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Euler's 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] |
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] |
Global 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] |
Global 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] |
Taylor's Theorem
In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order ''k'' of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial. Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1715, although an earlier version of the result was already mentioned in 1671 by James Gregory. Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathematical analysis. It gives simple arithmetic formula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivation
Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proceeding in United States patent law Music * The creation of a derived row, in the twelve-tone musical technique Science and mathematics * Derivation (differential algebra), a unary function satisfying the Leibniz product law * Formal proof or derivation, a sequence of sentences each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference * An after-the-fact justification for an action, in the work of sociologist Vilfredo Pareto See also *Derive (other), for meanings of "derive" and "derived" *Derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument ( ... [...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] |
Rectangle Method
In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lines on a graph, but also the length of curves and other approximations. The sum is calculated by partitioning the region into shapes (rectangles, trapezoids, parabolas, or cubics) that together form a region that is similar to the region being measured, then calculating the area for each of these shapes, and finally adding all of these small areas together. This approach can be used to find a numerical approximation for a definite integral even if the fundamental theorem of calculus does not make it easy to find a closed-form solution. Because the region by the small shapes is usually not exactly the same shape as the region being measured, the Riemann sum will differ from the area being measured. This error can be reduced by dividing u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). The two operations are inverses of each other apart from a constant value which depends on where one starts to compute area. The first part of the theorem, the first fundamental theorem of calculus, states that for a function , an antiderivative or indefinite integral may be obtained as the integral of over an interval with a variable upper bound. This implies the existence of antiderivatives for continuous functions. Conversely, the second part of the theorem, the second fundamental theorem of calculus, states that the integral of a function over a fixed interval is equal to the change of any antiderivative between the ends of the interval. This greatly simplifies the calculation of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Backward Euler Method
In numerical analysis and scientific computing, the backward Euler method (or implicit Euler method) is one of the most basic numerical methods for the solution of ordinary differential equations. It is similar to the (standard) Euler method, but differs in that it is an implicit method. The backward Euler method has error of order one in time. Description Consider the ordinary differential equation : \frac = f(t,y) with initial value y(t_0) = y_0. Here the function f and the initial data t_0 and y_0 are known; the function y depends on the real variable t and is unknown. A numerical method produces a sequence y_0, y_1, y_2, \ldots such that y_k approximates y(t_0+kh) , where h is called the step size. The backward Euler method computes the approximations using : y_ = y_k + h f(t_, y_). This differs from the (forward) Euler method in that the forward method uses f(t_k, y_k) in place of f(t_, y_). The backward Euler method is an implicit method: the new approxima ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Difference
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems. The difference operator, commonly denoted \Delta is the operator that maps a function to the function \Delta /math> defined by :\Delta x)= f(x+1)-f(x). A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives. There are many similarities between difference equations and differential equations, specially in the solving methods. Certain recurrence relations can be written as difference equations by replacing iteration notation with finite differences. In numerical analysis, finite differences are widely used for approximating derivatives, and the term " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Runge–Kutta Methods
In numerical analysis, the Runge–Kutta methods ( ) are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. These methods were developed around 1900 by the German mathematicians Carl Runge and Wilhelm Kutta. The Runge–Kutta method The most widely known member of the Runge–Kutta family is generally referred to as "RK4", the "classic Runge–Kutta method" or simply as "the Runge–Kutta method". Let an initial value problem be specified as follows: : \frac = f(t, y), \quad y(t_0) = y_0. Here y is an unknown function (scalar or vector) of time t, which we would like to approximate; we are told that \frac, the rate at which y changes, is a function of t and of y itself. At the initial time t_0 the corresponding y value is y_0. The function f and the initial conditions t_0, y_0 are given. Now we pick a step-size ''h'' > 0 and define: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
