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Polynomial Eigenvalue Problem
In mathematics, a nonlinear eigenproblem, sometimes nonlinear eigenvalue problem, is a generalization of the (ordinary) Eigenvalue, eigenvector and eigenspace, eigenvalue problem to equations that depend nonlinearly on the eigenvalue. Specifically, it refers to equations of the form : M (\lambda) x = 0 , where x\neq0 is a Vector (mathematics), vector, and ''M'' is a Matrix (mathematics), matrix-valued Function (mathematics), function of the number \lambda. The number \lambda is known as the (nonlinear) eigenvalue, the vector x as the (nonlinear) eigenvector, and (\lambda,x) as the eigenpair. The matrix M (\lambda) is singular at an eigenvalue \lambda. Definition In the discipline of numerical linear algebra the following definition is typically used. Let \Omega \subseteq \Complex, and let M : \Omega \rightarrow \Complex^ be a function that maps scalars to matrices. A scalar \lambda \in \Complex is called an ''eigenvalue'', and a nonzero vector x \in \Complex^n is called a ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Eigenvalue Problem
In mathematics, the quadratic eigenvalue problemF. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM Rev., 43 (2001), pp. 235–286. (QEP), is to find scalar eigenvalues \lambda, left eigenvectors y and right eigenvectors x such that : Q(\lambda)x = 0 ~ \text ~ y^\ast Q(\lambda) = 0, where Q(\lambda)=\lambda^2 M + \lambda C + K, with matrix coefficients M, \, C, K \in \mathbb^ and we require that M\,\neq 0, (so that we have a nonzero leading coefficient). There are 2n eigenvalues that may be ''infinite'' or finite, and possibly zero. This is a special case of a nonlinear eigenproblem. Q(\lambda) is also known as a quadratic polynomial matrix. Spectral theory A QEP is said to be regular if \text (Q(\lambda)) \not \equiv 0 identically. The coefficient of the \lambda^ term in \text(Q(\lambda)) is \text(M), implying that the QEP is regular if M is nonsingular. Eigenvalues at infinity and eigenvalues at 0 may be exchanged by considering the reversed polynomial, \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIAM Journal On Matrix Analysis And Applications
The ''SIAM Journal on Matrix Analysis and Applications'' is a peer-reviewed scientific journal covering matrix analysis and its applications. The relevant applications include signal processing, systems and control theory, statistics, Markov chains, mathematical biology, graph theory, and data science. The journal was originally established as the ''SIAM Journal on Algebraic and Discrete Methods'' in 1980, until it split into '' SIAM Journal on Discrete Mathematics'' and the current title in 1988. The journal is published by the Society for Industrial and Applied Mathematics. The founding editor-in-chief An editor-in-chief (EIC), also known as lead editor or chief editor, is a publication's editorial leader who has final responsibility for its operations and policies. The editor-in-chief heads all departments of the organization and is held accoun ... was Gene H. Golub, who established the journal in 1980. The current editor is Michele Benzi ( Scuola Normale Superiore). Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Computational And Applied Mathematics
The ''Journal of Computational and Applied Mathematics'' is a peer-reviewed scientific journal covering computational and applied mathematics. It was established in 1975 and is published biweekly by Elsevier. The editors-in-chief are Yalchin Efendiev (Texas A&M University), Taketomo Mitsui (Nagoya University), Michael Kwok-Po Ng (Hong Kong Baptist University) and Fatih Tank (Ankara University). According to the ''Journal Citation Reports'', the journal has a 2021 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... of 2.872. References External links * Biweekly journals Applied mathematics Applied mathematics journals Elsevier academic journals Academic journals established in 1975 English-language journals Computational mathematics {{Mathematics-journal-s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gene H
In biology, the word gene has two meanings. The Mendelian gene is a basic unit of heredity. The molecular gene is a sequence of nucleotides in DNA that is transcribed to produce a functional RNA. There are two types of molecular genes: protein-coding genes and non-coding genes. During gene expression (the synthesis of RNA or protein from a gene), DNA is first copied into RNA. RNA can be directly functional or be the intermediate template for the synthesis of a protein. The transmission of genes to an organism's offspring, is the basis of the inheritance of phenotypic traits from one generation to the next. These genes make up different DNA sequences, together called a genotype, that is specific to every given individual, within the gene pool of the population of a given species. The genotype, along with environmental and developmental factors, ultimately determines the phenotype of the individual. Most biological traits occur under the combined influence of polygenes (a se ... [...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] |
Françoise Tisseur
Françoise Tisseur is a numerical analyst and Professor of Numerical Analysis at the Department of Mathematics, University of Manchester, UK. She works in numerical linear algebra and in particular on nonlinear eigenvalue problems and structured matrix problems, including the development of algorithms and software. She is a graduate of the University of St-Etienne, France, from where she gained her Maitrise (Mathematical Engineering) in 1993, Diplome d'Etude Approfondie in 1994, and PhD (Numerical Analysis) in 1997. She has contributed software to LAPACK, ScaLAPACK, and the MATLAB distribution. Tisseur is a member of the editorial boards of the SIAM Journal on Matrix Analysis and Applications, the IMA Journal of Numerical Analysis and the Electronic Journal of Linear Algebra. Awards and honours Tisseur was awarded the 2010 Whitehead Prize by the London Mathematical Society for her research achievements in numerical linear algebra, including polynomial eigenvalue and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermitian Matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and : A \text \quad \iff \quad a_ = \overline or in matrix form: A \text \quad \iff \quad A = \overline . Hermitian matrices can be understood as the complex extension of real symmetric matrices. If the conjugate transpose of a matrix A is denoted by A^\mathsf, then the Hermitian property can be written concisely as A \text \quad \iff \quad A = A^\mathsf Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are A^\mathsf = A^\dagger = A^\ast, although in quantum mechanics, A^\ast typically means the complex conjugate onl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Julia (programming Language)
Julia is a high-level programming language, high-level, general-purpose programming language, general-purpose dynamic programming language, dynamic programming language, designed to be fast and productive, for e.g. data science, artificial intelligence, machine learning, modeling and simulation, most commonly used for numerical analysis and computational science. Distinctive aspects of Julia's design include a type system with parametric polymorphism and the use of multiple dispatch as a core programming paradigm, a default just-in-time compilation, just-in-time (JIT) compiler (with support for ahead-of-time compilation) and an tracing garbage collection, efficient (multi-threaded) garbage collection implementation. Notably Julia does not support classes with encapsulated methods and instead it relies on structs with generic methods/functions not tied to them. By default, Julia is run similarly to scripting languages, using its runtime, and allows for read–eval–print loop, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages. Although MATLAB is intended primarily for numeric computing, an optional toolbox uses the MuPAD symbolic engine allowing access to symbolic computing abilities. An additional package, Simulink, adds graphical multi-domain simulation and model-based design for dynamic and embedded systems. , MATLAB has more than four million users worldwide. They come from various backgrounds of engineering, science, and economics. , more than 5000 global colleges and universities use MATLAB to support instruction and research. History Origins MATLAB was invented by mathematician and computer programmer Cleve Moler. The idea for MATLAB was base ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SLEPc
SLEPc is a software library for the parallel computation of eigenvalues and eigenvectors of large, sparse matrices. It can be seen as a module of PETSc that provides solvers for different types of eigenproblems, including linear (standard and generalized) and nonlinear ( quadratic, polynomial and general), as well as the SVD. Recent versions also include support for matrix functions. It uses the MPI standard for parallelization. Both real and complex arithmetic are supported, with single, double and quadruple precision. When using SLEPc, the application programmer can use any of the PETSc's data structures and solvers. Other PETSc features are incorporated into SLEPc as well, such as command-line option setting, automatic profiling, error checking, portability to virtually all computing platforms, etc. Components EPS provides iterative algorithms for linear eigenvalue problems. * Krylov methods such as Krylov-Schur, Arnoldi and Lanczos. * Davidson methods such as Generalized ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero Of A Function
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 a 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=(x-2)(x-3) 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 n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
