Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. Numerical analysis naturally finds application in all fields of engineering and the physical sciences, but in the 21st century also the life sciences, social sciences, medicine, business and even the arts have adopted elements of scientific computations. The growth in computing power has revolutionized the use of realistic mathematical models in science and engineering, and subtle numerical analysis is required to implement these detailed models of the world. For example, ordinary differential equations appear in celestial mechanics ; numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.
In mathematics, computer science and operations research, mathematical optimization or mathematical programming is the selection of a best element from some set of available alternatives.
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the residuals made in the results of every single equation.
In mathematics and computing, a root-finding algorithm is an algorithm for finding roots of continuous functions. A root of a function f, from the real numbers to real numbers or from the complex numbers to the complex numbers, is a number x such that f(x) = 0. As, generally, the roots of a function cannot be computed exactly, nor expressed in closed form, root-finding algorithms provide approximations to roots, expressed either as floating point numbers or as small isolating intervals, or disks for complex roots.
Exponential growth is exhibited when the rate of change—the change per instant or unit of time—of the value of a mathematical function of time is proportional to the function's current value, resulting in its value at any time being an exponential function of time, i.e., a function in which the time value is the exponent. Exponential decay occurs in the same way when the growth rate is negative. In the case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay, the function values forming a geometric progression. In either exponential growth or exponential decay, the ratio of the rate of change of the quantity to its current size remains constant over time.
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation.
In mathematical optimization, Dantzig's simplex algorithm is a popular algorithm for linear programming.
In computer science, the time complexity is the computational complexity that describes the amount of time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to differ by at most a constant factor.
In mathematics, to solve an equation is to find its solutions, which are the values that fulfill the condition stated by the equation, consisting generally of two expressions related by an equality sign. When seeking a solution, one or more free variables are designated as unknowns. A solution is an assignment of expressions to the unknown variables that makes the equality in the equation true. In other words, a solution is an expression or a collection of expressions such that, when substituted for the unknowns, the equation becomes an identity. A solution of an equation is often also called a root of the equation, particularly but not only for algebraic or numerical equations.
In mathematics, extrapolation is the process of estimating, beyond the original observation range, the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between known observations, but extrapolation is subject to greater uncertainty and a higher risk of producing meaningless results. Extrapolation may also mean extension of a method, assuming similar methods will be applicable. Extrapolation may also apply to human experience to project, extend, or expand known experience into an area not known or previously experienced so as to arrive at a knowledge of the unknown. The extrapolation method can be applied in the interior reconstruction problem.
The Gauss–Newton algorithm is used to solve non-linear least squares problems. It is a modification of Newton's method for finding a minimum of a function. Unlike Newton's method, the Gauss–Newton algorithm can only be used to minimize a sum of squared function values, but it has the advantage that second derivatives, which can be challenging to compute, are not required.
In physics and mathematics, an ansatz is an educated guess that is verified later by its results.
Numerical linear algebra is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to mathematical questions. It is a subfield of numerical analysis, and a type of linear algebra. Because computers use floating-point arithmetic, they cannot exactly represent irrational data, and many algorithms increase that imprecision when implemented by a computer. Numerical linear algebra uses properties of vectors and matrices to develop computer algorithms that minimize computer error while retaining efficiency and precision.
Non-linear least squares is the form of least squares analysis used to fit a set of m observations with a model that is non-linear in n unknown parameters (m ≥ n). It is used in some forms of nonlinear regression. The basis of the method is to approximate the model by a linear one and to refine the parameters by successive iterations. There are many similarities to linear least squares, but also some significant differences. Examples of NLLS are (i) the Probit Regression, (ii) Threshold Regression, (iii) Smooth Regression, (iv) Logistic Link Regression, (v) Box-Cox Transformed Regressors, and many others in Economic Theory.
The finite element method (FEM), is a numerical method for solving problems of engineering and mathematical physics. Typical problem areas of interest include structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The analytical solution of these problems generally require the solution to boundary value problems for partial differential equations. The finite element method formulation of the problem results in a system of algebraic equations. The method approximates the unknown function over the domain. To solve the problem, it subdivides a large system into smaller, simpler parts that are called finite elements. The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. FEM then uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function.
Deterministic global optimization is a branch of numerical optimization which focuses on finding the global solutions of an optimization problem whilst providing theoretical guarantees that the reported solution is indeed the global one, within some predefined tolerance. The term "deterministic global optimization" typically refers to complete or rigorous optimization methods. Rigorous methods converge to the global optimum in finite time. Deterministic global optimization methods are typically used when locating the global solution is a necessity, when it is extremely difficult to find a feasible solution, or simply when the user desires to locate the best possible solution to a problem.
