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Local Linearization Method
In numerical analysis, the local linearization (LL) method is a general strategy for designing Numerical integration, numerical integrators for differential equations based on a local (piecewise) linearization of the given equation on consecutive time intervals. The numerical integrators are then iteratively defined as the solution of the resulting piecewise linear equation at the end of each consecutive interval. The LL method has been developed for a variety of equations such as the Ordinary differential equation, ordinary, Delay differential equation, delayed, random and Stochastic differential equation, stochastic differential equations. The LL integrators are key component in the implementation of Estimation theory, inference methods for the estimation of unknown parameters and unobserved variables of differential equations given time series of (potentially noisy) observations. The LL schemes are ideals to deal with complex models in a variety of fields as neuroscience, finance, ... [...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] |
Rate Of Convergence
In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. A sequence (x_n) that converges to x^* is said to have ''order of convergence'' q \geq 1 and ''rate of convergence'' \mu if : \lim _ \frac=\mu. The rate of convergence \mu is also called the ''asymptotic error constant''. Note that this terminology is not standardized and some authors will use ''rate'' where this article uses ''order'' (e.g., ). In practice, the rate and order of convergence provide useful insights when using iterative methods for calculating numerical approximations. If the order of convergence is higher, then typically fewer iterations are necessary to yield a useful approximation. Strictly speaking, however, the asymptotic behavior of a sequence does not give conclusive information about any finite part of the sequence. Similar concepts are used for discretization methods. The solutio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Geometry
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed differential form, closed, nondegenerate form, nondegenerate differential form, 2-form. Symplectic geometry has its origins in the Hamiltonian mechanics, Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold. The term "symplectic", introduced by Weyl, is a calque of "complex"; previously, the "symplectic group" had been called the "line complex group". "Complex" comes from the Latin ''com-plexus'', meaning "braided together" (co- + plexus), while symplectic comes from the corresponding Greek ''sym-plektikos'' (συμπλεκτικός); in both cases the stem comes from the Indo-European root wiktionary:Reconstruction:Proto-Indo-European/pleḱ-, *pleḱ- The name reflects the deep connections between complex and sym ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floating-point Arithmetic
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be represented as a base-ten floating-point number: 12.345 = \underbrace_\text \times \underbrace_\text\!\!\!\!\!\!^ In practice, most floating-point systems use base two, though base ten (decimal floating point) is also common. The term ''floating point'' refers to the fact that the number's radix point can "float" anywhere to the left, right, or between the significant digits of the number. This position is indicated by the exponent, so floating point can be considered a form of scientific notation. A floating-point system can be used to represent, with a fixed number of digits, numbers of very different orders of magnitude — such as the number of meters between galaxies or between protons in an atom. For this reason, floating-poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L-stability
Within mathematics regarding differential equations, L-stability is a special case of A-stability, a property of Runge–Kutta methods for solving ordinary differential equations. A method is L-stable if it is A-stable and \phi(z) \to 0 as z \to \infty , where \phi is the stability function of the method (the stability function of a Runge–Kutta method is a rational function and thus the limit as z \to +\infty is the same as the limit as z \to -\infty). L-stable methods are in general very good at integrating stiff equation In mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. It has proven difficult to formulate a precise ...s. References * . Numerical differential equations {{mathapplied-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stiff Equation
In mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. It has proven difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes some terms that can lead to rapid variation in the solution. When integrating a differential equation numerically, one would expect the requisite step size to be relatively small in a region where the solution curve displays much variation and to be relatively large where the solution curve straightens out to approach a line with slope nearly zero. For some problems this is not the case. In order for a numerical method to give a reliable solution to the differential system sometimes the step size is required to be at an unacceptably small level in a region where the solution curve is very smooth. The phenomenon is known as ''stiffness''. In some cases there may be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Figure ODE
Figure may refer to: General *A shape, drawing, depiction, or geometric configuration *Figure (wood), wood appearance *Figure (music), distinguished from musical motif *Noise figure, in telecommunication *Dance figure, an elementary dance pattern *A person's figure, human physical appearance Arts *Figurine, a miniature statuette representation of a creature *Action figure, a posable jointed solid plastic character figurine *Figure painting, realistic representation, especially of the human form *Figure drawing *Model figure, a scale model of a creature Writing *figure, in writing, a type of floating block (text, table, or graphic separate from the main text) *Figure of speech, also called a rhetorical figure *Christ figure, a type of character * in typesetting, text figures and lining figures Accounting *Figure, a synonym for number *Significant figures in a decimal number Science *Figure of the Earth, the size and shape of the Earth in geodesy Sports *Figure (horse), a sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dormand–Prince Method
In numerical analysis, the Dormand–Prince (RKDP) method or DOPRI method, is an embedded method for solving ordinary differential equations . The method is a member of the Runge–Kutta family of ODE solvers. More specifically, it uses six function evaluations to calculate fourth- and fifth-order accurate solutions. The difference between these solutions is then taken to be the error of the (fourth-order) solution. This error estimate is very convenient for adaptive stepsize integration algorithms. Other similar integration methods are Fehlberg (RKF) and Cash–Karp (RKCK). The Dormand–Prince method has seven stages, but it uses only six function evaluations per step because it has the FSAL (First Same As Last) property: the last stage is evaluated at the same point as the first stage of the next step. Dormand and Prince chose the coefficients of their method to minimize the error of the fifth-order solution. This is the main difference with the Fehlberg method, which was con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Autonomous System (mathematics)
In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable is time, they are also called time-invariant systems. Many laws in physics, where the independent variable is usually assumed to be time, are expressed as autonomous systems because it is assumed the laws of nature which hold now are identical to those for any point in the past or future. Definition An autonomous system is a system of ordinary differential equations of the form \fracx(t)=f(x(t)) where takes values in -dimensional Euclidean space; is often interpreted as time. It is distinguished from systems of differential equations of the form \fracx(t)=g(x(t),t) in which the law governing the evolution of the system does not depend solely on the system's current state but also the parameter , again often interpreted as time; such systems are by definition not autonomous. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krylov Subspace
In linear algebra, the order-''r'' Krylov subspace generated by an ''n''-by-''n'' matrix ''A'' and a vector ''b'' of dimension ''n'' is the linear subspace spanned by the images of ''b'' under the first ''r'' powers of ''A'' (starting from A^0=I), that is, :\mathcal_r(A,b) = \operatorname \, \. Background The concept is named after Russian applied mathematician and naval engineer Alexei Krylov, who published a paper about it in 1931. Properties * \mathcal_r(A,b),A\mathcal_r(A,b)\subset \mathcal_(A,b). * Vectors \ are linearly independent until r, where p(A) is the minimal polynomial of A. Furthermore, there exists a b such that r_0 = \deg (A)/math>. * \mathcal_r(A,b) is a cyclic submodule generated by b of the torsion k /math>-module (k^n)^A, where k^n is the linear space on k. * k^n can be decomposed as the direct sum of Krylov subspaces. Use Krylov subspaces are used in algorithms for finding approximate solutions to high-dimensional linear algebra problems. Many linear dyn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Padé Approximant
In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is approximating. The technique was developed around 1890 by Henri Padé, but goes back to Georg Frobenius, who introduced the idea and investigated the features of rational approximations of power series. The Padé approximant often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge. For these reasons Padé approximants are used extensively in computer calculations. They have also been used as auxiliary functions in Diophantine approximation and transcendental number theory, though for sharp results ad hoc methods— in some sense inspired by the Padé theory— typically replace them. Since Padé approximant is a rational function, an artificial singul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite Polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well as in connection with Brownian motion; * combinatorics, as an example of an Appell sequence, obeying the umbral calculus; * numerical analysis as Gaussian quadrature; * physics, where they give rise to the eigenstates of the quantum harmonic oscillator; and they also occur in some cases of the heat equation (when the term \beginxu_\end is present); * systems theory in connection with nonlinear operations on Gaussian noise. * random matrix theory in Gaussian ensembles. Hermite polynomials were defined by Pierre-Simon Laplace in 1810, though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in 1864, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
