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Residual (numerical Analysis)
Loosely speaking, a residual is the error in a result. To be precise, suppose we want to find ''x'' such that : f(x)=b. Given an approximation ''x''0 of ''x'', the residual is : b - f(x_0) that is, "what is left of the right hand side" after subtracting ''f''(''x''0)" (thus, the name "residual": what is left, the rest). On the other hand, the error is : x - x_0 If the exact value of ''x'' is not known, the residual can be computed, whereas the error cannot. Residual of the approximation of a function Similar terminology is used dealing with differential, integral and functional equations. For the approximation f_\text of the solution f of the equation : T(f)(x)=g(x) \, , the residual can either be the function : ~g(x)~ - ~T(f_\text)(x), or can be said to be the maximum of the norm of this difference : \max_ , g(x)-T(f_\text)(x), over the domain \mathcal X, where the function f_\text is expected to approximate the solution f , or some integral of a function of the diffe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Error
An error (from the Latin , meaning 'to wander'Oxford English Dictionary, s.v. “error (n.), Etymology,” September 2023, .) is an inaccurate or incorrect action, thought, or judgement. In statistics, "error" refers to the difference between the value which has been computed and the correct value. An error could result in failure or in a Deviation (statistics), deviation from the intended performance or behavior. Human behavior One reference differentiates between "error" and "mistake" as follows: In human behavior the norms or expectations for behavior or its consequences can be derived from the intention of the actor or from the expectations of other individuals or from a social grouping or from social norms. (See deviance (sociology), deviance.) Gaffes and faux pas can be labels for certain instances of this kind of error. More serious departures from social norms carry labels such as misbehavior and labels from the legal system, such as misdemeanor and crime. Departures f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Equation
In mathematical analysis, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,\ldots,x_n ; u(x_1,x_2,x_3,\ldots,x_n) ; I^1 (u), I^2(u), I^3(u), \ldots, I^m(u)) = 0 where I^i(u) is an integral operator acting on ''u.'' Hence, integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals. A direct comparison can be seen with the mathematical form of the general integral equation above with the general form of a differential equation which may be expressed as follows:f(x_1,x_2,x_3,\ldots,x_n ; u(x_1,x_2,x_3,\ldots,x_n) ; D^1 (u), D^2(u), D^3(u), \ldots, D^m(u)) = 0where D^i(u) may be viewed as a differential operator of order ''i''. Due to this close connection between differential and integral equations, one can often convert between the two. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Equation
In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning is often used, where a ''functional equation'' is an equation that relates several values of the same function. For example, the logarithm functions are essentially characterized by the ''logarithmic functional equation'' \log(xy)=\log(x) + \log(y). If the domain of the unknown function is supposed to be the natural numbers, the function is generally viewed as a sequence, and, in this case, a functional equation (in the narrower meaning) is called a recurrence relation. Thus the term ''functional equation'' is used mainly for real functions and complex functions. Moreover a smoothness condition is often assumed for the solutions, since without such a condition, most functional equations have very irregular solutions. For example, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterative Solver
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''i''-th approximation (called an "iterate") is derived from the previous ones. A specific implementation with termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or quasi-Newton methods like BFGS, is an algorithm of an iterative method or a method of successive approximation. An iterative method is called '' convergent'' if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, heuristic-based iterative methods are also common. In contrast, direct methods attempt to solve the problem by a finite sequence of operations. In the absence of rounding errors, direct methods would deliver an exact solution (for example ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Minimal Residual Method
In mathematics, the generalized minimal residual method (GMRES) is an iterative method for the numerical solution of an indefinite nonsymmetric system of linear equations. The method approximates the solution by the vector in a Krylov subspace with minimal residual. The Arnoldi iteration is used to find this vector. The GMRES method was developed by Yousef Saad and Martin H. Schultz in 1986. It is a generalization and improvement of the MINRES method due to Paige and Saunders in 1975. The MINRES method requires that the matrix is symmetric, but has the advantage that it only requires handling of three vectors. GMRES is a special case of the DIIS method developed by Peter Pulay in 1980. DIIS is applicable to non-linear systems. The method Denote the Euclidean norm of any vector v by \, v\, . Denote the (square) system of linear equations to be solved by Ax = b. The matrix ''A'' is assumed to be invertible of size ''m''-by-''m''. Furthermore, it is assumed that b is normali ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
