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Aberth Method
The Aberth method, or Aberth–Ehrlich method or Ehrlich–Aberth method, named after Oliver Aberth and Louis W. Ehrlich, is a root-finding algorithm developed in 1967 for simultaneous approximation of all the roots of a univariate polynomial. This method converges cubically, an improvement over the Durand–Kerner method, another algorithm for approximating all roots at once, which converges quadratically. (However, both algorithms converge linearly at multiple zeros.) This method is used in MPSolve, which is the reference software for approximating all roots of a polynomial to an arbitrary precision. Description Let p(x)=p_nx^n+p_x^+\cdots+p_1x+p_0 be a univariate polynomial of degree ''n'' with real or complex coefficients. Then there exist complex numbers z^*_1,\,z^*_2,\dots,z^*_n, the roots of ''p(x)'', that give the factorization: :p(x)=p_n\cdot(x-z^*_1)\cdot(x-z^*_2)\cdots(x-z^*_n). Although those numbers are unknown, upper and lower bounds for their absolute values ... [...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] |
Univariate Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' joins tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Durand–Kerner Method
In numerical analysis, the Weierstrass method or Durand–Kerner method, discovered by Karl Weierstrass in 1891 and rediscovered independently by Durand in 1960 and Kerner in 1966, is a root-finding algorithm for solving polynomial equations. In other words, the method can be used to solve numerically the equation : ''f''(''x'') = 0, where ''f'' is a given polynomial, which can be taken to be scaled so that the leading coefficient is 1. Explanation This explanation considers equations of degree four. It is easily generalized to other degrees. Let the polynomial ''f'' be defined by : f(x) = x^4 + ax^3 + bx^2 + cx + d for all ''x''. The known numbers ''a'', ''b'', ''c'', ''d'' are the coefficients. Let the (complex) numbers ''P'', ''Q'', ''R'', ''S'' be the roots of this polynomial ''f''. Then : f(x) = (x - P)(x - Q)(x - R)(x - S) for all ''x''. One can isolate the value ''P'' from this equation: : P = x - \frac. So if used as a fixed-point iteration : x_1 := x_0 - \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicity (mathematics)
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] |
MPSolve
MPSolve (Multiprecision Polynomial Solver) is a package for the Root-finding algorithm, approximation of the roots of a polynomial, univariate polynomial. It uses the Aberth method, combined with a careful use of multiprecision. "Mpsolve takes advantage of Sparse matrix, sparsity, and has special hooking, hooks for polynomials that can be evaluated efficiently by straight-line programs" Implementation The program is written mostly in C (programming language), ANSI C and makes use of the GNU Multi-Precision Library. It uses a command-line interface (CLI) and, starting from version 3.1.0 has also a Graphical user interface, GUI and interfaces for MATLAB and GNU Octave, GNU/Octave. Usage The executable program of the package is called mpsolve. It can be execution (computers), run from command line in terminal emulator, console. The executable file for the graphical user interface is called xmpsolve, and the MATLAB and Octave functions are called mps_roots. They behave similarly t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Univariate
In mathematics, a univariate object is an expression, equation, function or polynomial involving only one variable. Objects involving more than one variable are multivariate. In some cases the distinction between the univariate and multivariate cases is fundamental; for example, the fundamental theorem of algebra and Euclid's algorithm for polynomials are fundamental properties of univariate polynomials that cannot be generalized to multivariate polynomials. In statistics, a univariate distribution characterizes one variable, although it can be applied in other ways as well. For example, univariate data are composed of a single scalar component. In time series analysis, the whole time series is the "variable": a univariate time series is the series of values over time of a single quantity. Correspondingly, a "multivariate time series" characterizes the changing values over time of several quantities. In some cases, the terminology is ambiguous, since the values within a univariate t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorisation
In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several ''factors'', usually smaller or simpler objects of the same kind. For example, is a factorization of the integer , and is a factorization of the polynomial . Factorization is not usually considered meaningful within number systems possessing division, such as the real or complex numbers, since any x can be trivially written as (xy)\times(1/y) whenever y is not zero. However, a meaningful factorization for a rational number or a rational function can be obtained by writing it in lowest terms and separately factoring its numerator and denominator. Factorization was first considered by ancient Greek mathematicians in the case of integers. They proved the fundamental theorem of arithmetic, which asserts that every positive integer may be factored into a product of prime numbers, which cannot be further ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Properties Of Polynomial Roots
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property. Property may also refer to: Mathematics * Property (mathematics) Philosophy and science * Property (philosophy), in philosophy and logic, an abstraction characterizing an object *Material properties, properties by which the benefits of one material versus another can be assessed *Chemical property, a material's properties that becomes evident during a chemical reaction *Physical property, any property that is measurable whose value describes a state of a physical system *Semantic property *Thermodynamic properties, in thermodynamics and materials science, intensive and extensive physical properties of substances *Mental property, a property of the mind studied by many sciences and parasciences Computer science * Property (programming), a type of class member in object-oriented programming * .properties, a Java Properties File to store program settings as name-value p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electrostatic
Electrostatics is a branch of physics that studies electric charges at rest (static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber, (), was thus the source of the word 'electricity'. Electrostatic phenomena arise from the forces that electric charges exert on each other. Such forces are described by Coulomb's law. Even though electrostatically induced forces seem to be rather weak, some electrostatic forces are relatively large. The force between an electron and a proton, which together make up a hydrogen atom, is about 36 orders of magnitude stronger than the gravitational force acting between them. There are many examples of electrostatic phenomena, from those as simple as the attraction of plastic wrap to one's hand after it is removed from a package, to the apparently spontaneous explosion of grain silos, the damage of electronic components during manufacturi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point Particle
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up space. A point particle is an appropriate representation of any object whenever its size, shape, and structure are irrelevant in a given context. For example, from far enough away, any finite-size object will look and behave as a point-like object. Point masses and point charges, discussed below, are two common cases. When a point particle has an additive property, such as mass or charge, it is often represented mathematically by a Dirac delta function. In quantum mechanics, the concept of a point particle is complicated by the Heisenberg uncertainty principle, because even an elementary particle, with no internal structure, occupies a nonzero volume. For example, the atomic orbit of an electron in the hydrogen atom occupies a volume of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas Joannes Stieltjes
Thomas Joannes Stieltjes (, 29 December 1856 – 31 December 1894) was a Dutch mathematician. He was a pioneer in the field of moment problems and contributed to the study of continued fractions. The Thomas Stieltjes Institute for Mathematics at Leiden University, dissolved in 2011, was named after him, as is the Riemann–Stieltjes integral. Biography Stieltjes was born in Zwolle on 29 December 1856. His father (who had the same first names) was a civil engineer and politician. Stieltjes Sr. was responsible for the construction of various harbours around Rotterdam, and also seated in the Dutch parliament. Stieltjes Jr. went to university at the Polytechnical School in Delft in 1873. Instead of attending lectures, he spent his student years reading the works of Gauss and Jacobi — the consequence of this being he failed his examinations. There were 2 further failures (in 1875 and 1876), and his father despaired. His father was friends with H. G. van de Sande Bakhuyzen (who was t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton's Method
In numerical analysis, Newton's method, also known as the Newton–Raphson 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 single-variable function defined for a real variable , the function's derivative , and an initial guess for a root of . If the function satisfies sufficient assumptions and the initial guess is close, then :x_ = x_0 - \frac is a better approximation of the root than . Geometrically, is the intersection of the -axis and the tangent of the graph of at : that is, the improved guess is the unique root of the linear approximation at the initial point. The process is repeated as :x_ = x_n - \frac until a sufficiently precise value is reached. This algorithm is first in the class of Householder's methods, succeeded by Halley's method. The method can also be extended to complex functions an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
