successive approach method

Introduction

The concept of continuous approximation was initially developed by electrical engineers seeking to analyze behavior of circuit systems. This method of analysis seeks to minimize errors in modeling by assuming that the system is infinite and infinitely differentiable, avoiding separate discrete steps. This approximation is also known as a differential approximation, and its main advantage lies in its ability to provide a completely accurate representation of a system without the cumbersome process of analyzing it discretely.

Definition

Put simply, continuous approximation is the process of making an attempt to accurately predict the behavior of a system by using an infinite number of sample points. It can be seen as a type of mathematical modeling, where the system is taken to be an continuous function rather than a discrete entity. The sample points used in this approximation can be created with such tools as finite difference, finite elements, and boundary elements. In essence, the continuous approximation seeks to reduce the number of discrete steps needed to analyze a given system by simplifying the complexity of the system into a linear, continuous graph.

In practice, this continuous approximation works best when its behavior is primarily determined by a small number of parameters. The smaller the number of parameters, the more accurately the approximation will be able to represent the behavior of the real system.

Application

Continuous approximation is widely used in a variety of fields, including physical sciences, engineering, economics, and finance. In physical sciences, it can be used to analyze the behavior of electrical circuits and mechanical systems. This method was also widely used in mechanical engineering, where it was used to calculate the behavior of machines without having to solve the equations of motion discretely.

In economics, continuous approximation was used to analyze the changes in prices of commodities, or the behavior of stock markets. One well-known example is the Black-Scholes equation, which is a tool used to price European-style stock options. This equation uses the continuous approximation to analyze a system of different variables related to the stock market to come up with a valuation for stock options.

In finance, continuous approximation is a widely used tool for analyzing portfolio performance. This method takes into account the different characteristics of the investments in the portfolio to produce a more accurate analysis as compared to if each asset was examined discretely.

History

The concept of continuous approximation has its roots in the work of analysts such as William Thomson, James Clerk Maxwell, Augustus De Morgan, Pierre-Simon Laplace and George Boole. These analysts developed several basic tools for modeling and analyzing physical systems, tools that are still widely used today.

Later on, this concept was further developed by mathematicians such as Joseph Fourier, who sought to solve differential equations without having to solve them discretely. This was followed by the development of the Black-Scholes equation in 1973 by Fischer Black and Myron Scholes. This groundbreaking equation used the continuous approximation to value the performance of stock options and is still widely used today in financial analysis.

Conclusion

Continuous approximation is a method of analyzing behavior of system without having to solve the underlying equations discretely. This method is based on the assumption that the system is infinitely differentiable and is widely used in fields such as engineering, economics and finance. Its roots can be traced back to the work of analysts such as William Thomson, James Clerk Maxwell and George Boole, and it was popularized by mathematicians such as Joseph Fourier and further developed by Fischer Black and Myron Scholes with the creation of the Black-Scholes equation. Continuous approximation has the potential to reduce errors in analyzing a system and provide more accurate results.

The Continuous Approach is a method of problem solving employed primarily in mathematics and computer science contexts. It uses numerical data and algorithms for the introduction of a problem by way of a continuous approximation, as opposed to discrete input. This method is useful for analytical computations and requires only knowledge of certain types of equations to be successful.

The continuous approach is often employed in optimization problems, where it is used to find the most energy efficient solution to a given problem. This method begins by introducing the problem into a finite series of equations and then applying numerical algorithms to these equations. As this occurs, the equations are continually tested with varying levels of accuracy until a solution is found. Aside from being used in optimization, the continuous approach can also be used to approximate solutions to estimation and integration problems.

In optimization, the continuous approach can provide greater accuracy than alternative methods of higher accuracy. It often requires fewer calculations, as well as lesser knowledge of the system being analyzed. Another advantage to the continuous approach is that it can also be used to compute discrete functions, though not as accurately as a dedicated discrete method.

The continuous approach is widely employed in other areas, such as curve fitting and numerical optimization. It is often chosen over other methods when an approximate rather than exact solution is required. Due to its nature and its accuracy in the areas of optimization, estimation, and fitting, the continuous approach remains incredibly important to the fields of mathematics and computer science.