Study On The Applications Of Numerical Analysis Computer Science Essay

Study On The Applications Of Numerical Analysis Computer Science EssayIt finds applications in alone heavenss of engineering and the physical sciences, nevertheless in the 21stcentury, the life sciences and even the arts have adopted elements of scientific computations.Ordinary differential equationsappear in themovement of heavenly bodies (planets, stars and galaxies) optimisationoccurs in portfolio managementnumerical bilinear algebrais significant for data analysisstochastic differential equationsandMarkov chains atomic number 18 immanent in simulating living cells for treat and biology.Before the advent of modern computers numerical modes practically depended on hand interpolationin turgid printed tables. Since the mid 20th century, computers calculate the required lasts instead. The interpolationalgorithmsnevertheless may be accustomd as part of the software for solvingdifferential equations.INTRODUCTION TO NUMERICAL ANALYSIS AND METHODSThe overall goal of the doma in of a function of numerical analysis is the design and analysis of techniques to give suppose provided accurate solutions to hard conundrums, the variety of which is suggested by the interest.Advanced numerical methods are essential in makingnumerical weather predictionfeasible.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 incomplete differential equationsnumerically.Hedge funds(private investment funds) use hammers from all fields of numerical analysis to calculate the value of stocks and derivatives more precisely than about other food market participants.Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs. This field is also calledoperations research.Insurance companie s use numerical programs foractuarialanalysis.The rest of this section outlines several important themes of numerical analysis.History of Numerical AnalysisThe field of numerical analysis predates the invention of modern computers by umpteen centuries.Linear interpolationwas already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, as is pellucid from the names of important algorithms likeNewtons method,Lagrange interpolation polynomial,Gaussian elimination, orEulers method.To facilitate computations by hand, self-aggrandising books were produced with formulas and tables of data such as interpolation billets and function coefficients. Using these tables, often calculated out to 16 decimal places or more for some functions, one could look up values to plug into the formulas given and progress to precise good numerical estimates of some functions. The canonical work in the field is theNISTpublication edited byAbramowit z and Stegun, a 1000-plus page book of a truly large do of commonly utilise formulas and functions and their values at many points. The function values are no longer actually useful when a computer is available, but the large listing of formulas can still be very handy.Themechanical calculatorwas also developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s, and it was then found that these computers were also useful for administrative purposes. But the invention of the computer also influenced the field of numerical analysis, since straightway longer and more complicated calculations could be done.Direct and iterative methodsDirect methods compute the solution to a paradox in a finite number of steps. These methods would give the precise answer if they were performed ininfinite precision arithmetical. Examples includeGaussian elimination, theQRfactorization method for solvingsystems of linear equations, and thesimplex methodof linear programming. In practice,finite precisionis used and the result is an approximation of the true solution (assumingstability).In contrast to direct methods,iterative methodsare not judge to terminate in a number of steps. Starting from an initial guess, iterative methods form successive approximations thatconvergeto the exact solution only in the limit. Aconvergence testis specified in order to decide when a sufficiently accurate solution has (hopefully) been found. Even using infinite precision arithmetic these methods would not reach the solution within a finite number of steps (in general). Examples includeNewtons method, thebisection method, andJacobi iteration. In computational matrix algebra, iterative methods are in general needed for large problems.Iterative methods are more common than direct methods in numerical analysis. Some methods are direct in principle but are usually used as though they were not, e.g.GMRESand theconjugate gradient method. For these methods t he number of steps needed to obtain the exact solution is so large that an approximation is accepted in the same manner as for an iterative method.DiscretizationFurthermore, continuous problems must sometimes be replaced by a discrete problem whose solution is known to approximate that of the continuous problem this process is calleddiscretization. For example, the solution of adifferential equationis a function. This function must be be by a finite amount of data, for instance by its value at a finite number of points at its public, even though this domain is a continuum.Different Areas And Methods under Numerical AnalysisThe field of numerical analysis is divided into different disciplines according to the problem that is to be solved.One of the simplest problems is the evaluation of a function at a given point. The most straightforward approach, of just plugging in the number in the formula is sometimes not very efficient. For polynomials, a better approach is using theHorner scheme, since it reduces the necessary number of multiplications and additions. Generally, it is important to estimate and controlround-off errorsarising from the use offloating pointarithmetic.Interpolation, extrapolation, and regressionInterpolationsolves the following problem given the value of some unknown function at a number of points, what value does that function have at some other point between the given points?Extrapolationis very similar to interpolation, except that now we want to find the value of the unknown function at a point which is outside the given points.Regressionis also similar, but it takes into account that the data is imprecise. presumptuousness some points, and a measurement of the value of some function at these points (with an error), we want to determine the unknown function. Theleast squares-method is one favourite way to achieve this.Solving equations and systems of equationsAnother fundamental problem is computing the solution of some given equatio n. Two cases are commonly distinguished, depending on whether the equation is linear or not. For instance, the equation2x+ 5 = 3is linear while22+ 5 = 3is not.Much effort has been put in the development of methods for solvingsystems of linear equations. Standard direct methods, i.e., methods that use somematrix decompositionareGaussian elimination,LU decomposition,Cholesky decompositionforsymmetric(orhermitian) andpositive-definite matrix, andQR decompositionfor non-square matrices.Iterative methodssuch as theJacobi method,Gauss-Seidel method,successive over-relaxationandconjugate gradient methodare usually preferable for large systems.Root-finding algorithmsare used to solve nonlinear equations (they are so named since a root of a function is an argument for which the function yields zero). If the function isdifferentiableand the derivative is known, thenNewtons methodis a popular choice.Linearizationis another technique for solving nonlinear equations.Solving eigenvalue or singul ar value problemsSeveral important problems can be phrased in terms ofeigenvalue decompositionsorsingular value decompositions. For instance, thespectral image compressionalgorithmis based on the singular value decomposition. The check tool in statistics is calledprincipal component analysis.OptimizationOptimization problems ask for the point at which a given function is maximized (or minimized). Often, the point also has to satisfy someconstraints.The field of optimization is further split in several subfields, depending on the form of the objective function and the constraint. For instance,linear programmingdeals with the case that both the objective function and the constraints are linear. A famous method in linear programming is thesimplex method.The method ofLagrange multiplierscan be used to reduce optimization problems with constraints to free optimization problems.Evaluating integralsNumerical integration, in some instances also known as numericalquadrature, asks for the v alue of a definiteintegral. Popular methods use one of theNewton-Cotes formulas(like the centre of attention rule orSimpsons rule) orGaussian quadrature. These methods rely on a divide and conquer strategy, whereby an integral on a relatively large square up is broken down into integrals on smaller sets. In higher dimensions, where these methods become prohibitively expensive in terms of computational effort, one may useMonte Carloorquasi-Monte Carlo methods(seeMonte Carlo integration), or, in modestly large dimensions, the method ofsparse grids.Differential equationsNumerical analysis is also concerned with computing (in an approximate way) the solution ofdifferential equations, both ordinary differential equations andpartial differential equations.Partial differential equations are solved by first discretizing the equation, bringing it into a finite-dimensional subspace. This can be done by afinite element method, afinite differencemethod, or (particularly in engineering) afinit e volume method. The theory-based justification of these methods often involves theorems fromfunctional analysis. This reduces the problem to the solution of an algebraic equation.Applications Of Numerical Analysis Methods and Its substantial Life Implementations, Advantages Etc.NEWTON RAPHSON METHOD devote OF point of intersection 2 ADVANTAGES 1. The advantage of the method is its order of convergence is quadratic. 2. Convergence rate is one of the fastest when it does converges 3. Linear convergence near multiple roots.REGULA FALSI METHOD ORDER OF CONVERGENCE 1.618 ADVANTAGES 1. Better-than-linear convergence near simple root 2. Linear convergence near multiple root 3. No derivative needed DISADVANTAGES 1. Iterates may mold 2. No practical rigorous error boundGAUSS ELIMINATION METHODADVANTAGESIt is the direct method of solving linear simultaneous equations. 2. It uses back substitution. 3. It is reduced to like upper triangular matrix. 1. It requires right vectors to be know n.GAUSS JORDAN ADVANTAGES 1. It is direct method. 2. The roots of the equation are found immediately without using back substitution.. It is reduced to equivalent personal identity matrix. The additional steps increase round off errors. 2. It requires right vectors to be known.GAUSS JACOBI METHOD1. It is iterative method. 2. The system of equations must be diagonally dominant. 3. It suits better for large numbers of unknowns 4. It is self correcting method.GAUSS SEIDEL METHOD1. It is iterative method. 2. The system of equations must be diagonally dominant. 3. It suits better for large numbers of unknowns 4. It is self correcting method. 5. The number of iterations is less than Jacobi method.Real life ApplicationsArea of mathematics and computer science.Applications of algebraGeometryCalculusVariables which vary continuously.Problems(application areas)1. Natural sciences2. Social sciences3. Engineering4. Medicine5. Business.(in financial industry)Tools of numerical analysisMost reg ent(postnominal) tools of numerical analysisComputer graphicsSymbolic mathematical computationsGraphical user interfacesNumerical analysis is needed to solve engineering problems that lead to equations that cannot be solved analytically with simple formulas.Examples are solutions of largesystemsof algebraic equations, evaluation of integrals, and solution of differential equations. The finite element method is a numerical method that is in widespread use to solve partial differential equations in a variety of engineering fields including stress analysis, fluid dynamics, heat transfer, and electro-magnetic fields.In hydro static pressure processingIn high hydrostatic pressure (HHP) processing, food and biotechnological substances are compressed up to 1000 M Pa to achieve various pressure-induced conversions such as microbial and enzyme inactivations, phase transitions of proteins, and solid-liquid state transitions.From the point of view of thermodynamics, Heat transfer leads to spac e-time-dependent temperature fields that affect many pressure-induced conversions and produce unsought process non uniformitiesEffects related to HHP processing can be studied appropriately by use of numerical analysis because in situ measurement techniques are barely available, optical accessibility is hardly possible, and technical equipment is expensive.This reports on two examples, where numerical analysis is applied successfully and delivers substantial insights into the phenomenon of high-pressure processing.CalculationE.g TSP problem (traveling salesman problem)to travel no. of cities in such a way that the expenses on traveling are minimized. NP-complete problem. optimal solution we have to go through all possible routes numbers of routes increases exponential with the numbers of cities.Modern Applications and Computer SoftwareSophisticated numerical analysis software is being embedded in popular software packagese.g. spreadsheet programs.Buisness Applications-Modern busine ss makes much use of optimization methods in deciding how to allocate resources most efficiently. These include problems such as inventory control,scheduling, how outperform to locate manufacturing storage facilities, investment strategies,and others.In Financial IndustryQuantitative analysts underdeveloped financial applications have specialized expertise in their area of analysis.Algorithms used for numerical analysis range from basic numerical functions to calculate interest income to advanced functions that offer specialized optimization and forecasting techniques.Sample Finance Applications iii common examples from the financial services industry that require numerical algorithms are Portfolio selection Option pricing Risk managementIn marketGiven the broad range of numerical tools available a financial services provider can develop targeted applications that address specific market needs. For example, quantitative analysts developing financial applications have specialized e xpertise in their area of analysis.