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Broyden's Method
In numerical analysis, Broyden's method is a quasi-Newton method Quasi-Newton methods are methods used to either find zeroes or local maxima and minima of functions, as an alternative to Newton's method. They can be used if the Jacobian or Hessian is unavailable or is too expensive to compute at every iteration. ... for root-finding algorithm, finding roots in variables. It was originally described by Charles George Broyden, C. G. Broyden in 1965. Newton's method for solving uses the Jacobian matrix and determinant, Jacobian matrix, , at every iteration. However, computing this Jacobian is a difficult and expensive operation. The idea behind Broyden's method is to compute the whole Jacobian 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 me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-Newton Method
Quasi-Newton methods are methods used to either find zeroes or local maxima and minima of functions, as an alternative to Newton's method. They can be used if the Jacobian or Hessian is unavailable or is too expensive to compute at every iteration. The "full" Newton's method requires the Jacobian in order to search for zeros, or the Hessian for finding extrema. 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 of the Jacobian matrix J_g(x_n) of g evaluated for x_n. Strictly speaking, any method that replaces the exact Jacobian J_g(x_n) with an approximation is a quasi-Newton method. For instance, the chord method (where J_g(x_n) is replaced by J_g(x_0) for all iterations) is a simple example. The methods given below for optimization refer to an important subclass of quasi-Newton methods, secant methods. Using methods developed to find extrema in order to fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimization (mathematics)
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a define ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton's Method In Optimization
In calculus, Newton's method is an iterative method for finding the roots of a differentiable function , which are solutions to the equation . As such, Newton's method can be applied to the derivative of a twice-differentiable function to find the roots of the derivative (solutions to ), also known as the critical points of . These solutions may be minima, maxima, or saddle points; see section "Several variables" in Critical point (mathematics) and also section "Geometric interpretation" in this article. This is relevant in optimization, which aims to find (global) minima of the function . Newton's method The central problem of optimization is minimization of functions. Let us first consider the case of univariate functions, i.e., functions of a single real variable. We will later consider the more general and more practically useful multivariate case. Given a twice differentiable function f:\mathbb\to \mathbb, we seek to solve the optimization problem : \min_ f(x) . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-Newton Method
Quasi-Newton methods are methods used to either find zeroes or local maxima and minima of functions, as an alternative to Newton's method. They can be used if the Jacobian or Hessian is unavailable or is too expensive to compute at every iteration. The "full" Newton's method requires the Jacobian in order to search for zeros, or the Hessian for finding extrema. 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 of the Jacobian matrix J_g(x_n) of g evaluated for x_n. Strictly speaking, any method that replaces the exact Jacobian J_g(x_n) with an approximation is a quasi-Newton method. For instance, the chord method (where J_g(x_n) is replaced by J_g(x_0) for all iterations) is a simple example. The methods given below for optimization refer to an important subclass of quasi-Newton methods, secant methods. Using methods developed to find extrema in order to fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secant Method
In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function ''f''. The secant method can be thought of as a finite-difference approximation of Newton's method. However, the secant method predates Newton's method by over 3000 years. The method For finding a zero of a function , the secant method is defined by the recurrence relation. : x_n = x_ - f(x_) \frac = \frac. As can be seen from this formula, two initial values and are required. Ideally, they should be chosen close to the desired zero. Derivation of the method Starting with initial values and , we construct a line through the points and , as shown in the picture above. In slope–intercept form, the equation of this line is :y = \frac(x - x_1) + f(x_1). The root of this linear function, that is the value of such that is :x = x_1 - f(x_1) \frac. We then use this new value of as and repeat the process, u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Davidon–Fletcher–Powell Formula
The Davidon–Fletcher–Powell formula (or DFP; named after William C. Davidon, Roger Fletcher (mathematician), Roger Fletcher, and Michael J. D. Powell) finds the solution to the secant equation that is closest to the current estimate and satisfies the curvature condition. It was the first quasi-Newton method to generalize the secant method to a multidimensional problem. This update maintains the symmetry and positive definiteness of the Hessian matrix. Given a function f(x), its gradient (\nabla f), and positive-definite matrix, positive-definite Hessian matrix B, the Taylor series is :f(x_k+s_k) = f(x_k) + \nabla f(x_k)^T s_k + \frac s^T_k s_k + \dots, and the Taylor series of the gradient itself (secant equation) :\nabla f(x_k+s_k) = \nabla f(x_k) + B s_k + \dots is used to update B. The DFP formula finds a solution that is symmetric, positive-definite and closest to the current approximate value of B_k: :B_= (I - \gamma_k y_k s_k^T) B_k (I - \gamma_k s_k y_k^T) + \gamm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hessian Matrix
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants". Definitions and properties Suppose f : \R^n \to \R is a function taking as input a vector \mathbf \in \R^n and outputting a scalar f(\mathbf) \in \R. If all second-order partial derivatives of f exist, then the Hessian matrix \mathbf of f is a square n \times n matrix, usually defined and arranged as follows: \mathbf H_f= \begin \dfrac & \dfrac & \cdots & \dfrac \\ .2ex \dfrac & \dfrac & \cdots & \dfrac \\ .2ex \vdots & \vdots & \ddots & \vdots \\ .2ex \dfrac & \dfrac & \cdots & \dfrac \end, or, by stating an equation for the coefficients using indices i and j, (\mathbf H_f)_ = \fra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradient of a function is non-zero at a point , the direction of the gradient is the direction in which the function increases most quickly from , and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to maximize a function by gradient ascent. In coordinate-free terms, the gradient of a function f(\bf) may be defined by: :df=\nabla f \cdot d\bf where ''df'' is the total infinitesimal change in ''f'' for an infinitesimal displacement d\bf, and is seen to be maximal when d\bf is in the direction of the gradi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sherman–Morrison Formula
In mathematics, in particular linear algebra, the Sherman–Morrison formula, named after Jack Sherman and Winifred J. Morrison, computes the inverse of the sum of an invertible matrix A and the outer product, u v^\textsf, of vectors u and v. The Sherman–Morrison formula is a special case of the Woodbury formula. Though named after Sherman and Morrison, it appeared already in earlier publications. Statement Suppose A\in\mathbb^ is an invertible square matrix and u,v\in\mathbb^n are column vectors. Then A + uv^\textsf is invertible iff 1 + v^\textsf A^u \neq 0. In this case, :\left(A + uv^\textsf\right)^ = A^ - . Here, uv^\textsf is the outer product of two vectors u and v. The general form shown here is the one published by Bartlett. Proof (\Leftarrow) To prove that the backward direction 1 + v^\textsfA^u \neq 0 \Rightarrow A + uv^\textsf is invertible with inverse given as above) is true, we verify the properties of the inverse. A matrix Y (in this case the right-ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root-finding Algorithm
In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex numbers, 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, expressed either as floating-point numbers or as small isolating intervals, or disks for complex roots (an interval or disk output being equivalent to an approximate output together with an error bound). Solving an equation is the same as finding the roots of the function . Thus root-finding algorithms allow solving any equation defined by continuous functions. However, most root-finding algorithms do not guarantee that they will find all the roots; in particular, if such an algorithm does not find any root, that does not mean that no root exists. Most nume ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Norm
In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions). Preliminaries Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m rows and n columns and entries in the field K. A matrix norm is a norm on K^. This article will always write such norms with double vertical bars (like so: \, A\, ). Thus, the matrix norm is a function \, \cdot\, : K^ \to \R that must satisfy the following properties: For all scalars \alpha \in K and matrices A, B \in K^, *\, A\, \ge 0 (''positive-valued'') *\, A\, = 0 \iff A=0_ (''definite'') *\left\, \alpha A\right\, =\left, \alpha\ \left\, A\right\, (''absolutely homogeneous'') *\, A+B\, \le \, A\, +\, B\, (''sub-additive'' or satisfying the ''triangle inequality'') The only feature distinguishing matrices from rearranged vectors is multiplication. Matrix norms are particularly useful if they are also sub-multiplicative: *\left\, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Underdetermined System
In mathematics, a system of linear equations or a system of polynomial equations is considered underdetermined if there are fewer equations than unknowns (in contrast to an overdetermined system, where there are more equations than unknowns). The terminology can be explained using the concept of constraint counting. Each unknown can be seen as an available degree of freedom. Each equation introduced into the system can be viewed as a constraint that restricts one degree of freedom. Therefore, the critical case (between overdetermined and underdetermined) occurs when the number of equations and the number of free variables are equal. For every variable giving a degree of freedom, there exists a corresponding constraint removing a degree of freedom. The underdetermined case, by contrast, occurs when the system has been underconstrained—that is, when the unknowns outnumber the equations. Solutions of underdetermined systems An underdetermined linear system has either no sol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
