call for scientific computing to analyze the reflection of signals from the upper soil layer to the lower layers to deduce the approximate nature of the geological layers, the existence of water, minerals, rocks, oil .... Approximation of the velocity of groundwater flow. Provide indicators to the risk of flooding compared to the amount of rainfall and the geographical coordinates. Operation calculated to derive the digital ecosystem balance. The digital analysis of data from the population census to anticipate sustainable development strategies. In fact we can develop several numerical analysis methods to model phenomena of the natural order, social, environmental, economy, to have an approximate vision of the role of computers that calculated through the following software mathematical algorithms developed for scientific calculus. At the end of this paper we will address questions bellow: Is it possible to defragment scientific calculations on multiple computers and then gather the results for a central computer? If so, what is the algorithm to take to manage a computer network in parallel way to solve complex numerical problems with very huge matrices? Is it possible to exploit the numerical analysis by data sampling to reduce the time for global calculus? If so, then how quality approximation to approach reality? Conclusion : The numerical analysis methods for modeling complex problems, and simulation of multiple and varied phenomenal status, remain an important and vital topic that promises an intelligent development of optimization techniques for numerical calculus witch give profitability in several sectors like engineering and scientific research.
Journal of Computational and Applied Mathematics, 2000
This volume contains contributions in the area of di erential equations and integral equations. The editors wish to thank the numerous authors, referees, and fellow editors Claude Brezinski and Luc Wuytack, who have made this volume a possibility; it has been a major but personally rewarding e ort to compile it. Due to the limited number of pages we were obliged to make a selection when composing this volume. At an early stage it was agreed that, despite the connections between the subject areas, it would be beneÿcial to allocate the area of partial di erential equations to a volume for that area alone.
Numerical analysis is a branch of mathematics devoted to the development of iterative matrix calculation techniques. We are searching for operations optimization as objective to calculate and solve systems of equations of order n with time and energy saving for computers that are conducted to calculate big data for solving matrix equations sizes. Furthermore, this scientific discipline is producing results with a margin of error of approximation called rates. Thus, the results obtained from the numerical analysis techniques that are held on computer software such as MATLAB or Simulink offers a preliminary diagnosis of the situation of the environment or space targets. By this we can offer technical procedures needed for engineering or scientific studies exploitable by engineers and scientists. We will propose in this paper the following scientific applications: The call for scientific computing to analyze the reflection of signals from the upper soil layer to the lower layers to deduce the approximate nature of the geological layers, the existence of water, minerals, rocks, oil .... Approximation of the velocity of groundwater flow. Provide indicators to the risk of flooding compared to the amount of rainfall and the geographical coordinates. Operation calculated to derive the digital ecosystem balance. The digital analysis of data from the population census to anticipate sustainable development strategies. In fact we can develop several numerical analysis methods to model phenomena of the natural order, social, environmental, economy, to have an approximate vision of the role of computers that calculated through the following software mathematical algorithms developed for scientific calculus. At the end of this paper we will address questions bellow: Is it possible to defragment scientific calculations on multiple computers and then gather the results for a central computer? If so, what is the algorithm to take to manage a computer network in parallel way to solve complex numerical problems with very huge matrices? Is it possible to exploit the numerical analysis by data sampling to reduce the time for global calculus? If so, then how quality approximation to approach reality? Conclusion : The numerical analysis methods for solving complex problems, modeling and simulation of multiple and varied phenomenal status, remain an important and vital topic that promises an intelligent development of optimization techniques for numerical calculus witch give profitability in several sectors like engineering and scientific research.
Numerical algorithms for scientific and engineering applications
Journal of Computational and Applied Mathematics, 2017
Numerical algorithms for scientific and engineering applications The development of the modern society depends crucially on the successful solution of numerous problems, which are often both very challenging and extremely difficult. Scientists and engineers are using complicated and robust mathematical models in the attempts to resolve successfully these problems. Many physical and chemical processes must be incorporated correctly and very carefully in these models in order to increase the reliability and the applicability of the obtained results. This leads to the treatment of complex mathematical tasks, described very often with systems of non-linear partial differential equations. In most of the cases, it is not possible to find the exact solution of these systems. Therefore, one must use different numerical algorithms in the treatment of the discretized systems and, furthermore, to run the selected algorithms on high-speed computers. Many people believe that the modern fast supercomputers will always enable them to resolve successfully even the largest and the most complicated tasks. This is unfortunately not necessarily true. Arthur Jaffe predicted, [1], more than thirty years ago, the fact that the scientists and the engineers will always or at least very often have great problems with the treatment of their models on computers. He wrote in 1984 that: ''Although the fastest computers can execute millions of operations in one second, they are always too slow. This may seem a paradox, but the heart of the matter is: the bigger and better computers become, the larger are the problems scientists and engineers want to solve''.
Numerical Analysis and Optimization methods to solve practical problems in computer science, business, engineering and science. Practical problem solving based on analyzing empirical, experimental or measured data where the precise mathematical model is approximated or not necessarily known. Limitations, trade-offs and margins of error are evaluated for various practical examples such as network traffic, engineering, science and business applications. MATLAB and/or C++ are used for computational problem solving. Suitable for computer science, mathematics, engineering, and business majors.