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Kreiss Matrix Theorem
In matrix analysis, Kreiss matrix theorem relates the so-called Kreiss constant of a matrix with the power iterates of this matrix. It was originally introduced by Heinz-Otto Kreiss to analyze the stability of finite difference methods for partial difference equations. Kreiss constant of a matrix Given a matrix ''A'', the Kreiss constant 𝒦(''A'') (with respect to the closed unit circle) of ''A'' is defined as \mathcal(\mathbf)=\sup _(, z, -1)\left\, (z-\mathbf)^\right\, , while the Kreiss constant 𝒦(''A'') with respect to the left-half plane is given by \mathcal_(\mathbf)=\sup _(\Re(z))\left\, (z-\mathbf)^\right\, . Properties * For any matrix ''A'', one has that 𝒦(''A'') ≥ 1 and 𝒦(''A'') ≥ 1. In particular, 𝒦(''A'') (resp. 𝒦(''A'')) are finite only if the matrix ''A'' is Schur stable (resp. Hurwitz stable). * Kreiss constant can be interpreted as a measure of normality of a matrix. In particular, for normal matrices ''A'' with spectral radius les ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Analysis
In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. Some particular topics out of many include; operations defined on matrices (such as matrix addition, matrix multiplication and operations derived from these), functions of matrices (such as matrix exponentiation and matrix logarithm, and even sines and cosines etc. of matrices), and the eigenvalues of matrices (eigendecomposition of a matrix, eigenvalue perturbation theory). Matrix spaces The set of all ''m'' × ''n'' matrices over a field ''F'' denoted in this article ''M''''mn''(''F'') form a vector space. Examples of ''F'' include the set of rational numbers \mathbb, the real numbers \mathbb, and set of complex numbers \mathbb. The spaces ''M''''mn''(''F'') and ''M''''pq''(''F'') are different spaces if ''m'' and ''p'' are unequal, and if ''n'' and ''q'' are unequal; for instance ''M''32(''F'') ≠ ''M''23(''F''). Two ''m''&th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heinz-Otto Kreiss
Heinz-Otto Kreiss (14 September 1930 – 16 December 2015) was a German mathematician in the fields of numerical analysis, applied mathematics, and what was the new area of computing in the early 1960s. Born in Hamburg, Germany, he earned his Ph.D. at Kungliga Tekniska Högskolan in 1959. Over the course of his long career, Kreiss wrote a number of books in addition to the purely academic journal articles he authored across several disciplines. He was professor at Uppsala University, California Institute of Technology and University of California, Los Angeles (UCLA). He was also a member of the Royal Swedish Academy of Sciences. At the time of his death, Kreiss was a Swedish citizen, living in Stockholm. He died in Stockholm in 2015, aged 85. Kreiss did research on the initial value problem for partial differential equations, numerical treatment of partial differential equations, difference equations, and applications to hydrodynamics and meteorology. In 1974, he delivered a pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Difference Methods
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time domain (if applicable) are discretized, or broken into a finite number of intervals, and the values of the solution at the end points of the intervals are approximated by solving algebraic equations containing finite differences and values from nearby points. Finite difference methods convert ordinary differential equations (ODE) or partial differential equations (PDE), which may be nonlinear, into a system of linear equations that can be solved by matrix algebra techniques. Modern computers can perform these linear algebra computations efficiently, and this, along with their relative ease of implementation, has led to the widespread use of FDM in modern numerical analysis. Today, FDMs are one of the most common approaches to the numerical solution of PDE, along with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Difference Equation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Polynomial
In the context of the characteristic polynomial of a differential equation or difference equation, a polynomial is said to be stable if either: * all its roots lie in the open left half-plane, or * all its roots lie in the open unit disk. The first condition provides stability for continuous-time linear systems, and the second case relates to stability of discrete-time linear systems. A polynomial with the first property is called at times a Hurwitz-stable polynomial and with the second property a Schur-stable polynomial. Stable polynomials arise in control theory and in mathematical theory of differential and difference equations. A linear, time-invariant system (see LTI system theory) is said to be BIBO stable if every bounded input produces bounded output. A linear system is BIBO stable if its characteristic polynomial is stable. The denominator is required to be Hurwitz stable if the system is in continuous-time and Schur stable if it is in discrete-time. In practic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hurwitz-stable Matrix
In mathematics, a Hurwitz-stable matrix, or more commonly simply Hurwitz matrix, is a square matrix whose eigenvalues all have strictly negative real part. Some authors also use the term stability matrix. Such matrices play an important role in control theory. Definition A square matrix A is called a Hurwitz matrix if every eigenvalue of A has strictly negative real part, that is, :\operatorname lambda_i< 0\, for each eigenvalue . is also called a stable matrix, because then the differential equation : is , that is, as If is a (matrix-valued) |
Normal Matrix
In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose : :A \text \iff A^*A = AA^* . The concept of normal matrices can be extended to normal operators on infinite-dimensional normed spaces and to normal elements in C*-algebras. As in the matrix case, normality means commutativity is preserved, to the extent possible, in the noncommutative setting. This makes normal operators, and normal elements of C*-algebras, more amenable to analysis. The spectral theorem states that a matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix satisfying the equation is diagonalizable. (The converse does not hold because diagonalizable matrices may have non-orthogonal eigenspaces.) Thus A = U D U^* and A^* = U D^* U^*where D is a diagonal matrix whose diagonal values are in general complex. The left and right singular vectors in the singular value decomposition of a normal matrix A = U D V^* dif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Radius
''Spectral'' is a 2016 Hungarian-American military science fiction action film co-written and directed by Nic Mathieu. Written with Ian Fried (screenwriter), Ian Fried & George Nolfi, the film stars James Badge Dale as DARPA research scientist Mark Clyne, with Max Martini, Emily Mortimer, Clayne Crawford, and Bruce Greenwood in supporting roles. The film is set in a civil war-ridden Moldova as invisible entities slaughter any living being caught in their path. The film was released worldwide on December 9, 2016 on Netflix. On February 1, 2017, Netflix released a prequel graphic novel of the film called ''Spectral: Ghosts of War'' which was made available digitally through the website ComiXology. Plot DARPA researcher Mark Clyne is sent to a United States, US United States Armed Forces, military Air base, airbase on the outskirts of Chișinău, to consult his created line of hyperspectral imaging goggles issued to United States Army, US Army United States Army Special Forces, S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudospectrum
In mathematics, the pseudospectrum of an operator is a set containing the spectrum of the operator and the numbers that are "almost" eigenvalues. Knowledge of the pseudospectrum can be particularly useful for understanding non-normal operators and their eigenfunctions. The ε-pseudospectrum of a matrix ''A'' consists of all eigenvalues of matrices which are ε-close to ''A'': :\Lambda_\epsilon(A) = \. Numerical algorithms which calculate the eigenvalues of a matrix give only approximate results due to rounding and other errors. These errors can be described with the matrix ''E''. More generally, for Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...s X,Y and operators A: X \to Y , one can define the \epsilon-pseudospectrum of A (typically denoted b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robust Control
In control theory, robust control is an approach to controller design that explicitly deals with uncertainty. Robust control methods are designed to function properly provided that uncertain parameters or disturbances are found within some (typically compact) set. Robust methods aim to achieve robust performance and/or stability in the presence of bounded modelling errors. The early methods of Bode and others were fairly robust; the state-space methods invented in the 1960s and 1970s were sometimes found to lack robustness, prompting research to improve them. This was the start of the theory of robust control, which took shape in the 1980s and 1990s and is still active today. In contrast with an adaptive control policy, a robust control policy is static, rather than adapting to measurements of variations, the controller is designed to work assuming that certain variables will be unknown but bounded. (Section 1.5) In German; an English version is also available Criteria for robust ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E (mathematical Constant)
The number is a mathematical constant approximately equal to 2.71828 that is the base of a logarithm, base of the natural logarithm and exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted \gamma. Alternatively, can be called Napier's constant after John Napier. The Swiss mathematician Jacob Bernoulli discovered the constant while studying compound interest. The number is of great importance in mathematics, alongside 0, 1, Pi, , and . All five appear in one formulation of Euler's identity e^+1=0 and play important and recurring roles across mathematics. Like the constant , is Irrational number, irrational, meaning that it cannot be represented as a ratio of integers, and moreover it is Transcendental number, transcendental, meaning that it is not a root of any non-zero polynomial with rational coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spijker's Lemma
In mathematics, Spijker's lemma is a result in the theory of rational mappings of the Riemann sphere. It states that the image of a circle under a complex rational map with numerator and denominator having degree at most ''n'' has length at most 2''nπ''. Applications Spijker's lemma can be used to derive a sharp bound version of Kreiss matrix theorem In matrix analysis, Kreiss matrix theorem relates the so-called Kreiss constant of a matrix with the power iterates of this matrix. It was originally introduced by Heinz-Otto Kreiss to analyze the stability of finite difference methods for partia .... See also * Buffon's needle External links * References *{{cite journal, last = Wegert, first = Elias, author2=Trefethen, Lloyd N. , title = From the Buffon Needle Problem to the Kreiss Matrix Theorem, journal = The American Mathematical Monthly, volume = 101, issue = 2, pages = 132–139, date=February 1994, doi = 10.2307/2324361, jstor=2324361, url = https://ecommons.cornell. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
