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Polynomial Eigenvalue Problem
In mathematics, a nonlinear eigenproblem, sometimes nonlinear eigenvalue problem, is a generalization of the (ordinary) 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, and ''M'' is a matrix-valued 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 ''right eigevector'' if M (\lambda) x = 0. Moreover, a nonzero vector y \in \Complex^n is called a ''left ei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 A_2 + \lambda A_1 + A_0, with matrix coefficients A_2, \, A_1, A_0 \in \mathbb^ and we require that A_2\,\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. Applications A QEP can result in part of the dynamic analysis of structures discretized by the finite element method. In this case the quadratic, Q(\lambda) has the form Q(\lambda)=\lambda^2 M + \lambda C + K, where M is the mass matrix, C is the damping matrix and K is the stiffness matrix. Other applicat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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SIAM Journal On Matrix Analysis And Applications
The ''SIAM Journal on Matrix Analysis and Applications'' (until 1989: ''SIAM Journal on Algebraic and Discrete Methods'') 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 is published by the Society for Industrial and Applied Mathematics. The founding editor-in-chief was Gene H. Golub, who established the journal in 1980. The current editor is Michele Benzi ( Scuola Normale Superiore). See also *Michele Benzi Michele Benzi (born 1962 in Bologna) is an Italian mathematician who works as a full professor in the Scuola Normale Superiore in Pisa. He is known for his contributions to numerical linear algebra and its applications, especially to the solut ... External links * Mathematics journals Publications established in 1980 English-language journals Quarterly jo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... of 2.872. References External links * {{Mathematics-journal-stub Biweekly journals Applied mathematics Mathematics journals Elsevier academic journals Publications established in 1975 English-language journals Computational mathematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Gene H
In biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditar ..., the word gene (from , ; "...Wilhelm Johannsen coined the word gene to describe the Mendelian inheritance#History, Mendelian units of heredity..." meaning ''generation'' or ''birth'' or ''gender'') can have several different meanings. The Mendelian gene is a basic unit of heredity and 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 noncoding genes. During gene expression, the DNA is first Transcription (biology), copied into RNA. The RNA can be Non-coding RNA, directly functional or be the intermediate Protein biosynthesis, template for a protein that performs a function. The transmission of genes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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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 s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 : 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 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 note that in quantum mechanics, A^\ast typically means the complex conjugate only, and not the conjugate transpose. Alternative characterizations Her ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Julia (programming Language)
Julia is a high-level, dynamic programming language. Its features are well suited for numerical analysis and computational science. Distinctive aspects of Julia's design include a type system with parametric polymorphism in a dynamic programming language; with multiple dispatch as its core programming paradigm. Julia supports concurrent, (composable) parallel and distributed computing (with or without using MPI or the built-in corresponding to "OpenMP-style" threads), and direct calling of C and Fortran libraries without glue code. Julia uses a just-in-time (JIT) compiler that is referred to as "just- ahead-of-time" (JAOT) in the Julia community, as Julia compiles all code (by default) to machine code before running it. Julia is garbage-collected, uses eager evaluation, and includes efficient libraries for floating-point calculations, linear algebra, random number generation, and regular expression matching. Many libraries are available, including some (e.g., for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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. As of 2020, MATLAB has more than 4 million users worldwide. They come from various backgrounds of engineering, science, and economics. History Origins MATLAB was invented by mathematician and computer programmer Cleve Moler. The idea for MATLAB was based on his 1960s PhD thesis. Moler became a math professor at the University of New Mexico an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 Generali ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |