|
Anderson Acceleration
In mathematics, Anderson acceleration, also called Anderson mixing, is a method for the acceleration of the convergence rate of fixed-point iterations. Introduced by Donald G. Anderson, this technique can be used to find the solution to fixed point equations f(x) = x often arising in the field of computational science. Definition Given a function f:\mathbb^n \to \mathbb^n, consider the problem of finding a fixed point of f, which is a solution to the equation f(x) = x. A classical approach to the problem is to employ a fixed-point iteration scheme; that is, given an initial guess x_0 for the solution, to compute the sequence x_ = f(x_i) until some convergence criterion is met. However, the convergence of such a scheme is not guaranteed in general; moreover, the rate of convergence is usually linear, which can become too slow if the evaluation of the function f is computationally expensive. Anderson acceleration is a method to accelerate the convergence of the fixed-point sequenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed-point Iteration
In numerical analysis, fixed-point iteration is a method of computing fixed points of a function. More specifically, given a function f defined on the real numbers with real values and given a point x_0 in the domain of f, the fixed-point iteration is x_=f(x_n), \, n=0, 1, 2, \dots which gives rise to the sequence x_0, x_1, x_2, \dots of iterated function applications x_0, f(x_0), f(f(x_0)), \dots which is hoped to converge to a point x_\text. If f is continuous, then one can prove that the obtained x_\text is a fixed point of f, i.e., f(x_\text)=x_\text . More generally, the function f can be defined on any metric space with values in that same space. Examples * A first simple and useful example is the Babylonian method for computing the square root of , which consists in taking f(x) = \frac 1 2 \left(\frac a x + x\right), i.e. the mean value of and , to approach the limit x = \sqrt a (from whatever starting point x_0 \gg 0 ). This is a special case of Newton's method quo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton's Method
In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a real-valued function , its derivative , and an initial guess for a root of . If satisfies certain assumptions and the initial guess is close, then x_ = x_0 - \frac is a better approximation of the root than . Geometrically, is the x-intercept of the tangent of the graph of at : that is, the improved guess, , is the unique root of the linear approximation of at the initial guess, . The process is repeated as x_ = x_n - \frac until a sufficiently precise value is reached. The number of correct digits roughly doubles with each step. This algorithm is first in the class of Householder's methods, and was succeeded by Halley's method. The method can also be extended t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, 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 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 the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root-finding Algorithms
In numerical analysis, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function is a number such that . As, generally, the zeros of a function cannot be computed exactly nor expressed in closed form, root-finding algorithms provide approximations to zeros. For functions from the real numbers to real numbers or from the complex numbers to the complex numbers, these are expressed either as floating-point numbers without error bounds or as floating-point values together with error bounds. The latter, approximations with error bounds, are equivalent to small isolating intervals for real roots or disks for complex roots. Solving an equation is the same as finding the roots of the function . Thus root-finding algorithms can be used to solve any equation of continuous functions. However, most root-finding algorithms do not guarantee that they will find all roots of a function, and if such an algorithm does not fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed-point Iteration
In numerical analysis, fixed-point iteration is a method of computing fixed points of a function. More specifically, given a function f defined on the real numbers with real values and given a point x_0 in the domain of f, the fixed-point iteration is x_=f(x_n), \, n=0, 1, 2, \dots which gives rise to the sequence x_0, x_1, x_2, \dots of iterated function applications x_0, f(x_0), f(f(x_0)), \dots which is hoped to converge to a point x_\text. If f is continuous, then one can prove that the obtained x_\text is a fixed point of f, i.e., f(x_\text)=x_\text . More generally, the function f can be defined on any metric space with values in that same space. Examples * A first simple and useful example is the Babylonian method for computing the square root of , which consists in taking f(x) = \frac 1 2 \left(\frac a x + x\right), i.e. the mean value of and , to approach the limit x = \sqrt a (from whatever starting point x_0 \gg 0 ). This is a special case of Newton's method quo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages. Although MATLAB is intended primarily for numeric computing, an optional toolbox uses the MuPAD symbolic engine allowing access to symbolic computing abilities. An additional package, Simulink, adds graphical multi-domain simulation and model-based design for dynamic and embedded systems. , MATLAB has more than four million users worldwide. They come from various backgrounds of engineering, science, and economics. , more than 5000 global colleges and universities use MATLAB to support instruction and research. History Origins MATLAB was invented by mathematician and computer programmer Cleve Moler. The idea for MATLAB was base ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GMRES
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] |
Broyden's Method
In numerical analysis, Broyden's method is a quasi-Newton method for finding roots in variables. It was originally described by C. G. Broyden in 1965. Newton's method for solving uses the Jacobian matrix, , at every iteration. However, computing this Jacobian can be a difficult and expensive operation; for large problems such as those involving solving the Kohn–Sham equations in quantum mechanics the number of variables can be in the hundreds of thousands. The idea behind Broyden's method is to compute the whole Jacobian at most only at the first iteration, and to do rank-one updates at other iterations. In 1979 Gay proved that when Broyden's method is applied to a linear system of size , it terminates in steps, although like all quasi-Newton methods, it may not converge for nonlinear systems. Description of the method Solving single-variable nonlinear equation In the secant method, we replace the first derivative at with the finite-difference approximation: : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-Newton Method
In numerical analysis, a quasi-Newton method is an iterative numerical method used either to find zeroes or to find local maxima and minima of functions via an iterative recurrence formula much like the one for Newton's method, except using approximations of the derivatives of the functions in place of exact derivatives. Newton's method requires the Jacobian matrix of all partial derivatives of a multivariate function when used to search for zeros or the Hessian matrix when used for finding extrema. Quasi-Newton methods, on the other hand, can be used when the Jacobian matrices or Hessian matrices are unavailable or are impractical to compute at every iteration. Some iterative methods that reduce to Newton's method, such as sequential quadratic programming, may also be considered quasi-Newton methods. Search for zeros: root finding Newton's method to find zeroes of a function g of multiple variables is given by x_ = x_n - _g(x_n) g(x_n), where _g(x_n) is the left inverse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Difference
A finite difference is a mathematical expression of the form . Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation. The difference operator, commonly denoted \Delta, is the operator (mathematics), operator that maps a function to the function \Delta[f] defined by \Delta[f](x) = f(x+1)-f(x). A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives. There are many similarities between difference equations and differential equations. Certain Recurrence relation#Relationship to difference equations narrowly defined, recurrence relations can be written as difference equations by replacing iteration notation with finite differences. In numerical analysis, finite differences are widely used for #Relation with derivatives, approximating derivatives, and the term "finite difference" is often used a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Choice Of M
A choice is the range of different things from which a being can choose. The arrival at a choice may incorporate motivators and models. Freedom of choice is generally cherished, whereas a severely limited or artificially restricted choice can lead to discomfort with choosing, and possibly an unsatisfactory outcome. In contrast, a choice with excessively numerous options may lead to confusion, reduced satisfaction, regret of the alternatives not taken, and indifference in an unstructured existence; and the illusion that choosing an object or a course, necessarily leads to the control of that object or course, can cause psychological problems. Types One can distinguish four or five main types of decisions, although they can be expressed in different ways. Brian Tracy breaks them down into: # command decisions, which can only be made by you, as the "Commander in Chief", or owner of a company # delegated decisions, which may be made by anyone. Decisions for example can be: The c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
