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Trapezoidal Rule (differential Equations)
In numerical analysis and scientific computing, the trapezoidal rule is a numerical method to solve ordinary differential equations derived from the trapezoidal rule for computing integrals. The trapezoidal rule is an implicit second-order method, which can be considered as both a Runge–Kutta method and a linear multistep method. Method Suppose that we want to solve the differential equation : y' = f(t,y). The trapezoidal rule is given by the formula : y_ = y_n + \tfrac12 h \Big( f(t_n,y_n) + f(t_,y_) \Big), where h = t_ - t_n is the step size. This is an implicit method: the value y_ appears on both sides of the equation, and to actually calculate it, we have to solve an equation which will usually be nonlinear. One possible method for solving this equation is Newton's method. We can use the Euler method to get a fairly good estimate for the solution, which can be used as the initial guess of Newton's method. Cutting short, using only the guess from Eulers method is e ... [...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] |
Euler Method
In mathematics and computational science, the Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who treated it in his book ''Institutionum calculi integralis'' (published 1768–1870). The Euler method is a first-order method, which means that the local error (error per step) is proportional to the square of the step size, and the global error (error at a given time) is proportional to the step size. The Euler method often serves as the basis to construct more complex methods, e.g., predictor–corrector method. Informal geometrical description Consider the problem of calculating the shape of an unknown curve which starts at a given point and satisfies a given differential equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...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] |
Stability Region For Trapezoidal Method
Stability may refer to: Mathematics *Stability theory, the study of the stability of solutions to differential equations and dynamical systems **Asymptotic stability **Linear stability **Lyapunov stability **Orbital stability **Structural stability *Stability (probability), a property of probability distributions *Stability (learning theory), a property of machine learning algorithms *Stability, a property of sorting algorithms *Numerical stability, a property of numerical algorithms which describes how errors in the input data propagate through the algorithm *Stability radius, a property of continuous polynomial functions *Stable theory, concerned with the notion of stability in model theory *Stability, a property of points in geometric invariant theory *K-Stability, a stability condition for algebraic varieties. *Bridgeland stability conditions, a class of stability conditions on elements of a triangulated category. *Stability (algebraic geometry) Engineering *In atmospheric flui ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for ''Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Truncation Error
Truncation errors in numerical integration are of two kinds: * ''local truncation errors'' – the error caused by one iteration, and * ''global truncation errors'' – the cumulative error caused by many iterations. Definitions Suppose we have a continuous differential equation : y' = f(t,y), \qquad y(t_0) = y_0, \qquad t \geq t_0 and we wish to compute an approximation y_n of the true solution y(t_n) at discrete time steps t_1,t_2,\ldots,t_N . For simplicity, assume the time steps are equally spaced: : h = t_n - t_, \qquad n=1,2,\ldots,N. Suppose we compute the sequence y_n with a one-step method of the form : y_n = y_ + h A(t_, y_, h, f). The function A is called the ''increment function'', and can be interpreted as an estimate of the slope \frac . Local truncation error The local truncation error \tau_n is the error that our increment function, A , causes during a single iteration, assuming perfect knowledge of the true solution at the previous ite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heun's Method
In mathematics and computational science, Heun's method may refer to the improved or modified Euler's method (that is, the explicit trapezoidal rule), or a similar two-stage Runge–Kutta method. It is named after Karl Heun and is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. Both variants can be seen as extensions of the Euler method into two-stage second-order Runge–Kutta methods. The procedure for calculating the numerical solution to the initial value problem: :y'(t) = f(t,y(t)), \qquad \qquad y(t_0)=y_0, by way of Heun's method, is to first calculate the intermediate value \tilde_ and then the final approximation y_ at the next integration point. :\tilde_ = y_i + h f(t_i,y_i) :y_ = y_i + \frac (t_i, y_i) + f(t_,\tilde_) : where h is the step size and t_=t_i+h. Description Euler's method is used as the foundation for Heun's method. Euler's method uses the line tangent to the function at the beginning of the interval ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton's Method
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a single-variable function defined for a real variable , the function's derivative , and an initial guess for a root of . If the function satisfies sufficient assumptions and the initial guess is close, then :x_ = x_0 - \frac is a better approximation of the root than . Geometrically, is the intersection of the -axis and the tangent of the graph of at : that is, the improved guess is the unique root of the linear approximation at the initial point. The process is repeated as :x_ = x_n - \frac until a sufficiently precise value is reached. This algorithm is first in the class of Householder's methods, succeeded by Halley's method. The method can also be extended to complex functions an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scientific Computing
Computational science, also known as scientific computing or scientific computation (SC), is a field in mathematics that uses advanced computing capabilities to understand and solve complex problems. It is an area of science that spans many disciplines, but at its core, it involves the development of models and simulations to understand natural systems. * Algorithms ( numerical and non-numerical): mathematical models, computational models, and computer simulations developed to solve science (e.g., biological, physical, and social), engineering, and humanities problems * Computer hardware that develops and optimizes the advanced system hardware, firmware, networking, and data management components needed to solve computationally demanding problems * The computing infrastructure that supports both the science and engineering problem solving and the developmental computer and information science In practical use, it is typically the application of computer simulation and other fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Multistep Method
Linear multistep methods are used for the numerical solution of ordinary differential equations. Conceptually, a numerical method starts from an initial point and then takes a short step forward in time to find the next solution point. The process continues with subsequent steps to map out the solution. Single-step methods (such as Euler's method) refer to only one previous point and its derivative to determine the current value. Methods such as Runge–Kutta take some intermediate steps (for example, a half-step) to obtain a higher order method, but then discard all previous information before taking a second step. Multistep methods attempt to gain efficiency by keeping and using the information from previous steps rather than discarding it. Consequently, multistep methods refer to several previous points and derivative values. In the case of ''linear'' multistep methods, a linear combination of the previous points and derivative values is used. Definitions Numerical methods ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
