Study On The Applications Of Numerical Analysis Computer Science Essay

Study On The Applications Of Numerical Analysis Computer Science Essay It 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. INTRODUCTION TO NUMERICAL ANALYSIS AND METHODS 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. History of Numerical Analysis The field of numerical analysis predates the invention of modern computers by many centuries.  Linear interpolation  was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, as is obvious from the names of important algorithms like  Newtons method,  Lagrange interpolation polynomial,Gaussian elimination, or  Eulers method. To facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points 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 achieve very good numerical estimates of some functions. The canonical work in the field is the  NIST  publication edited by  Abramowitz and Stegun, a 1000-plus page book of a very large number of commonly used formulas and functions and their values at many points. The function values are no longer very useful when a computer is available, but the large listing of formulas can still be very handy. The  mechanical calculator  was 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 now longer and more complicated calculations could be done. Direct and iterative methods Direct methods compute the solution to a problem in a finite number of steps. These methods would give the precise answer if they were performed in  infinite precision arithmetic. Examples include  Gaussian elimination, the  QR  factorization method for solving  systems of linear equations, and the  simplex method  of  linear programming. In practice,  finite precision  is used and the result is an approximation of the true solution (assuming  stability). In contrast to direct methods,  iterative methods  are not expected to terminate in a number of steps. Starting from an initial guess, iterative methods form successive approximations that  converge  to the exact solution only in the limit. A  convergence test  is 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 include  Newtons method, the  bisection method, and  Jacobi iteration. In computational matrix algebra, iterative methods are generally 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.  GMRES  and the  conjugate gradient method. For these methods the 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. Discretization Furthermore, continuous problems must sometimes be replaced by a discrete problem whose solution is known to approximate that of the continuous problem; this process is called  discretization. For example, the solution of a  differential equation  is a function. This function must be represented by a finite amount of data, for instance by its value at a finite number of points at its domain, even though this domain is a continuum. Different Areas And Methods under Numerical Analysis The 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 the  Horner scheme, since it reduces the necessary number of multiplications and additions. Generally, it is important to estimate and control  round-off errors  arising from the use of  floating point  arithmetic. Interpolation, extrapolation, and regression Interpolation  solves 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? Extrapolation  is 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. Regression  is also similar, but it takes into account that the data is imprecise. Given some points, and a measurement of the value of some function at these points (with an error), we want to determine the unknown function. The  least squares-method is one popular way to achieve this. Solving equations and systems of equations Another fundamental problem is computing the solution of some given equation. Two cases are commonly distinguished, depending on whether the equation is linear or not. For instance, the equation  2x  + 5 = 3  is linear while  22  + 5 = 3  is not. Much effort has been put in the development of methods for solving  systems of linear equations. Standard direct methods, i.e., methods that use some  matrix decomposition  are  Gaussian elimination,  LU decomposition,  Cholesky decomposition  for  symmetric  (or  hermitian) and  positive-definite matrix, and  QR decomposition  for non-square matrices.  Iterative methods  such as the  Jacobi method,  Gauss-Seidel method,  successive over-relaxation  and  conjugate gradient method  are usually preferred for large systems. Root-finding algorithms  are 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 is  differentiable  and the derivative is known, then  Newtons method  is a popular choice.  Linearization  is another technique for solving nonlinear equations. Solving eigenvalue or singular value problems Several important problems can be phrased in terms of  eigenvalue decompositions  or  singular value decompositions. For instance, thespectral image compression  algorithm  is based on the singular value decomposition. The corresponding tool in statistics is calledprincipal component analysis. Optimization Optimization 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 programming  deals with the case that both the objective function and the constraints are linear. A famous method in linear programming is the  simplex method. The method of  Lagrange multipliers  can be used to reduce optimization problems with constraints to unconstrained optimization problems. Evaluating integrals Numerical integration, in some instances also known as numerical  quadrature, asks for the value of a definite  integral. Popular methods use one of the  Newton-Cotes formulas  (like the midpoint rule or  Simpsons rule) or  Gaussian quadrature. These methods rely on a divide and conquer strategy, whereby an integral on a relatively large set is broken down into integrals on smaller sets. In higher dimensions, where these methods become prohibitively expensive in terms of computational effort, one may use  Monte Carlo  or  quasi-Monte Carlo methods  (see  Monte Carlo integration), or, in modestly large dimensions, the method of  sparse grids. Differential equations Numerical analysis is also concerned with computing (in an approximate way) the solution of  differential equations, both ordinary differential equations and  partial differential equations. Partial differential equations are solved by first discretizing the equation, bringing it into a finite-dimensional subspace. This can be done by a  finite element method, a  finite differencemethod, or (particularly in engineering) a  finite volume method. The theoretical justification of these methods often involves theorems from  functional analysis. This reduces the problem to the solution of an algebraic equation. Applications Of Numerical Analysis Methods and Its Real Life Implementations, Advantages Etc. NEWTON RAPHSON METHOD: ORDER OF CONVERGENCE: 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 diverge 2. No practical rigorous error bound GAUSS ELIMINATION METHOD: ADVANTAGES: It is the direct method of solving linear simultaneous equations. 2. It uses back substitution. 3. It is reduced to equivalent upper triangular matrix.: 1. It requires right vectors to be known. 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 identity matrix. The additional steps increase round off errors. 2. It requires right vectors to be known. GAUSS JACOBI METHOD: 1. 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 METHOD: 1. 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 Applications Area of mathematics and computer science. Applications of algebra Geometry Calculus Variables which vary continuously. Problems(application areas) 1. Natural sciences 2. Social sciences 3. Engineering 4. Medicine 5. Business.(in financial industry) Tools of numerical analysis Most powerful tools of numerical analysis à  Computer graphics à  Symbolic mathematical computations à  Graphical user interfaces Numerical analysis is needed to solve engineering problems that lead to equations that cannot be solved analytically with simple formulas. Examples are solutions of large  systems  of 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 processing In 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 space-time-dependent temperature fields that affect many pressure-induced conversions and produce undesired process non uniformities Effects 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. Calculation E.g TSP problem (traveling salesman problem) to travel no. of cities in such a way that the expenses on traveling are minimized. à   NP-complete problem. à   optimal solution we have to go through all possible routes à   numbers of routes increases exponential with the numbers of cities. Modern Applications and Computer Software Sophisticated numerical analysis software is being embedded in popular software packages e.g. spreadsheet programs. Buisness Applications:- Modern business makes much use of optimization methods in deciding how to allocate resources most efficiently. These include problems such as inventory control,scheduling, how best to locate manufacturing storage facilities, investment strategies,and others. In Financial Industry Quantitative analysts developing 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 Three common examples from the financial services industry that require numerical algorithms are: à ¢Ã¢â€š ¬Ã‚ ¢ Portfolio selection à ¢Ã¢â€š ¬Ã‚ ¢ Option pricing à ¢Ã¢â€š ¬Ã‚ ¢ Risk management   In market Given 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 expertise in their area of analysis.