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Laguerre's Method
In numerical analysis, Laguerre's method is a root-finding algorithm tailored to polynomials. In other words, Laguerre's method can be used to numerically solve the equation for a given polynomial . One of the most useful properties of this method is that it is, from extensive empirical study, very close to being a "sure-fire" method, meaning that it is almost guaranteed to always converge to ''some'' root of the polynomial, no matter what initial guess is chosen. However, for computer computation, more efficient methods are known, with which it is guaranteed to find all roots (see ) or all real roots (see Real-root isolation). This method is named in honour of the French mathematician, Edmond Laguerre. Definition The algorithm of the Laguerre method to find one root of a polynomial of degree is: * Choose an initial guess * For ** If p(x_k) is very small, exit the loop ** Calculate G = \frac ** Calculate H = G^2 - \frac ** Calculate a = \frac, where the sign is chosen to give ... [...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] |
Graeffe's Method
In mathematics, Graeffe's method or Dandelin–Lobachesky–Graeffe method is an Root-finding algorithm, algorithm for finding all of the roots of a polynomial. It was developed independently by Germinal Pierre Dandelin in 1826 and Nikolai Lobachevsky, Lobachevsky in 1834. In 1837 Karl Heinrich Gräffe also discovered the principal idea of the method. The method separates the roots of a polynomial by squaring them repeatedly. This squaring of the roots is done implicitly, that is, only working on the coefficients of the polynomial. Finally, Viète's formulas are used in order to approximate the roots. Dandelin–Graeffe iteration Let be a polynomial of degree :p(x) = (x-x_1)\cdots(x-x_n). Then :p(-x) = (-1)^n (x+x_1)\cdots(x+x_n). Let be the polynomial which has the squares x_1^2, \cdots, x_n^2 as its roots, :q(x)= \left (x-x_1^2 \right )\cdots \left (x-x_n^2 \right ). Then we can write: :\begin q(x^2) & = \left (x^2-x_1^2 \right )\cdots \left (x^2-x_n^2 \r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIAM Review
Society for Industrial and Applied Mathematics (SIAM) is a professional society dedicated to applied mathematics, computational science, and data science through research, publications, and community. SIAM is the world's largest scientific society devoted to applied mathematics, and roughly two-thirds of its membership resides within the United States. Founded in 1951, the organization began holding annual national meetings in 1954, and now hosts conferences, publishes books and scholarly journals, and engages in advocacy in issues of interest to its membership. Members include engineers, scientists, and mathematicians, both those employed in academia and those working in industry. The society supports educational institutions promoting applied mathematics. SIAM is one of the four member organizations of the Joint Policy Board for Mathematics. Membership Membership is open to both individuals and organizations. By the end of its first full year of operation, SIAM had 130 membe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIAM Journal On Scientific Computing
The ''SIAM Journal on Scientific Computing'' (''SISC''), formerly ''SIAM Journal on Scientific & Statistical Computing'', is a scientific journal focusing on the research articles on numerical methods and techniques for scientific computation. It is published by the Society for Industrial and Applied Mathematics (SIAM). Hans De Sterck is the current editor-in-chief, assuming the role in January 2022. The impact factor is currently around 2. This journal papers address computational issues relevant to solution of scientific or engineering problems and include computational results demonstrating the effectiveness of proposed techniques. They are classified into three categories: 1) Methods and Algorithms for Scientific Computing. 2) Computational Methods in Science and Engineering. 3) Software and High-Performance Computing. The first type papers focus on theoretical analysis, provided that relevance to applications in science and engineering is demonstrated. They are supposed to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jenkins–Traub Algorithm
The Jenkins–Traub algorithm for polynomial zeros is a fast globally convergent iterative polynomial root-finding method published in 1970 by Michael A. Jenkins and Joseph F. Traub. They gave two variants, one for general polynomials with complex coefficients, commonly known as the "CPOLY" algorithm, and a more complicated variant for the special case of polynomials with real coefficients, commonly known as the "RPOLY" algorithm. The latter is "practically a standard in black-box polynomial root-finders". This article describes the complex variant. Given a polynomial ''P'', P(z) = \sum_^na_iz^, \quad a_0=1,\quad a_n\ne 0 with complex coefficients it computes approximations to the ''n'' zeros \alpha_1,\alpha_2,\dots,\alpha_n of ''P''(''z''), one at a time in roughly increasing order of magnitude. After each root is computed, its linear factor is removed from the polynomial. Using this ''deflation'' guarantees that each root is computed only once and that all roots are found. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stephensen's Method
In numerical analysis, Steffensen's method is an iterative method for numerical root-finding named after Johan Frederik Steffensen that is similar to the secant method and to Newton's method. Steffensen's method achieves a quadratic order of convergence without using derivatives, whereas the more familiar Newton's method also converges quadratically, but requires derivatives and the secant method does not require derivatives but also converges less quickly than quadratically. Steffensen's method has the drawback that it requires two function evaluations per step, whereas the secant method requires only one evaluation per step, so it is not necessarily most efficient in terms of computational cost, depending on the number of iterations each requires. Newton's method also requires evaluating two functions per step – for the function and for its derivative – and its computational cost varies between being the same as Steffensen's method (for most functions, where calculation of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiple Root
In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root. The notion of multiplicity is important to be able to count correctly without specifying exceptions (for example, ''double roots'' counted twice). Hence the expression, "counted with multiplicity". If multiplicity is ignored, this may be emphasized by counting the number of ''distinct'' elements, as in "the number of distinct roots". However, whenever a set (as opposed to multiset) is formed, multiplicity is automatically ignored, without requiring use of the term "distinct". Multiplicity of a prime factor In prime factorization, the multiplicity of a prime factor is its p-adic valuation. For example, the prime factorization of the integer is : the multiplicity of the prime factor is , while the multiplicity of each of the prime factors and is . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rate Of Convergence
In mathematical analysis, particularly numerical analysis, the rate of convergence and order of convergence of a sequence that converges to a limit are any of several characterizations of how quickly that sequence approaches its limit. These are broadly divided into rates and orders of convergence that describe how quickly a sequence further approaches its limit once it is already close to it, called asymptotic rates and orders of convergence, and those that describe how quickly sequences approach their limits from starting points that are not necessarily close to their limits, called non-asymptotic rates and orders of convergence. Asymptotic behavior is particularly useful for deciding when to stop a sequence of numerical computations, for instance once a target precision has been reached with an iterative root-finding algorithm, but pre-asymptotic behavior is often crucial for determining whether to begin a sequence of computations at all, since it may be impossible or imprac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Attraction Zones Of Laguerre's
Attraction may refer to: * Interpersonal attraction, the attraction between people which leads to friendships, platonic and romantic relationships. ** Physical attractiveness, attraction on the basis of beauty ** Sexual attraction * Object or event that is attractive ** Tourist attraction, a place of interest where tourists visit *** Amusement park attraction *Attraction in physics ** Electromagnetic attraction *** Magnetism ** Gravity ** Strong nuclear force ** Weak nuclear force Other uses * Attraction basin (a.k.a. attractor), in dynamical systems * Attraction (grammar), the process by which a relative pronoun takes on the case of its antecedent * Attraction (horse) (foaled 2001) * Attraction (2017 film), a Russian science fiction action film focusing upon an extraterrestrial spaceship crash-landing * Attraction (2018 film), a Bulgarian romantic comedy film See also * * * Attractive nuisance doctrine * Attract (other) * Law of attraction (other) ... [...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] |
Halley's Method
In numerical analysis, Halley's method is a root-finding algorithm used for functions of one real variable with a continuous second derivative. Edmond Halley was an English mathematician and astronomer who introduced the method now called by his name. The algorithm is second in the class of Householder's methods, after Newton's method. Like the latter, it iteratively produces a sequence of approximations to the root; their rate of convergence to the root is cubic. Multidimensional versions of this method exist. Halley's method exactly finds the roots of a linear-over-linear Padé approximation to the function, in contrast to Newton's method or the Secant method which approximate the function linearly, or Muller's method which approximates the function quadratically. There is also Halley's irrational method, described below. Method Halley's method is a numerical algorithm for solving the nonlinear equation In this case, the function has to be a function of one real varia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root-finding Algorithm
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 f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
