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Polynomial Root-finding Algorithms
Finding polynomial roots is a long-standing problem that has been the object of much research throughout history. A testament to this is that up until the 19th century, algebra meant essentially theory of polynomial equations. Principles Finding the root of a linear polynomial (degree one) is easy and needs only one division: the general equation ax + b = 0 has solution x = -b/a. For quadratic polynomials (degree two), the quadratic formula produces a solution, but its numerical evaluation may require some care for ensuring numerical stability. For degrees three and four, there are closed-form solutions in terms of radicals, which are generally not convenient for numerical evaluation, as being too complicated and involving the computation of several th roots whose computation is not easier than the direct computation of the roots of the polynomial (for example the expression of the real roots of a cubic polynomial may involve non-real cube roots). For polynomials of degree fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Root
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equivalently, x is the solution to the equation f(x) = 0. A "zero" of a function is thus an input value that produces an output of 0. A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f of degree two, defined by f(x)=x^2-5x+6 has the two roots (or zeros) that are 2 and 3. f(2)=2^2-5\times 2+6= 0\textf(3)=3^2-5\times 3+6=0. If the function maps real numbers to real numbers, then it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
QR Algorithm
In numerical linear algebra, the QR algorithm or QR iteration is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix. The QR algorithm was developed in the late 1950s by John G. F. Francis and by Vera N. Kublanovskaya, working independently. The basic idea is to perform a QR decomposition, writing the matrix as a product of an orthogonal matrix and an upper triangular matrix, multiply the factors in the reverse order, and iterate. The practical QR algorithm Formally, let ''A'' be a real matrix of which we want to compute the eigenvalues, and let ''A''0:=''A''. At the ''k''-th step (starting with ''k'' = 0), we compute the QR decomposition ''A''''k''=''Q''''k''''R''''k'' where ''Q''''k'' is an orthogonal matrix (i.e., ''Q''''T'' = ''Q''−1) and ''R''''k'' is an upper triangular matrix. We then form ''A''''k''+1 = ''R''''k''''Q''''k''. Note that : A_ = R_k Q_k = Q_k^ Q_k R_k Q_k = Q_k^ A_k Q_k = Q_k^ A_k Q_k, so all the ''A''''k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Algebra System
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The development of the computer algebra systems in the second half of the 20th century is part of the discipline of "computer algebra" or "symbolic computation", which has spurred work in algorithms over mathematical objects such as polynomials. Computer algebra systems may be divided into two classes: specialized and general-purpose. The specialized ones are devoted to a specific part of mathematics, such as number theory, group theory, or teaching of elementary mathematics. General-purpose computer algebra systems aim to be useful to a user working in any scientific field that requires manipulation of mathematical expressions. To be useful, a general-purpose computer algebra system must include various features such as: *a user interface allo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vincent's Theorem
In mathematics, Vincent's theorem—named after Alexandre Joseph Hidulphe Vincent—is a theorem that isolates the real roots of polynomials with rational coefficients. Even though Vincent's theorem is the basis of the fastest method for the isolation of the real roots of polynomials, it was almost totally forgotten, having been overshadowed by Sturm's theorem; consequently, it does not appear in any of the classical books on the theory of equations (of the 20th century), except for Uspensky's book. Two variants of this theorem are presented, along with several (continued fractions and bisection) real root isolation methods derived from them. Sign variation :Let ''c''0, ''c''1, ''c''2, ... be a finite or infinite sequence of real numbers. Suppose ''l'' 1. To obtain an arbitrary positive root we need to assume that a_1 \ge 0. * Negative roots are obtained by replacing ''x'' by −''x'', in which case the negative roots become positive. Vincent's theorem: Bisection version (Alesina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Budan's Theorem
In mathematics, Budan's theorem is a theorem for bounding the number of real roots of a polynomial in an interval, and computing the parity of this number. It was published in 1807 by François Budan de Boislaurent. A similar theorem was published independently by Joseph Fourier in 1820. Each of these theorems is a corollary of the other. Fourier's statement appears more often in the literature of 19th century and has been referred to as Fourier's, Budan–Fourier, Fourier–Budan, and even Budan's theorem Budan's original formulation is used in fast modern algorithms for real-root isolation of polynomials. Sign variation Let c_0, c_1, c_2, \ldots c_k be a finite sequence of real numbers. A ''sign variation'' or ''sign change'' in the sequence is a pair of indices such that c_ic_j < 0, and either or for all such that . In other words, a sign variation occurs in the sequence at each place where the signs change, when ignoring zeros. For studyin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descartes' Rule Of Signs
In mathematics, Descartes' rule of signs, first described by René Descartes in his work ''La Géométrie'', is a technique for getting information on the number of positive real roots of a polynomial. It asserts that the number of positive roots is at most the number of sign changes in the sequence of polynomial's coefficients (omitting the zero coefficients), and that the difference between these two numbers is always even. This implies, in particular, that if the number of sign changes is zero or one, then there are exactly zero or one positive roots, respectively. By a homographic transformation of the variable, one may use Descartes' rule of signs for getting a similar information on the number of roots in any interval. This is the basic idea of Budan's theorem and Budan–Fourier theorem. By repeating the division of an interval into two intervals, one gets eventually a list of disjoint intervals containing together all real roots of the polynomial, and containing each exactly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sturm's Theorem
In mathematics, the Sturm sequence of a univariate polynomial is a sequence of polynomials associated with and its derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem expresses the number of distinct real roots of located in an interval in terms of the number of changes of signs of the values of the Sturm sequence at the bounds of the interval. Applied to the interval of all the real numbers, it gives the total number of real roots of . Whereas the fundamental theorem of algebra readily yields the overall number of complex roots, counted with multiplicity, it does not provide a procedure for calculating them. Sturm's theorem counts the number of distinct real roots and locates them in intervals. By subdividing the intervals containing some roots, it can isolate the roots into arbitrarily small intervals, each containing exactly one root. This yields the oldest real-root isolation algorithm, and arbitrary-precision root-finding algorithm for univariate ... [...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] |
Free Software
Free software or libre software is computer software distributed under terms that allow users to run the software for any purpose as well as to study, change, and distribute it and any adapted versions. Free software is a matter of liberty, not price; all users are legally free to do what they want with their copies of a free software (including profiting from them) regardless of how much is paid to obtain the program.Selling Free Software (gnu.org) Computer programs are deemed "free" if they give end-users (not just the developer) ultimate control over the software and, subsequently, over their devices. The right to study and modify a computer program entails that |
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] |
Wilkinson's Polynomial
In numerical analysis, Wilkinson's polynomial is a specific polynomial which was used by James H. Wilkinson in 1963 to illustrate a difficulty when finding the root of a polynomial: the location of the roots can be very sensitive to perturbations in the coefficients of the polynomial. The polynomial is : w(x) = \prod_^ (x - i) = (x-1)(x-2) \cdots (x-20). Sometimes, the term ''Wilkinson's polynomial'' is also used to refer to some other polynomials appearing in Wilkinson's discussion. Background Wilkinson's polynomial arose in the study of algorithms for finding the roots of a polynomial : p(x) = \sum_^n c_i x^i. It is a natural question in numerical analysis to ask whether the problem of finding the roots of ''p'' from the coefficients ''c''''i'' is well-conditioned. That is, we hope that a small change in the coefficients will lead to a small change in the roots. Unfortunately, this is not the case here. The problem is ill-conditioned when the polynomial has a multiple ro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Instability
In the mathematics, mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues. On the other hand, in numerical algorithms for differential equations the concern is the growth of round-off errors and/or small fluctuations in initial data which might cause a large deviation of final answer from the exact solution. Some numerical algorithms may damp out the small fluctuations (errors) in the input data; others might magnify such errors. Calculations that can be proven not to magnify approximation errors are called ''numerically stable''. One of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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