Numerical analysis
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Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics).
For thousands of years, man has used mathematics for construction, warfare, engineering, accounting and many other puposes. The earliest mathematical writing is perhaps the famous Babylonian tablet Plimpton 322, dating from approximately 1800 BC. On it one can read a list of pythagorean triples: triples of numbers, like (3,4,5), which are the lengths of the sides of a right-angle triangle. The Babylonian tablet YBC 7289 gives an approximation of <math>\sqrt{2}</math> ,
<ref>The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures.
Photograph, illustration, and description of the root(2) tablet from the Yale Babylonian Collection</ref>
which is the length of the diagonal of a square whose side measures one unit of length. 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. If the roof of a house makes a right angle isosceles triangle whose side is 3 meters long, then the central support beam must be <math>\sqrt{18}\approx4.2426</math> meters longer than the side beams.
Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation to <math>\sqrt{2}</math>, modern numerical analysis does not seek exact answers, because exact answers are 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 essential to quantitative psychology; 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. Nowadays (after mid 20th century) these tables have fallen into disuse, because computers can calculate the required functions. The interpolation algorithms nevertheless may be used as part of the software for solving differential equations and the like.
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[edit] General introduction
We will now outline several important themes of numerical analysis. The overall goal is the design and analysis of techniques to give approximate solutions to hard problems. To fix ideas, the reader might consider the following problems and methods:
- If a company wants to put a toothpaste commercial on television, it might produce five commercials and then choose the best one by testing each one on a focus group. This would be an example of a Monte Carlo optimization method.
- To send a rocket to the moon, rocket scientists will need a rocket simulator. This simulator will essentially be an integrator for an ordinary differential equation.
- Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. These simulations are essentially solving partial differential equations numerically.
- Hedge funds (secretive financial companies) 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.
[edit] History
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.
[edit] Direct and iterative methods
Direct methods compute an approximate solution to a problem in a exact number of steps. Examples include Gaussian elimination and the QR factorization for solving systems of linear equations. In constrast, iterative methods take a variable number of steps. Starting from a initial guess, iterative methods form successive approximations that converge to the exact solution only in the limit.
[edit] 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.
[edit] The generation and propagation of errors
The study of errors forms an important part of numerical analysis. There are several ways in which error can be introduced in the solution of the problem.
[edit] Round-off
Round-off errors arise because it is impossible to represent all real numbers exactly on a finite-state machine (which is what all practical digital computers are).
On a pocket calculator, if one enters 0.0000000000001 (or the maximum number of zeros possible), then a +, and then 100000000000000 (again, the maximum number of zeros possible), one will obtain the number 100000000000000 again, and not 100000000000000.0000000000001. The calculator's answer is incorrect because of roundoff in the calculation.
[edit] Truncation and discretization error
Truncation errors are committed when an iterative method is terminated and the approximate solution differs from the exact solution. Similarly, discretization induces a discretization error because the solution of the discrete problem does not coincide with the solution of the continuous problem. For instance, in the iteration above to compute the solution of <math>3x^3+4=28</math>, after 10 or so iterations, we conclude that the root is roughly 1.99 (for example). We therefore have a truncation error of 0.01.
Once an error is generated, it will generally propagate through the calculation. For instance, we have already noted that the operation + on a calculator (or a computer) is inexact. It follows that a calculation of the type a+b+c+d+e is even more inexact.
[edit] Numerical stability and well posedness
This leads to the notion of numerical stability: an algorithm is numerically stable if an error, once it is generated, does not grow too much during the calculation. This is only possible if the problem is well-conditioned, meaning that the solution changes by only a small amount if the problem data are changed by a small amount. Indeed, if a problem is ill-conditioned, then any error in the data will grow a lot.
However, an algorithm that solves a well-conditioned problem may or may not be numerically stable. An art of numerical analysis is to find a stable algorithm for solving a well-posed mathematical problem.
[edit] Areas of study
The field of numerical analysis is divided in different disciplines according to the problem that is to be solved.
[edit] Computing values of functions
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.
[edit] 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? A very simple method is to use linear interpolation, which assumes that the unknown function is linear between every pair of successive points. This can be generalized to polynomial interpolation, which is sometimes more accurate but suffers from Runge's phenomenon. Other interpolation methods use localized functions like splines or wavelets.
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.
[edit] 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 <math>2x+5=3</math> is linear while <math>2x^2+5=3</math> 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.
[edit] Solving eigenvalue or singular value problems
Several important problems can be phrased in terms of eigenvalue decompositions or singular value decompositions. For instance, the spectral image compression algorithm <ref>[1]</ref> is based on the singular value decomposition. The corresponding tool in statistics is called principal component analysis. One application is to automatically find the 100 top subjects of discussion on the web, and to then classify each web page according to which subject it belongs to.
[edit] Optimization
Optimization problems ask for the point at which a given function is maximized (or minimized). Often, the point also has to satisfy some constraints.
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.
[edit] 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 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, or, in modestly large dimensions, the method of sparse grids.
[edit] 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 difference method, 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.
[edit] Software
Since the late twentieth century, most algorithms are implemented and run on a computer. The Netlib repository contains various collections of software routines for numerical problems, mostly in Fortran and C. Commercial products implementing many different numerical algorithms include the IMSL and NAG libraries; a free alternative is the GNU Scientific Library.
MATLAB is a popular commercial programming language for numerical scientific calculations, but there are commercial alternatives such as S-PLUS and IDL, as well as free and open source alternatives such as FreeMat, GNU Octave (similar to Matlab), R (similar to S-PLUS) and certain variants of Python. Performance varies widely: while vector and matrix operations are usually fast, scalar loops vary in speed by more than an order of magnitude. <ref>Speed comparison of various number crunching packages</ref>
Many computer algebra systems such as Mathematica or Maple (free software systems include SAGE, Maxima, Axiom, calc and Yacas), can also be used for numerical computations. However, their strength typically lies in symbolic computations. Also, any spreadsheet software can be used to solve simple problems relating to numerical analysis.
[edit] See also
- Scientific computing
- Numerical linear algebra
- List of numerical analysis topics
- Gram-Schmidt process
- Important publications in numerical analysis
- Halting problem
- Numerical integration
- Numerical differentiation
[edit] Notes
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[edit] References
- Gilat, Amos (2004). MATLAB: An Introduction with Applications 2nd Edition. John Wiley & Sons. ISBN 0471694207.
- Hildebrand, F. B. (1987 (repr. of 1974 ed.)). Introduction to Numerical Analysis, 2nd Ed.. Dover.
- Leader, Jeffery J. (2004). Numerical Analysis and Scientific Computation. Addison Wesley.
- Wolfram, Stephen (1999). The Mathematica Book, Fourth Ed.. Cambridge University Press.
[edit] External links
- Lloyd N. Trefethen, "Numerical analysis", May 2006, 20 pages, to appear in: Timothy Gowers and June Barrow-Green (editors), Princeton Companion of Mathematics, Princeton University Press.
- Numerische Mathematik, volumes 1-66, Springer, 1959-1994 (searchable; pages are images). (English) (German)
- Scientific computing FAQ
- Numerical analysis DMOZ category
- Lists of free software for scientific computing and numerical analysis (English) (French)
- Numerical Recipes Homepage - with free, complete downloadable books
- Alternatives to Numerical Recipes
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