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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.
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 Newton’s method, Lagrange interpolation polynomial,Gaussian elimination, or Euler’s 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 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 Newton’s 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.
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.
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 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.
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 2×2 + 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 Newton’s method is a popular choice. Linearization is another technique for solving nonlinear equations.
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 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.
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 Simpson’s 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.
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.
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
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.
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.
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.
Area of mathematics and computer science.
Applications of algebra
Variables which vary continuously.
1. Natural sciences
2. Social sciences
5. Business.(in financial industry)
Tools of numerical analysis
àSymbolic mathematical computations
àGraphical 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 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 inactivation’s, 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.
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.
Sophisticated numerical analysis software is being embedded in popular software packages
e.g. spreadsheet programs.
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.
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.
Three common examples from the financial services industry that require numerical algorithms are:
â€¢ Portfolio selection
â€¢ Option pricing
â€¢ Risk management
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.
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