Error Analysis In Numerical Analysis Computer Science Essay

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Numerical analysis involves the study of methods of computing numerical data. In many problems this implies producing a sequence of approximations; thus the questions involve the rate of convergence, the accuracy (or even validity) of the answer, and the completeness of the response. (With many problems it is difficult to decide from a program's termination whether other solutions exist.) Since many problems across mathematics can be reduced to linear algebra, this too is studied numerically; here there are significant problems with the amount of time necessary to process the initial data. Numerical solutions to differential equations require the determination not of a few numbers but of an entire function; in particular, convergence must be judged by some global criterion. Other topics include numerical simulation, optimization, and graphical analysis, and the development of robust working code.

Numerical linear algebra topics: solutions of linear systems AX = B, eigenvalues and eigenvectors, matrix factorizations. Calculus topics: numerical differentiation and integration, interpolation, solutions of nonlinear equations f(x) = 0. Statistical topics: polynomial approximation, curve fitting.

Error analysis is the study of kind and quantity of error that occurs, particularly in the fields of applied mathematics (particularly numerical analysis), applied linguistics and statistics.

General introduction:

The overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to hard problems, the variety of which is suggested by the following.

Advanced numerical methods are essential in making numerical weather prediction feasible.

Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations.

Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving partial differential equations numerically.

Hedge funds (private investment funds) use tools from all fields of numerical analysis to calculate the value of stocks and derivatives more precisely than other market participants.

Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs. This field is also called operations research.

Insurance companies use numerical programs for actuarial analysis.

The rest of this section outlines several important themes of numerical analysis.

Error analysis in numerical modeling

Backward error analysis involves the analysis of the approximation function , to determine the bounds on the parameters  such that the result .[2] In numerical simulation or modeling of real systems, error analysis is concerned with the changes in the output of the model as the parameters to the model vary about a mean.

For instance, in a system modeled as a function of two variables . Error analysis deals with the propagation of the numerical errors in  and  (around mean values  and ) to error in (around a mean ).[1]

In numerical analysis, error analysis comprises both forward error analysis and backward error analysis. Forward error analysis involves the analysis of a function  which is an approximation (usually a finite polynomial) to a function  to determine the bounds on the error in the approximation; i.e., to find  such that .

In numerical simulation or modeling of real systems, error analysis is concerned with the changes in the output of the model as the parameters to the model vary about a mean.

For instance, in a system modeled as a function of two variables . Error analysis deals with the propagation of the numerical errors in and (around mean values and ) to error in (around a mean ).[1]

In numerical analysis, error analysis comprises both forward error analysis and backward error analysis. Forward error analysis involves the analysis of a function which is an approximation (usually a finite polynomial) to a function to determine the bounds on the error in the approximation; i.e., to find such that . Backward error analysis involves the analysis of the approximation function , to determine the bounds on the parameters such that the result .[2]

Areas of numerical analysis:

A rough categorization of the principal areas of numerical analysis is given below, keeping in mind that there is often a great deal of overlap between the listed areas. In addition, the numerical solution of many mathematical problems involves some combination of some of these areas, possibly all of them. There are also a few problems which do not fit neatly into any of the following categories.

Types Of Error Analysis in numerical analysis:

ï‚· In the world of math, the practice of numerical analysis is well known for focusing on algorithms as they are used to solve issues in continuous math. The practice is familiar territory for engineers and those who work with physical science, but it is beginning to expand further into liberal arts areas as well. This can be seen in astrology, stock portfolio analysis, data analysis and medicine. Part of the application of numerical analysis involves the use of errors. Specific errors are sought out and applied to arrive at mathematical conclusions.

The Round-Off Error:

ï‚· The round-off error is used because it a representation of every number as a real number is not possible. So rounding is introduced adjust for this situation. A round-off error, represents the numerical amount between what a figure actually is versus its closest real number value, depending on how the round is applied. For instance, rounding to the nearest whole number means you round up or down to what is the closest whole figure. So if your result is 3.31 then you would round to 3. Rounding the highest amount would be a bit different. In this approach, if your figure is 3.31, your rounding would be to 4. In terms of numerical analysis the round-off error is an attempt to identify what the rounding distance is when it comes up in algorithms. It's also known as a quantization error.

The Truncation Error:

ï‚· A truncation error occurs when approximation is involved in numerical analysis. The error factor is related to how much the approximate value is a variance from the actual value in a formula or math result. For example, take the formula of 3 times 3 plus 4. The calculation equals 28. Now, break it down and the root is close to 1.99. The truncation error value is equal to 0.01.

The Discretization Error:

ï‚· As a type of truncation error, the discretization error focuses on how much a discrete math problem is not consistent with a continuous math problem.

Numerical Stability Errors:

ï‚· If an error stays at one point in an algorithm and doesn't aggregate further as the calculation continues, then it is considered a numerically stable error. This happens when the error causes only a very small variation in the formula result. If the opposite occurs, and the error propagates bigger as the calculation continues, then it is considered numerically unstable.

Approximation theory:

Use computable functions  to approximate the values of functions  that are not easily computable or use approximations to simplify dealing with such functions. The most popular types of computable functions  are polynomials, rational functions, and piecewise versions of them, for example spline functions. Trigonometric polynomials are also a very useful choice.

Best approximations. Here a given function  is approximated within a given finite-dimensional family of computable functions. The quality of the approximation is expressed by a functional, usually the maximum absolute value of the approximation error or an integral involving the error. Least squares approximations and minimax approximations are the most popular choices.

Interpolation. A computable function  is to be chosen to agree with a given  at a given finite set of points . The study of determining and analyzing such interpolation functions is still an active area of research, particularly when  is a multivariate polynomial.

Fourier series. A function  is decomposed into orthogonal components based on a given orthogonal basis , and then  is approximated by using only the largest of such components. The convergence of Fourier series is a classical area of mathematics, and it is very important in many fields of application. The development of the Fast Fourier Transform in 1965 spawned a rapid progress in digital technology. In the 1990s wavelets became an important tool in this area.

Numerical integration and differentiation. Most integrals cannot be evaluated directly in terms of elementary functions, and instead they must be approximated numerically. Most functions can be differentiated analytically, but there is still a need for numerical differentiation, both to approximate the derivative of numerical data and to obtain approximations for discretizing differential equations.

we studied three numerical methods for the discretization in time of a fractional-order evolution equation in a Banach space framework. Each of the methods applied a quadrature rule to a contour integral representation of the solution in the complex plane, where for each quadrature point an elliptic boundary-value problem had to be solved to determine the value of the integrand. The first two methods involved the Laplace transform of the forcing term, but the third did not. We analysed both the quadrature error and the error arising from a spatial discretization by finite elements, measured in the L2-norm. The present work extends our earlier results by proving error bounds in the technically more complicated case of the maximum norm. We also establish new regularity properties for the exact solution that are needed for our analysis.

Development of numerical methods

Numerical analysts and applied mathematicians have a variety of tools which they use in developing numerical methods for solving mathematical problems. An important perspective, one mentioned earlier, which cuts across all types of mathematical problems is that of replacing the given problem with a 'nearby problem' which can be solved more easily. There are other perspectives which vary with the type of mathematical problem being solved.

Numerical solution of systems of linear equations:

Linear systems arise in many of the problems of numerical analysis, a reflection of the approximation of mathematical problems using linearization. This leads to diversity in the characteristics of linear systems, and for this reason there are numerous approaches to solving linear systems. As an example, numerical methods for solving partial differential equations often lead to very large 'sparse' linear systems in which most coefficients are zero. Solving such sparse systems requires methods that are quite different from those used to solve more moderate sized 'dense' linear systems in which most coefficients are non-zero.

There are 'direct methods' and 'iterative methods' for solving all types of linear systems, and the method of choice depends on the characteristics of both the linear system and on the computer hardware being used. For example, some sparse systems can be solved by direct methods, whereas others are better solved using iteration. With iteration methods, the linear system is sometimes transformed to an equivalent form that is more amenable to being solved by iteration; this is often called 'pre-conditioning' of the linear system.

With the matrix eigenvalue problem , it is standard to transform the matrix to a simpler form, one for which the eigenvalue problem can be solved more easily and/or cheaply. A favorite choice are 'orthogonal transformations' because they are a simple and stable way to convert the given matrix . Orthogonal transformations are also very useful in transforming other problems in numerical linear algebra. Of particular importance in this regard is the least squares solution of over-determined linear systems.

The linear programming problem was solved principally by the 'simplex method' until new approaches were developed in the 1980s, and it remains an important method of solution. The simplex method is a direct method that uses tools from the numerical solution of linear systems.

Numerical solution of systems of nonlinear equations

With a single equation, and having an initial estimate of the root, approximate by its tangent line at the point. Find the root of this tangent line as an approximation to the root of the original equation. This leads to 'Newton's iteration method',

Other linear and higher degree approximations can be used, and these lead to alternative iteration methods. An important derivative-free approximation of Newton's method is the 'secant method'.

For a system of nonlinear equations for a solution vector in, we approximate by its linear Taylor approximation about the initial estimate . This leads to Newton's method for nonlinear systems,

In which denotes the Jacobean matrix, of order for .

In practice, the Jacobean matrix for is often too complicated to compute directly; instead the partial derivatives in the Jacobean matrix are approximated using 'finite differences'. This leads to a 'finite difference Newton method'. As an alternative strategy and in analogy with the development of the secant method for the single variable problem, there is a similar root finding iteration method for solving nonlinear systems. It is called 'Broyden's method' and it uses finite difference approximations of the derivatives in the Jacobean matrix, avoiding the evaluation of the partial derivatives of .

Numerical methods for solving differential and integral equations:

With such equations, there are usually at least two general steps involved in obtaining a nearby problem from which a numerical approximation can be computed; this is often referred to as 'discretization' of the original problem. The given equation will have a domain on which the unknown function is defined, perhaps an interval in one dimension and maybe a rectangle, ellipse, or other simply connected bounded region in two dimensions. Many numerical methods begin by introducing a mesh or grid on this domain, and the solution is to be approximated using this grid. Following this, there are several common approaches.

One approach approximates the equation with a simpler equation defined on the mesh. For example, consider approximating the boundary value problem

Introduce a set of mesh points , , with for some given . Approximate the boundary value problem by

The second derivative in the original problem has been replaced by a numerical approximation to the second derivative. The new problem is a finite system of nonlinear equations, presumably amenable to solution by known techniques. The solution to this new problem is , and it is defined on only the mesh points .

A second approach to discretizing differential and integral equations is as follows. Choose a finite-dimensional family of functions, denoted here by , with which to approximate the unknown solution function . Write the given differential or integral equation as , with a function for any function , perhaps over a restricted class of functions . The numerical method consists of selecting a function such that is a small function in some sense. The various ways of doing this lead to 'Galerkin methods', 'collocation methods', and 'least square methods'.

Yet another approach is to reformulate the equation as an optimization problem. Such reformulations are a part of the classical area of mathematics known as the 'calculus of variations', a subject that reflects the importance in physics of minimization principles. The well-known 'finite element method' for solving elliptic partial differential equations is obtained in this way, although it often coincides with a Galerkin method.

The approximating functions in are often chosen as piecewise polynomial functions which are polynomial over the elements of the mesh chosen earlier. Such methods are sometimes called 'local methods'. When the approximating functions are defined without reference to a grid, then the methods are sometimes called 'global methods' or 'spectral methods'. Examples of such are sets of polynomials or trigonometric functions of some finite degree or less.

With all three approaches to solving differential or integral equations, the intent is that the resulting solution be close to the desired solution . The business of theoretical numerical analysis is to analyze such an algorithm and investigate the size of .

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of continuous mathematics (as distinguished from discrete mathematics).

One of the earliest mathematical writings is the Babylonian tablet BC 7289, which gives a sexagesimal numerical approximation of , the length of the diagonal in a unit square. Being able to compute the sides of a triangle (and hence, being able to compute square roots) is extremely important, for instance, in carpentry and construction.[3]

Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation of , modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors.

Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st century, the life sciences and even the arts have adopted elements of scientific computations. Ordinary differential equations appear in the movement of heavenly bodies (planets, stars and galaxies); optimization occurs in portfolio management; numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.

Before the advent of modern computers numerical methods often depended on hand interpolation in large printed tables. Since the mid 20th century, computers calculate the required functions instead. The interpolation algorithms nevertheless may be used as part of the software for solving differential equations.

Conclusion:

ï‚· Math errors, unlike the inference of their name, come in useful in statistics, computer programming, advanced mathematics and much more. The error evaluation provides significantly useful information, especially when probability is required.

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