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What is Numerical Analysis?
Numerical analysis is the use of mathematical techniques to solve problems. The field of numerical analysis encompasses the development and analysis of algorithms for efficient computation. Numerical analysis (and scientific computing) focuses on obtaining approximate solutions to technical problems when it is unfeasible or too time-consuming to get an exact solution. Many numerical methods assume that the problem can be reduced to another finite problem in a computational domain with some known properties and structures.
Importance of Numerical Analysis
The development of a theory of computation is not the only component for developing numerical analysis, but it plays a major role in helping to develop efficient algorithms. The numerical analysis provides tools that allow us to approximate solutions to difficult problems and then use these approximations (along with other methods) to help solve other problems. Many computational methods are based on the assumption that if we know how accurately we can solve one problem, we can efficiently solve many related problems. Therefore, there is always a tradeoff between accuracy and efficiency when applying these methods.
An algorithm is an efficiently implemented process or set of operations used to achieve some goal—which might be as simple as sorting data or as complex as finding a cure for cancer. The study of algorithms and their complexity is a major field in computer science that has developed thousands of such processes, many more than can be listed here.
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Topics Covered in Our Numerical Analysis Assignment Help
Series and sequence:
The terms series and sequence are often used interchangeably in mathematics. Both have an order, though the term ‘series’ is generally used to refer to a finite sequence and ‘sequence’ for infinite sequences. The elements of a sequence may also be called its terms.
An index n usually denotes a real-valued sequence. It indicates the position of that element in the nth position counting from zero as origin in most programming languages or scientific calculators. This strict definition allows us to implicitly indicate any integer n as well as negative integers, thus allowing zero-based counting, including negative indices that make arithmetic operations easier. However, some authors use other notations, such as (a 0, a 1, …).
Interpolation is the process of filling in “missing” data points along a curve. It is a very useful technique when constructing smooth curves or surfaces from discrete sample points obtained by measurement and scanning. To interpolate means to get an estimate of a parameter for values where no exact measurement is available. The term interpolation refers to the fact that these parameters connect the given discrete data points on a function; by contrast, extrapolation refers to estimating values outside these known data points, using other information about the function.
A nonlinear equation is an algebraic equation containing more terms, not all of the same order. They can be solved in much the same way, but they present a greater challenge than linear equations since their solutions tend to be more complex. Nonlinear problems may also have multiple solutions, peculiar behavior, and higher dimensions, requiring computer assistance for visualization.
Fourier analysis, named after Joseph Fourier who was a French mathematician and physicist born in 1768 , refers to several related concepts in mathematics and theoretical physics: discrete-time or continuous-time frequency transformations, such as Fourier series or Fourier transform; infinite sequences/forms called “Fouriers” of functions; and operators acting on functions, which are central to the modern theory. These concepts have widespread use in science and mathematics, such as signal processing, cosmology, quantum mechanics, image processing, and statistics.
An integral is a fundamental concept in calculus. It can define a function (an indefinite integral) or the area under a curve (a definite integral). Elementary algebraic methods compute the integrals of most familiar functions, but general forms are usually handled with higher degree integrals and require numerical integration.
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