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Alternating Direction Implicit Method
In numerical linear algebra, the alternating-direction implicit (ADI) method is an iterative method used to solve Sylvester matrix equations. It is a popular method for solving the large matrix equations that arise in systems theory and control, and can be formulated to construct solutions in a memory-efficient, factored form. It is also used to numerically solve parabolic and elliptic partial differential equations, and is a classic method used for modeling heat conduction and solving the diffusion equation in two or more dimensions.. It is an example of an operator splitting method. ADI for matrix equations The method The ADI method is a two step iteration process that alternately updates the column and row spaces of an approximate solution to AX - XB = C. One ADI iteration consists of the following steps:1. Solve for X^, where \left( A - \beta_ I\right) X^ = X^\left( B - \beta_ I \right) + C. 2. Solve for X^, where X^\left( B - \alpha_ I \right) = \left( A - \alpha_ I\r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Linear Algebra
Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematics. It is a subfield of numerical analysis, and a type of linear algebra. Computers use floating-point arithmetic and cannot exactly represent irrational data, so when a computer algorithm is applied to a matrix of data, it can sometimes increase the difference between a number stored in the computer and the true number that it is an approximation of. Numerical linear algebra uses properties of vectors and matrices to develop computer algorithms that minimize the error introduced by the computer, and is also concerned with ensuring that the algorithm is as efficient as possible. Numerical linear algebra aims to solve problems of continuous mathematics using finite precision computers, so its applications to the natural and social sciences ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Equation
In control theory, the discrete Lyapunov equation is of the form :A X A^ - X + Q = 0 where Q is a Hermitian matrix and A^H is the conjugate transpose of A. The continuous Lyapunov equation is of the form :AX + XA^H + Q = 0. The Lyapunov equation occurs in many branches of control theory, such as stability analysis and optimal control. This and related equations are named after the Russian mathematician Aleksandr Lyapunov. Application to stability In the following theorems A, P, Q \in \mathbb^, and P and Q are symmetric. The notation P>0 means that the matrix P is positive definite. Theorem (continuous time version). Given any Q>0, there exists a unique P>0 satisfying A^T P + P A + Q = 0 if and only if the linear system \dot=A x is globally asymptotically stable. The quadratic function V(x)=x^T P x is a Lyapunov function that can be used to verify stability. Theorem (discrete time version). Given any Q>0, there exists a unique P>0 satisfying A^T P A -P + Q = 0 if and only if t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Electromagnetics
Computational electromagnetics (CEM), computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of electromagnetic fields with physical objects and the environment. It typically involves using computer programs to compute approximate solutions to Maxwell's equations to calculate antenna performance, electromagnetic compatibility, radar cross section and electromagnetic wave propagation when not in free space. A large subfield is ''antenna modeling'' computer programs, which calculate the radiation pattern and electrical properties of radio antennas, and are widely used to design antennas for specific applications. Background Several real-world electromagnetic problems like electromagnetic scattering, electromagnetic radiation, modeling of waveguides etc., are not analytically calculable, for the multitude of irregular geometries found in actual devices. Computational numerical techniques can overcome the inability to derive closed f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if a_ denotes the entry in the ith row and jth column then for all indices i and j. Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. In linear algebra, a real symmetric matrix represents a self-adjoint operator represented in an orthonormal basis over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incomplete Cholesky Factorization
In numerical analysis, an incomplete Cholesky factorization of a symmetric positive definite matrix is a sparse approximation of the Cholesky factorization. An incomplete Cholesky factorization is often used as a preconditioner for algorithms like the conjugate gradient method. The Cholesky factorization of a positive definite matrix ''A'' is ''A'' = ''LL''* where ''L'' is a lower triangular matrix In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are .... An incomplete Cholesky factorization is given by a sparse lower triangular matrix ''K'' that is in some sense close to ''L''. The corresponding preconditioner is ''KK''*. One popular way to find such a matrix ''K'' is to use the algorithm for finding the exact Cholesky decomposition in which ''K'' has the same sparsity pattern as ''A'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Gradient Method
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite. The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that are too large to be handled by a direct implementation or other direct methods such as the Cholesky decomposition. Large sparse systems often arise when numerically solving partial differential equations or optimization problems. The conjugate gradient method can also be used to solve unconstrained optimization problems such as energy minimization. It is commonly attributed to Magnus Hestenes and Eduard Stiefel, who programmed it on the Z4, and extensively researched it. The biconjugate gradient method provides a generalization to non-symmetric matrices. Various nonlinear conjugate gradient methods seek minima of nonlinear optimization problems. Description of the problem addressed by co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System Of Linear Equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variable (math), variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in the three variables . A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A Equation solving, solution to the system above is given by the Tuple, ordered triple :(x,y,z)=(1,-2,-2), since it makes all three equations valid. The word "system" indicates that the equations are to be considered collectively, rather than individually. In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Band Matrix
Band or BAND may refer to: Places *Bánd, a village in Hungary *Band, Iran, a village in Urmia County, West Azerbaijan Province, Iran *Band, Mureș, a commune in Romania * Band-e Majid Khan, a village in Bukan County, West Azerbaijan Province, Iran People *Band (surname), various people with the surname Arts, entertainment, and media Music *Musical ensemble, a group of people who perform instrumental or vocal music **Band (rock and pop), a small ensemble that plays rock or pop **Concert band, an ensemble of woodwind, brass, and percussion instruments **Dansband, band playing popular music for a partner-dancing audience **Jazz band, a musical ensemble that plays jazz music **Marching band, a group of instrumental musicians who generally perform outdoors **School band, a group of student musicians who rehearse and perform instrumental music * The Band, a Canadian-American rock and roll group ** ''The Band'' (album), The Band's eponymous 1969 album * "Bands" (song), by American rap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann Stability Analysis
The term ''von'' () is used in German language surnames either as a nobiliary particle indicating a noble patrilineality, or as a simple preposition used by commoners that means ''of'' or ''from''. Nobility directories like the ''Almanach de Gotha'' often abbreviate the noble term ''von'' to ''v.'' In medieval or early modern names, the ''von'' particle was at times added to commoners' names; thus, ''Hans von Duisburg'' meant "Hans from he city ofDuisburg". This meaning is preserved in Swiss toponymic surnames and in the Dutch or Afrikaans '' van'', which is a cognate of ''von'' but does not indicate nobility. Usage Germany and Austria The abolition of the monarchies in Germany and Austria in 1919 meant that neither state has a privileged nobility, and both have exclusively republican governments. In Germany, this means that legally ''von'' simply became an ordinary part of the surnames of the people who used it. There are no longer any legal privileges or constraints asso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tridiagonal Matrix Algorithm
In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations. A tridiagonal system for ''n'' unknowns may be written as :a_i x_ + b_i x_i + c_i x_ = d_i, where a_1 = 0 and c_n = 0. : \begin b_1 & c_1 & & & 0 \\ a_2 & b_2 & c_2 & & \\ & a_3 & b_3 & \ddots & \\ & & \ddots & \ddots & c_ \\ 0 & & & a_n & b_n \end \begin x_1 \\ x_2 \\ x_3 \\ \vdots \\ x_n \end = \begin d_1 \\ d_2 \\ d_3 \\ \vdots \\ d_n \end . For such systems, the solution can be obtained in O(n) operations instead of O(n^3) required by Gaussian elimination. A first sweep eliminates the a_i's, and then an (abbreviated) backward substitution produces the solution. Examples of such matrices commonly arise fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crank–Nicolson Method
In numerical analysis, the Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations. It is a second-order method in time. It is implicit in time, can be written as an implicit Runge–Kutta method, and it is numerically stable. The method was developed by John Crank and Phyllis Nicolson in the mid 20th century. For diffusion equations (and many other equations), it can be shown the Crank–Nicolson method is unconditionally stable. However, the approximate solutions can still contain (decaying) spurious oscillations if the ratio of time step \Delta t times the thermal diffusivity to the square of space step, \Delta x^2, is large (typically, larger than 1/2 per Von Neumann stability analysis). For this reason, whenever large time steps or high spatial resolution is necessary, the less accurate backward Euler method is often used, which is both stable and immune to oscillations. The method ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
