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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] |
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 at finding 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, 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 simulating living ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Symmetric Polynomial
In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary symmetric polynomials. That is, any symmetric polynomial is given by an expression involving only additions and multiplication of constants and elementary symmetric polynomials. There is one elementary symmetric polynomial of degree in variables for each positive integer , and it is formed by adding together all distinct products of distinct variables. Definition The elementary symmetric polynomials in variables , written for , are defined by :\begin e_1 (X_1, X_2, \dots,X_n) &= \sum_ X_j,\\ e_2 (X_1, X_2, \dots,X_n) &= \sum_ X_j X_k,\\ e_3 (X_1, X_2, \dots,X_n) &= \sum_ X_j X_k X_l,\\ \end and so forth, ending with : e_n (X_1, X_2, \dots,X_n) = X_1 X_2 \cdots X_n. In general, for we define : e_k (X_1 , \ldots , X_n )=\sum_ X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Java Programming Language
Java is a high-level, class-based, object-oriented programming language that is designed to have as few implementation dependencies as possible. It is a general-purpose programming language intended to let programmers ''write once, run anywhere'' ( WORA), meaning that compiled Java code can run on all platforms that support Java without the need to recompile. Java applications are typically compiled to bytecode that can run on any Java virtual machine (JVM) regardless of the underlying computer architecture. The syntax of Java is similar to C and C++, but has fewer low-level facilities than either of them. The Java runtime provides dynamic capabilities (such as reflection and runtime code modification) that are typically not available in traditional compiled languages. , Java was one of the most popular programming languages in use according to GitHub, particularly for client–server web applications, with a reported 9 million developers. Java was originally developed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open-Source
Open source is source code that is made freely available for possible modification and redistribution. Products include permission to use the source code, design documents, or content of the product. The open-source model is a decentralized software development model that encourages open collaboration. A main principle of open-source software development is peer production, with products such as source code, blueprints, and documentation freely available to the public. The open-source movement in software began as a response to the limitations of proprietary code. The model is used for projects such as in open-source appropriate technology, and open-source drug discovery. Open source promotes universal access via an open-source or free license to a product's design or blueprint, and universal redistribution of that design or blueprint. Before the phrase ''open source'' became widely adopted, developers and producers have used a variety of other terms. ''Open source'' gained ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ada Programming Language
Ada is a structured, statically typed, imperative, and object-oriented high-level programming language, extended from Pascal and other languages. It has built-in language support for '' design by contract'' (DbC), extremely strong typing, explicit concurrency, tasks, synchronous message passing, protected objects, and non-determinism. Ada improves code safety and maintainability by using the compiler to find errors in favor of runtime errors. Ada is an international technical standard, jointly defined by the International Organization for Standardization (ISO), and the International Electrotechnical Commission (IEC). , the standard, called Ada 2012 informally, is ISO/IEC 8652:2012. Ada was originally designed by a team led by French computer scientist Jean Ichbiah of CII Honeywell Bull under contract to the United States Department of Defense (DoD) from 1977 to 1983 to supersede over 450 programming languages used by the DoD at that time. Ada was named after Ada Lovelace (1815â ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open-source
Open source is source code that is made freely available for possible modification and redistribution. Products include permission to use the source code, design documents, or content of the product. The open-source model is a decentralized software development model that encourages open collaboration. A main principle of open-source software development is peer production, with products such as source code, blueprints, and documentation freely available to the public. The open-source movement in software began as a response to the limitations of proprietary code. The model is used for projects such as in open-source appropriate technology, and open-source drug discovery. Open source promotes universal access via an open-source or free license to a product's design or blueprint, and universal redistribution of that design or blueprint. Before the phrase ''open source'' became widely adopted, developers and producers have used a variety of other terms. ''Open source'' gained ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Victor Pan
Victor Yakovlevich Pan (russian: Пан Виктор Яковлевич) is a Soviet and American mathematician and computer scientist, known for his research on algorithms for polynomials and matrix multiplication. Education and career Pan earned his Ph.D. at Moscow University in 1964, under the supervision of Anatoli Georgievich Vitushkin, and continued his work at the Soviet Academy of Sciences. During that time, he published a number of significant papers and became known informally as "polynomial Pan" for his pioneering work in the area of polynomial computations. In late 1970s, he immigrated to the United States and held positions at several institutions including IBM Research. Since 1988, he has taught at Lehman College of the City University of New York. Contributions Victor Pan is an expert in computational complexity and has developed a number of new algorithms. One of his notable early results is a proof that the number of multiplications in Horner's method is optimal. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gershgorin Circle Theorem
In mathematics, the Gershgorin circle theorem may be used to bound the spectrum of a square matrix. It was first published by the Soviet mathematician Semyon Aronovich Gershgorin in 1931. Gershgorin's name has been transliterated in several different ways, including Geršgorin, Gerschgorin, Gershgorin, Hershhorn, and Hirschhorn. Statement and proof Let A be a complex n\times n matrix, with entries a_. For i \in\ let R_i be the sum of the absolute values of the non-diagonal entries in the i-th row: : R_i= \sum_ \left, a_\. Let D(a_, R_i) \subseteq \Complex be a closed disc centered at a_ with radius R_i. Such a disc is called a Gershgorin disc. :Theorem. Every eigenvalue of A lies within at least one of the Gershgorin discs D(a_,R_i). ''Proof.'' Let \lambda be an eigenvalue of A with corresponding eigenvector x = (x_j). Find ''i'' such that the element of ''x'' with the largest absolute value is x_i. Since Ax=\lambda x, in particular we take the ''i''th component of that e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Interpolation
In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a data set of coordinate pairs (x_j, y_j) with 0 \leq j \leq k, the x_j are called ''nodes'' and the y_j are called ''values''. The Lagrange polynomial L(x) has degree \leq k and assumes each value at the corresponding node, L(x_j) = y_j. Although named after Joseph-Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring. It is also an easy consequence of a formula published in 1783 by Leonhard Euler. Uses of Lagrange polynomials include the Newton–Cotes method of numerical integration and Shamir's secret sharing scheme in cryptography. For equispaced nodes, Lagrange interpolation is susceptible to Runge's phenomenon of large oscillation. Definition Given a set of k + 1 nodes \, which must all be distinct, x_j \neq x_m for indices j \neq m, the Lagrange basis for polynomials of degr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Companion Matrix
In linear algebra, the Frobenius companion matrix of the monic polynomial : p(t)=c_0 + c_1 t + \cdots + c_t^ + t^n ~, is the square matrix defined as :C(p)=\begin 0 & 0 & \dots & 0 & -c_0 \\ 1 & 0 & \dots & 0 & -c_1 \\ 0 & 1 & \dots & 0 & -c_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & -c_ \end. Some authors use the transpose of this matrix, which (dually) cycles coordinates, and is more convenient for some purposes, like linear recurrence relations. Characterization The characteristic polynomial as well as the minimal polynomial of are equal to . In this sense, the matrix is the "companion" of the polynomial . If is an ''n''-by-''n'' matrix with entries from some field , then the following statements are equivalent: * is similar to the companion matrix over of its characteristic polynomial * the characteristic polynomial of coincides with the minimal polynomial of , equivalently the minimal polynomial has degree * there exists a cycl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group is a group homomorphism . In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set ''S'' to itself. In any category, the composition of any two endomorphisms of is again an endomorphism of . It follows that the set of all endomorphisms of forms a monoid, the full transformation monoid, and denoted (or to emphasize the category ). Automorphisms An invertible endomorphism of is called an automorphism. The set of all automorphisms is a subset of with a group structure, called the automorphism group of and denoted . In the following diagram, the arrows denote implication: Endomorphism rings Any two endomorphisms of an abelian group, , can be added toge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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