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61-398: Schwarz–Christoffel mappings are used in potential theory and some of its applications, including minimal surfaces , hyperbolic art , and fluid dynamics . Consider a polygon in the complex plane. The Riemann mapping theorem implies that there is a biholomorphic mapping f from the upper half-plane to the interior of the polygon. The function f maps the real axis to the edges of

122-400: A discretization error because the solution of the discrete problem does not coincide with the solution of the continuous problem. In the example above to compute the solution of 3 x 3 + 4 = 28 {\displaystyle 3x^{3}+4=28} , after ten iterations, the calculated root is roughly 1.99. Therefore, the truncation error is roughly 0.01. Once an error

183-616: 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. 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

244-533: A triangle with vertices P = 0 , Q = π i , and R (with R real), as R tends to infinity. Now α = 0 and β = γ = π ⁄ 2 in the limit. Suppose we are looking for the mapping f with f (−1) = Q , f (1) = P , and f (∞) = R . Then f is given by Evaluation of this integral yields where C is a (complex) constant of integration. Requiring that f (−1) = Q and f (1) = P gives C = 0 and K = 1 . Hence

305-445: A ) = −24, f ( b ) = 57. From this table it can be concluded 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. Ill-conditioned problem: Take the function f ( x ) = 1/( x  − 1) . Note that f (1.1) = 10 and f (1.001) = 1000: 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

366-439: A feel for exactly what is special about complex analysis in two dimensions and what is simply the two-dimensional instance of more general results. An important topic in potential theory is the study of the local behavior of harmonic functions. Perhaps the most fundamental theorem about local behavior is the regularity theorem for Laplace's equation, which states that harmonic functions are analytic. There are results which describe

427-404: A finite sum of regions can be found, and hence the approximation of the exact solution. Similarly, to differentiate a function, the differential element approaches zero, but numerically only a nonzero value of the differential element can be chosen. An algorithm is called numerically stable if an error, whatever its cause, does not grow to be much larger during the calculation. This happens if

488-461: A multi-valued harmonic function as a single-valued function on a branched cover of R or one can regard harmonic functions which are invariant under a discrete subgroup of the conformal group as functions on a multiply connected manifold or orbifold . From the fact that the group of conformal transforms is infinite-dimensional in two dimensions and finite-dimensional for more than two dimensions, one can surmise that potential theory in two dimensions

549-473: A point which is outside the given points must be found. 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. Another fundamental problem is computing the solution of some given equation. Two cases are commonly distinguished, depending on whether

610-412: A specific polygon one needs to find the a < b < c < ⋯ {\displaystyle a<b<c<\cdots } values which generate the correct polygon side lengths. This requires solving a set of nonlinear equations, and in most cases can only be done numerically . Consider a semi-infinite strip in the z plane . This may be regarded as a limiting form of

671-551: A surprising fact is that many results and concepts originally discovered in complex analysis (such as Schwarz's theorem , Morera's theorem , the Weierstrass-Casorati theorem , Laurent series , and the classification of singularities as removable , poles and essential singularities ) generalize to results on harmonic functions in any dimension. By considering which theorems of complex analysis are special cases of theorems of potential theory in any dimension, one can obtain

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732-452: A well-conditioned problem may be either numerically stable or numerically unstable. An art of numerical analysis is to find a stable algorithm for solving a well-posed mathematical problem. The field of numerical analysis includes many sub-disciplines. Some of the major ones are: 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

793-405: Is also intimately connected with probability and the theory of Markov chains . In the continuous case, this is closely related to analytic theory. In the finite state space case, this connection can be introduced by introducing an electrical network on the state space, with resistance between points inversely proportional to transition probabilities and densities proportional to potentials. Even in

854-401: Is an ill-conditioned problem. Well-conditioned problem: By contrast, evaluating the same function f ( x ) = 1/( x  − 1) near x = 10 is a well-conditioned problem. For instance, f (10) = 1/9 ≈ 0.111 and f (11) = 0.1: a modest change in x leads to a modest change in f ( x ). Furthermore, continuous problems must sometimes be replaced by a discrete problem whose solution

915-424: Is called principal component analysis . 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

976-516: 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

1037-402: Is considerable overlap between potential theory and the theory of Poisson's equation to the extent that it is impossible to draw a distinction between these two fields. The difference is more one of emphasis than subject matter and rests on the following distinction: potential theory focuses on the properties of the functions as opposed to the properties of the equation. For example, a result about

1098-462: Is different from potential theory in other dimensions. This is correct and, in fact, when one realizes that any two-dimensional harmonic function is the real part of a complex analytic function , one sees that the subject of two-dimensional potential theory is substantially the same as that of complex analysis. For this reason, when speaking of potential theory, one focuses attention on theorems which hold in three or more dimensions. In this connection,

1159-440: Is generated, it propagates through the calculation. For example, the operation + on a computer is inexact. A calculation of the type ⁠ a + b + c + d + e {\displaystyle a+b+c+d+e} ⁠ is even more inexact. A truncation error is created when a mathematical procedure is approximated. To integrate a function exactly, an infinite sum of regions must be found, but numerically only

1220-406: 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 the value of the unknown function at

1281-472: 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 study of errors forms an important part of numerical analysis. There are several ways in which error can be introduced in

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1342-584: Is mapped to the square by where F is the incomplete elliptic integral of the first kind. The upper half-plane is mapped to a triangle with circular arcs for edges by the Schwarz triangle map . An analogue of SC mapping that works also for multiply-connected is presented in: Case, James (2008), "Breakthrough in Conformal Mapping" (PDF) , SIAM News , 41 (1) . Potential theory In mathematics and mathematical physics , potential theory

1403-654: Is obvious from the names of important algorithms like Newton's method , Lagrange interpolation polynomial , Gaussian elimination , or Euler's method . The origins of modern numerical analysis are often linked to a 1947 paper by John von Neumann and Herman Goldstine , but others consider modern numerical analysis to go back to work by E. T. Whittaker in 1912. 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

1464-473: 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

1525-534: Is the consideration of inequalities they satisfy. Perhaps the most basic such inequality, from which most other inequalities may be derived, is the maximum principle . Another important result is Liouville's theorem , which states the only bounded harmonic functions defined on the whole of R are, in fact, constant functions. In addition to these basic inequalities, one has Harnack's inequality , which states that positive harmonic functions on bounded domains are roughly constant. One important use of these inequalities

1586-420: Is the study of harmonic functions . The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravity and the electrostatic force, could be modeled using functions called the gravitational potential and electrostatic potential , both of which satisfy Poisson's equation —or in the vacuum, Laplace's equation . There

1647-582: Is to prove convergence of families of harmonic functions or sub-harmonic functions, see Harnack's theorem . These convergence theorems are used to prove the existence of harmonic functions with particular properties. Since the Laplace equation is linear, the set of harmonic functions defined on a given domain is, in fact, a vector space . By defining suitable norms and/or inner products , one can exhibit sets of harmonic functions which form Hilbert or Banach spaces . In this fashion, one obtains such spaces as

1708-415: 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 finite number of steps, even if infinite precision were possible. Starting from an initial guess, iterative methods form successive approximations that converge to the exact solution only in the limit. A convergence test, often involving

1769-472: The Hardy space , Bloch space , Bergman space and Sobolev space . Numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations ) for the problems of mathematical analysis (as distinguished from discrete mathematics ). It is the study of numerical methods that attempt to find approximate solutions of problems rather than

1830-1259: 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 , TK Solver , S-PLUS , and IDL as well as free and open-source alternatives such as FreeMat , Scilab , GNU Octave (similar to Matlab), and IT++ (a C++ library). There are also programming languages such as R (similar to S-PLUS), Julia , and Python with libraries such as NumPy , SciPy 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. Many computer algebra systems such as Mathematica also benefit from

1891-469: The Jacobi method , Gauss–Seidel method , successive over-relaxation and conjugate gradient method 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

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1952-498: The Kelvin transform and the method of images . Third, one can use conformal transforms to map harmonic functions in one domain to harmonic functions in another domain. The most common instance of such a construction is to relate harmonic functions on a disk to harmonic functions on a half-plane. Fourth, one can use conformal symmetry to extend harmonic functions to harmonic functions on conformally flat Riemannian manifolds . Perhaps

2013-415: The conformal symmetries of the n {\displaystyle n} -dimensional Euclidean space . This fact has several implications. First of all, one can consider harmonic functions which transform under irreducible representations of the conformal group or of its subgroups (such as the group of rotations or translations). Proceeding in this fashion, one systematically obtains the solutions of

2074-415: 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. As an example, consider the problem of solving for the unknown quantity x . For the iterative method, apply the bisection method to f ( x ) = 3 x − 24. The initial values are a = 0, b = 3, f (

2135-407: The singularities of harmonic functions would be said to belong to potential theory whilst a result on how the solution depends on the boundary data would be said to belong to the theory of Poisson's equation. This is not a hard and fast distinction, and in practice there is considerable overlap between the two fields, with methods and results from one being used in the other. Modern potential theory

2196-454: The Laplace equation is linear . This means that the fundamental object of study in potential theory is a linear space of functions. This observation will prove especially important when we consider function space approaches to the subject in a later section. As for symmetry in the usual sense of the term, we may start with the theorem that the symmetries of the n {\displaystyle n} -dimensional Laplace equation are exactly

2257-452: The Laplace equation which arise from separation of variables such as spherical harmonic solutions and Fourier series . By taking linear superpositions of these solutions, one can produce large classes of harmonic functions which can be shown to be dense in the space of all harmonic functions under suitable topologies. Second, one can use conformal symmetry to understand such classical tricks and techniques for generating harmonic functions as

2318-560: The Schwarz–Christoffel mapping is given by This transformation is sketched below. A mapping to a plane triangle with interior angles π a , π b {\displaystyle \pi a,\,\pi b} and π ( 1 − a − b ) {\displaystyle \pi (1-a-b)} is given by which can be expressed in terms of hypergeometric functions , more precisely incomplete beta functions . The upper half-plane

2379-472: 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

2440-399: 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, the spectral image compression algorithm is based on the singular value decomposition. The corresponding tool in statistics

2501-638: The equation is linear or not. For instance, the equation 2 x + 5 = 3 {\displaystyle 2x+5=3} is linear while 2 x 2 + 5 = 3 {\displaystyle 2x^{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

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2562-511: The exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting

2623-449: The field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to a wide variety of hard problems, many of which are infeasible to solve symbolically: 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

2684-457: The finite case, the analogue I-K of the Laplacian in potential theory has its own maximum principle, uniqueness principle, balance principle, and others. A useful starting point and organizing principle in the study of harmonic functions is a consideration of the symmetries of the Laplace equation. Although it is not a symmetry in the usual sense of the term, we can start with the observation that

2745-500: 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

2806-402: The local structure of level sets of harmonic functions. There is Bôcher's theorem , which characterizes the behavior of isolated singularities of positive harmonic functions. As alluded to in the last section, one can classify the isolated singularities of harmonic functions as removable singularities, poles, and essential singularities. A fruitful approach to the study of harmonic functions

2867-403: 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 ,

2928-445: 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,

2989-429: The motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and biology. Before modern computers, numerical methods often relied on hand interpolation formulas, using data from large printed tables. Since the mid 20th century, computers calculate the required functions instead, but many of

3050-411: The polygon. If the polygon has interior angles α , β , γ , … {\displaystyle \alpha ,\beta ,\gamma ,\ldots } , then this mapping is given by where K {\displaystyle K} is a constant , and a < b < c < ⋯ {\displaystyle a<b<c<\cdots } are

3111-438: 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. To the contrary, if a problem is 'ill-conditioned', then any small error in the data will grow to be a large error. Both the original problem and the algorithm used to solve that problem can be well-conditioned or ill-conditioned, and any combination is possible. So an algorithm that solves

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3172-599: The residual , 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

3233-403: 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

3294-641: The same formulas continue to be used in software algorithms. The numerical point of view goes back to the earliest mathematical writings. A tablet from the Yale Babylonian Collection ( YBC 7289 ), gives a sexagesimal numerical approximation of the square root of 2 , the length of the diagonal in a unit square . Numerical analysis continues this long tradition: rather than giving exact symbolic answers translated into digits and applicable only to real-world measurements, approximate solutions within specified error bounds are used. The overall goal of

3355-402: The simplest such extension is to consider a harmonic function defined on the whole of R (with the possible exception of a discrete set of singular points) as a harmonic function on the n {\displaystyle n} -dimensional sphere . More complicated situations can also happen. For instance, one can obtain a higher-dimensional analog of Riemann surface theory by expressing

3416-422: The solution of the problem. Round-off errors arise because it is impossible to represent all real numbers exactly on a machine with finite memory (which is what all practical digital computers are). Truncation errors are committed when an iterative method is terminated or a mathematical procedure is approximated and the approximate solution differs from the exact solution. Similarly, discretization induces

3477-442: The values, along the real axis of the ζ {\displaystyle \zeta } plane, of points corresponding to the vertices of the polygon in the z {\displaystyle z} plane. A transformation of this form is called a Schwarz–Christoffel mapping . The integral can be simplified by mapping the point at infinity of the ζ {\displaystyle \zeta } plane to one of

3538-408: The vertices of the z {\displaystyle z} plane polygon. By doing this, the first factor in the formula becomes constant and so can be absorbed into the constant K {\displaystyle K} . Conventionally, the point at infinity would be mapped to the vertex with angle α {\displaystyle \alpha } . In practice, to find a mapping to

3599-410: 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 be 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

3660-402: 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. The Leslie Fox Prize for Numerical Analysis

3721-550: Was initiated in 1985 by the Institute of Mathematics and its Applications . 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

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