Computing values of functions
Areas of study[edit]
The field of numerical analysis includes many sub-disciplines. Some of the major ones are:
Computing values of functions[edit]
Interpolation: Observing that the temperature varies from 20 degrees Celsius at 1:00 to 14 degrees at 3:00, a linear interpolation of this data would conclude that it was 17 degrees at 2:00 and 18.5 degrees at 1:30pm. Extrapolation: If the gross domestic product of a country has been growing an average of 5% per year and was 100 billion last year, it might extrapolated that it will be 105 billion this year.  Regression: In linear regression, given n points, a line is computed that passes as close as possible to those n points.  Optimization: Suppose lemonade is sold at a lemonade stand, at $1.00 per glass, that 197 glasses of lemonade can be sold per day, and that for each increase of $0.01, one less glass of lemonade will be sold per day. If $1.485 could be charged, profit would be maximized, but due to the constraint of having to charge a whole-cent amount, charging $1.48 or $1.49 per glass will both yield the maximum income of $220.52 per day.  Differential equation: If 100 fans are set up to blow air from one end of the room to the other and then a feather is dropped 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. |
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[edit]
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 the value of the unknown function at a point which is outside the given points must be found.[14]
Regression is also similar, but it takes into account that the data are imprecise. Given some points, and a measurement of the value of some function at these points (with an error), the unknown function can be found. The least squares-method is one way to achieve this.
Solving equations and systems of equations[edit]
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 is linear while 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[15] are usually preferred for large systems. General iterative methods can be developed using a matrix splitting.
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.[16][17] Linearization is another technique for solving nonlinear equations.
Solving eigenvalue or singular value problems[edit]
Several important problems can be phrased in terms of eigenvalue decompositions or singular value decompositions. For instance, the spectral image compression algorithm[18] is based on the singular value decomposition. The corresponding tool in statistics is called principal component analysis.
Optimization[edit]
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.
Evaluating integrals[edit]
Numerical integration, in some instances also known as numerical quadrature, asks for the value of a definite integral.[19] Popular methods use one of the Newton–Cotes formulas (like the midpoint rule or Simpson's rule) or Gaussian quadrature.[20] 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[21]), or, in modestly large dimensions, the method of sparse grids.
Differential equations[edit]
Numerical analysis is also concerned with computing (in an approximate way) the solution of differential equations, both ordinary differential equations and partial differential equations.[22]
Partial differential equations are solved by first discretizing the equation, bringing it into a finite-dimensional subspace.[23] This can be done by a finite element method,[24][25][26] a finite difference method,[27] or (particularly in engineering) a finite volume method.[28] The theoretical justification of these methods often involves theorems from functional analysis. This reduces the problem to the solution of an algebraic equation.
Software[edit]
Since the late twentieth century, most algorithms are implemented in a variety of programming languages. 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-software alternative is the GNU Scientific Library.
Over the years the Royal Statistical Society published numerous algorithms in its Applied Statistics (code for these "AS" functions is here); ACM similarly, in its Transactions on Mathematical Software ("TOMS" code is here). The Naval Surface Warfare Center several times published its Library of Mathematics Subroutines (code here).
There are several popular numerical computing applications such as MATLAB,[29][30][31] TK Solver, S-PLUS, and IDL[32] as well as free and open source alternatives such as FreeMat, Scilab,[33][34] GNU Octave (similar to Matlab), and IT++ (a C++ library). There are also programming languages such as R[35] (similar to S-PLUS), Julia,[36] and Python with libraries such as NumPy, SciPy[37][38][39] and SymPy. Performance varies widely: while vector and matrix operations are usually fast, scalar loops may vary in speed by more than an order of magnitude.[40][41]
Many computer algebra systems such as Mathematica also benefit from the availability of arbitrary-precision arithmetic which can provide more accurate results.[42][43][44][45]
Also, any spreadsheet software can be used to solve simple problems relating to numerical analysis. Excel, for example, has hundreds of available functions, including for matrices, which may be used in conjunction with its built in "solver".
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