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Encyclopedia > Numerical analysis

Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). Flowcharts are often used to represent algorithms. ... Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity. ...

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 $sqrt{2}$ ,  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 $sqrt{18}approx4.2426$ meters longer than the side beams. Of the approximately half million clay tablets excavated at the beginning of the 19th century, about 400 are of a mathematical nature. ... The Pythagorean theorem: a2 + b2 = c2 A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. ... Its very easy to find the area of a triangle the formula is Italic textbase times Italic texthight equals (area of a square). ...

Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation to $sqrt{2}$, 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. In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ... In mathematics, optimization is the discipline which is concerned with finding the maxima and minima of functions, possibly subject to constraints. ... Numerical linear algebra is often at the heart of many engineering and computational science problems, such as image and signal processing, computational finance, materials science simulations, structural biology, datamining, and bioinformatics just to name a few. ... A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process. ... In mathematics, a Markov chain is a discrete-time stochastic process with the Markov property named after Andrey Markov. ...

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. In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points from a discrete set of known data points. ... (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901&#8211;2000 in the sense of the Gregorian calendar (1900&#8211;1999... In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points from a discrete set of known data points. ... Flowcharts are often used to represent algorithms. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

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.

Monte Carlo methods are a widely used class of computational algorithms for simulating the behavior of various physical and mathematical systems. ... In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... Operations Research, or simply OR is an interdisciplinary science which deploys scientific methods like mathematical modeling, statistics, and algorithms to decision making in complex real-world problems which are concerned with coordination and execution of the operations within an organization. ... Damage from Hurricane Katrina. ...

### 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. Linear interpolation is a process employed in mathematics, and numerous applications including computer graphics. ... In numerical analysis, Newtons method (or the Newtonâ€“Raphson method or the Newtonâ€“Fourier method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ... In numerical analysis, a Lagrange polynomial, named after Joseph Louis Lagrange, is the interpolation polynomial for a given set of data points in the Lagrange form. ... In mathematics, Gaussian elimination or Gaussâ€“Jordan elimination, named after Carl Friedrich Gauss and Wilhelm Jordan (for many, Gaussian elimination is regarded as the front half of the complete Gaussâ€“Jordan elimination), is an algorithm in linear algebra for determining the solutions of a system of linear equations, for determining... Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations (ODEs). ...

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. As a non-regulatory agency of the United States Department of Commerce&#8217;s Technology Administration, the National Institute of Standards (NIST) develops and promotes measurement, standards, and technology to enhance productivity, facilitate trade, and improve the quality of life. ... Abramowitz and Stegun is the informal moniker of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards. ...

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. A mechanical calculator is a device that does computations without the aid of electricity. ... The 1940s decade ran from 1940 to 1949. ...

### Direct and iterative methods

Direct vs iterative methods

Consider the problem of solving

3x3 + 4 = 28

for the unknown quantity x.

 3x3 + 4 = 28. Subtract 4 3x3 = 24. Divide by 3 x3 = 8. Take cube roots x = 2.

For the iterative method, apply the bisection method to f(x) = 3x3 + 24. The initial values are a = 0,b = 3,f(a) = 4,f(b) = 85. A few steps of the bisection method applied over the starting range [a1;b1]. The red dot is the root of the function. ...

 a b mid f(mid) 0 3 1.5 14.125 1.5 3 2.25 38.17... 1.5 2.25 1.875 23.77... 1.875 2.25 2.0625 30.32...

We conclude from this table that the solution is between 1.875 and 2.0625. The algorithm might return any number in that range with an error less than 0.2.

Some problems can be solved exactly by an algorithm. These algorithms are called direct methods. Examples are Gaussian elimination for solving systems of linear equations and the simplex method in linear programming. In mathematics, Gaussian elimination or Gaussâ€“Jordan elimination, named after Carl Friedrich Gauss and Wilhelm Jordan (for many, Gaussian elimination is regarded as the front half of the complete Gaussâ€“Jordan elimination), is an algorithm in linear algebra for determining the solutions of a system of linear equations, for determining... In mathematics and linear algebra, a system of linear equations is a set of linear equations such as 3x1 + 2x2 âˆ’ x3 = 1 2x1 âˆ’ 2x2 + 4x3 = âˆ’2 âˆ’x1 + Â½x2 âˆ’ x3 = 0. ... In mathematical optimization theory, the simplex algorithm of George Dantzig is the fundamental technique for numerical solution of the linear programming problem. ... In mathematics, linear programming (LP) problems are optimization problems in which the objective function and the constraints are all linear. ...

However, no direct methods exist for most problems. In such cases it is sometimes possible to use an iterative method. Such a method starts from a guess and finds successive approximations that hopefully converge to the solution. Even when a direct method does exist, an iterative method may be preferable if it is more efficient or more stable. An iterative method attempts to solve a problem (for example an equation or system of equations) by finding successive approximations to the solution starting from an initial guess. ... In the absence of a more specific context, convergence denotes the approach toward a definite value, as time goes on; or to a definite point, a common view or opinion, or toward a fixed or equilibrium state. ... In the mathematical subfield of numerical analysis, numerical stability is a property of numerical algorithms. ...

### Discretization

 A race car drives along a track for two hours. What distance has it covered? If we know that the race car was going at 150Km/h after one hour, we might guess that the total distance travelled is 300Km. If we know in addition that the car was travelling at 140Km/h at the 20 minutes mark, and at 180Km/h at the 1:40 mark, then we can divide the two hours into three blocks of 40 minutes. In the first block of 40 minutes, the car travelled roughly 93.3Km, in the interval between 0:40 and 1:20 the car travelled roughly 100Km, and in the interval from 1:20 to 2:00, the car travelled roughly 120Km, for a total of 313.3Km. Essentially, we have taken the continuously varying speed v(t) and approximated it using a speed $hat v(t)$ which is constant on each of the three intervals of 40 minutes. How far is Schumacher driving?

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. Download high resolution version (1798x700, 1022 KB) Wikipedia does not have an article with this exact name. ... Download high resolution version (1798x700, 1022 KB) Wikipedia does not have an article with this exact name. ... An illustration of a differential equation. ...

### 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.

#### 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). A round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value. ... In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ... In the theory of computation, a finite state machine (FSM) or finite state automaton (FSA) is an abstract machine that has only a finite, constant amount of memory. ... ...

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.

#### 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 3x3 + 4 = 28, 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. In mathematics, truncation is the term used for reducing the number of digits right of the decimal point, by discarding the least significant ones. ... Termination as a technical term has different meanings. ... In numerical analysis, computational physics, and simulation, discretization error is error resulting from the fact that a function of a continuous variable is represented in the computer by a finite number of evaluations, for example, on a lattice. ...

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.

#### Numerical stability and well posedness

 Ill posed problem: Take the function f(x) = 1 / (x − 1). The data is x and the output is f(x). Note that f(1.1) = 10 and f(1.001) = 1000. So a change in x of less than 0.1 turns into a change in f(x) of nearly 1000. Evaluating f(x) near x = 1 is an ill-posed (or ill-conditioned) problem. Well-posed problem: By contrast, the function $f(x)=sqrt{x}$ is continuous so the problem of computing $sqrt{x}$ is well-posed. Numerically unstable method: Still, some algorithms which are meant to compute $sqrt{x}$ are fallible. Consider the following iteration Start with x1 an approximation of $sqrt{2}$ (for instance, take x1 = 1.4) and then use the iteration $x_{k+1}=(x_k^2-2)^2+x_k.$ Then the iterates are ` x1 = 1.4 x2 = 1.4016 x3 = 1.4028614885... x4 = 1.403884186... x1000000 = 1.414213437... ` On the other hand, if we start from x1 = 1.42, we obtain the iterates ` x2 = 1.42026896 x3 = 1.42056... x20 = 1.445069... x27 = 9.34181... x28 = 7280.2284... ` and the iteration diverges. Because this algorithm fails for certain initial guesses near $sqrt{2}$, we say that it is numerically unstable. Numerically stable method: The Newton method for $sqrt{a}$ is x1 = 1 and $x_{k+1}={x_k + a/x_k over 2}$ and is extremely stable. The first few iterations for a = 2 are In numerical analysis, Newtons method (or the Newton-Raphson method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ... ` x1 = 1 x2 = 1.5 x3 = 1.416... x4 = 1.414215... x5 = 1.41421356237469... x6 = 1.41421356237309..., ` which is essentially exact to the last displayed digit. The method will work well even if we change a, x1 and/or introduce small errors at every step of the computation. 

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. In the mathematical subfield of numerical analysis, numerical stability is a property of numerical algorithms. ... In numerical analysis, the condition number associated with a numerical problem is a measure of that quantitys amenability to digital computation, that is, how well-posed the problem is. ...

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.

## Areas of study

The field of numerical analysis is divided in different disciplines according to the problem that is to be solved.

### 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. In the mathematical subfield of numerical analysis the Horner scheme or Horner algorithm, named after William George Horner, is an algorithm for the efficient evaluation of polynomials in monomial form. ... A round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value. ... A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ...

### Interpolation, extrapolation and regression

 Interpolation: If the temperature at noon was 20 degrees centigrade and at 2pm we observe that the temperature is 14 degrees centigrade, we might guess that at 1pm the temparature was in fact the average of 14 and 20, which is 17 degrees centigrade. This would correspond to a linear interpolation of the temperature between the times of noon and 2pm. Extrapolation: If the gross domestic product of a country has been growing an average of 5% per year and was 100 billion dollars last year, we might extrapolate that it will be 105 billion dollars this year. IMF 2005 figures of GDP of nominal compared to PPP. A regions gross domestic product, or GDP, is one of the several measures of the size of its economy. ... Regression: Given two points on a piece of paper, there is a line that goes through both points. Given twenty points on a piece of paper, what is the line that goes through them? In most cases, there is no straight line going through the twenty points. In linear regression, given n points, we compute a line that passes as close as possible to those n points. A line through 20 points?

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. Image File history File links Linear-regression. ... In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points from a discrete set of known data points. ... Linear interpolation is a process employed in mathematics, and numerous applications including computer graphics. ... In the mathematical subfield of numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial. ... The red curve is the Runge function, the blue curve is a 5th-degree polynomial, while the green curve is a 9th-degree polynomial. ... One type of spline, a bÃ©zier curve In the mathematical subfield of numerical analysis, a spline is a special function defined piecewise by polynomials. ... In mathematics, wavelets, wavelet analysis, and the wavelet transform refers to the representation of a signal in terms of a finite length or fast decaying oscillating waveform (known as the mother wavelet). ...

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. In mathematics, extrapolation is a type of interpolation. ...

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. In statistics, regression analysis is used to model relationships between variables, determine the magnitude of the relationships between variables, and can be used to make predictions based on the models. ... Least squares is a mathematical optimization technique which, when given a series of measured data, attempts to find a function which closely approximates the data (a best fit). It attempts to minimize the sum of the squares of the ordinate differences (called residuals) between points generated by the function and...

### 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 2x2 + 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. In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. ... In mathematics, Gaussian elimination or Gaussâ€“Jordan elimination, named after Carl Friedrich Gauss and Wilhelm Jordan (for many, Gaussian elimination is regarded as the front half of the complete Gaussâ€“Jordan elimination), is an algorithm in linear algebra for determining the solutions of a system of linear equations, for determining... In linear algebra, the LU decomposition is a matrix decomposition which writes a matrix as the product of a lower and upper triangular matrix. ... In mathematics, the Cholesky decomposition, named after AndrÃ©-Louis Cholesky, is a matrix decomposition of a symmetric positive-definite matrix into a lower triangular matrix and the transpose of the lower triangular matrix. ... In linear algebra, a symmetric matrix is a matrix that is its own transpose. ... A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries which is equal to its own conjugate transpose â€” that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for... In linear algebra, a positive-definite matrix is a Hermitian matrix which in many ways is analogous to a positive real number. ... In linear algebra, the QR decomposition of a matrix is a decomposition of the matrix into an orthogonal and a triangular matrix. ... An iterative method attempts to solve a problem (for example an equation or system of equations) by finding successive approximations to the solution starting from an initial guess. ... The Jacobi method is an algorithm in linear algebra for determining the solutions of a system of linear equations with largest absolute values in each row and column dominated by the diagonal element. ... The Gauss-Seidel method is a technique used to solve a linear system of equations. ... Successive over-relaxation (SOR) is a numerical method used to speed up convergence of the Gaussâ€“Seidel method for solving a linear system of equations. ... In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positive definite. ...

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. A root-finding algorithm is a numerical method, or algorithm, for finding a value x such that f(x) = 0, for a given function f. ... In mathematics, a derivative is defined as the instantaneous rate of change of a function and the process of finding the derivative is called differentiation. ... In numerical analysis, Newtons method (or the Newtonâ€“Raphson method or the Newtonâ€“Fourier method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ... Linearization in mathematics and its applications in general refers to finding the linear approximation to a function at a given point. ...

### 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  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. In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. ... In linear algebra singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics. ... In statistics, principal components analysis (PCA) is a technique that can be used to simplify a dataset; more formally it is a linear transformation that chooses a new coordinate system for the data set such that the greatest variance by any projection of the data set comes to lie on...

### Optimization Optimization: Say you sell lemonade at a lemonade stand, and notice that at 1\$, you can sell 197 glasses of lemonade per day, and that for each increase of 0.01\$, you will sell one less lemonade per day. The optimization problem is to find the price of lemonade at which your profit per day is largest possible, ignoring production costs. If you could charge 1.485\$, you would maximize your profit, but due to the constraint of having to charge a whole cent amount, charging 1.49\$ per glass will yield the maximum profit of 220.52\$ per day. In mathematics, the term optimization refers to the study of problems that have the form Given: a function f : A R from some set A to the real numbers Sought: an element x0 in A such that f(x0) â‰¤ f(x) for all x in A (minimization) or such that... Image File history File linksMetadata LemonadeJuly2006. ... Differential equation: If you set up 100 fans to blow air from one end of the room to the other and then you drop a feather into the wind, what happens? The feather will follow the air currents, which may be very complex. One approximation is to measure the speed at which the air is blowing near the feather every second, and advance the simulated feather as if it were moving in a straight line at that same speed for one second, before measuring the wind speed again. This is called the Euler method for solving an ordinary differential equation. Image File history File links Wind-particle. ...

Optimization problems ask for the point at which a given function is maximized (or minimized). Often, the point also has to satisfy some constraints. A constraint is a limitation of possibilities. ...

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. In mathematics, linear programming (LP) problems are optimization problems in which the objective function and the constraints are all linear. ... In mathematical optimization theory, the simplex algorithm of George Dantzig is the fundamental technique for numerical solution of the linear programming problem. ...

The method of Lagrange multipliers can be used to reduce optimization problems with constraints to unconstrained optimization problems. Fig. ...

### Evaluating integrals

Main article: Numerical integration
 How much paint would you need to give the Statue of Liberty a fresh coat? She is 151 feet tall and her waist is 35' across, so a first approximation is that it would require the same amount of paint you would need to paint a 151x35x35 room. Counting the four walls and the ceiling, that would make a surface area of 22,365 square feet. One gallon of paint covers about 350 square feet, so by this estimate, we might require 64 gallons of paint. Numerical Integration with the Monte Carlo method: Nodes are random equally distributed. ... However, it may be more precise to estimate each piece on its own. We can approximate the parts below the neck with a 95 feet tall cylinder whose radius is 17 feet. The head is roughly a sphere of radius 15 feet. The arm is another 42' long cylinder with a radius of 6', the tablet is a 24'x14'x2' box. Adding all the surface areas gives roughly 15385 square feet, which requires an approximate 44 gallons of paint. Image File history File links Download high resolution version (1304x2592, 481 KB) Summary Cropped and Colored version of Image:Single shot of Statue of Liberty (NY). ... To further improve our estimate, we would measure the folds in her cloth, how non-spherical her head really is and so on. This process is called numerical integration.

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. In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe numerical algorithms for solving differential equations. ... In calculus, the integral of a function is an extension of the concept of a sum. ... In numerical analysis, the Newton-Cotes formulas, also called the Newton-Cotes rules, are a group of formulas for numerical integration (also called quadrature) based on evaluating the integrand at n+1 equally-spaced points. ... The function f(x) (in blue) is approximated by a quadratic function P(x) (in red). ... In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. ... Monte Carlo methods are a widely used class of computational algorithms for simulating the behavior of various physical and mathematical systems. ... In numerical analysis, a quasi-Monte Carlo method is a method for the computation of an integral (or some other problem) which is based on low-discrepancy sequences. ... Sparse grids are a numerical technique to represent, integrate or interpolate high dimensional functions. ...

### Differential equations

Main articles: Numerical ordinary differential equations, Numerical partial 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. Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations (ODEs). ... Numerical partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations. ... An illustration of a differential equation. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...

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. The introduction to this article is too long. ... In mathematics, a finite difference is like a differential quotient, except that it uses finite quantities instead of infinitesimal ones. ... The finite volume method is a method for representing and evaluating partial differential equations as algebraic equations. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...

## 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. Listed here are a number of computer programs used for performing numerical calculations: DADiSP is a commercial program focused on DSP that combines the numerical capability of MATLAB with a spreadsheet like interface. ... Netlib, www. ... Fortran (previously FORTRAN) is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing. ... Wikibooks has a book on the topic of C Programming The C programming language (often, just C) is a general-purpose, procedural, imperative computer programming language developed in the early 1970s by Dennis Ritchie for use on the Unix operating system. ... The IMSL Numerical Libraries are software libraries of numerical analysis functionality that are implemented in widely used computer programming languages of C, Java, C#.NET, and Fortran. ... NÃ£g is the Spirit, god, and/or god-like animal representative of power, inteligens, strength, speed, war, and death in the Reptianity religon, In the discription they give Nag is an amphibious theropod with powerful arms, armed hid, a cobra hud, wings, four tenicals, a dorsal fin, a shark... Wikipedia does not yet have an article with this exact name. ...

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.  MATLAB is a numerical computing environment and programming language. ... S is a statistical programming language developed by John Chambers of Bell Laboratories. ... IDL, short for interactive data language, is a programming language which is a popular data analysis language among scientists. ... FreeMat is a free numerical computing environment and programming language, similar to Matlab. ... For other uses of the word octave see Octave (disambiguation) Octave is a free computer program for performing numerical computations, which is mostly compatible with MATLAB. It is part of the GNU project. ... The R programming language, sometimes described as GNU S, is a mathematical language and environment used for statistical analysis and display. ... Python is an interpreted programming language created by Guido van Rossum in 1990. ...

Scientific computing (or Computational science) is the field of study concerned with constructing mathematical models and numerical solution techniques and using computers to analyze and solve scientific and engineering problems. ... Numerical linear algebra is often at the heart of many engineering and computational science problems, such as image and signal processing, computational finance, materials science simulations, structural biology, datamining, and bioinformatics just to name a few. ... This is a list of numerical analysis topics, by Wikipedia page. ... In mathematics and numerical analysis, the Gram-Schmidt process of linear algebra is a method of orthogonalizing a set of vectors in an inner product space, most commonly the Euclidean space Rn. ... This is a list of important publications in mathematics, organized by field. ... In computability theory the halting problem is a decision problem which can be informally stated as follows: Given a description of a program and its initial input, determine whether the program, when executed on this input, ever halts (completes). ... Numerical Integration with the Monte Carlo method: Nodes are random equally distributed. ... Numerical differentiation is a technique of numerical analysis to produce an estimate of the derivative of a mathematical function or function subroutine using values from the function and perhaps other knowledge about the function. ... Results from FactBites:

 Numerical analysis - Wikipedia, the free encyclopedia (1751 words) Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). The field of numerical analysis is divided in different disciplines according to the problem that is to be solved. Numerical analysis is also concerned with computing (in an approximate way) the solution of differential equations, both ordinary differential equations and partial differential equations.
 Numerical analysis - definition of Numerical analysis in Encyclopedia (1405 words) While numerical analysis employs mathematical axioms, theorems and proofs in theory, it may use empirical results of computation runs to probe new methods and analyze problems. The analogous concept for the numerical algorithm for solving the problem is that of numerical stability: an algorithm for solving a well-conditioned problem is numerically stable if the result of the algorithm changes only a small amount if the data change a little. Computers are an essential tool in numerical analysis, but the field predates computers by many centuries, and actually computers were invented to a large extent in order to solve numerical problems, not the other way around.
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