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Journal ArticleDOI

A family of embedded Runge-Kutta formulae

01 Mar 1980-Journal of Computational and Applied Mathematics (North-Holland)-Vol. 6, Iss: 1, pp 19-26
TL;DR: In this article, a family of embedded Runge-Kutta formulae RK5 (4) are derived from these and a small principal truncation term in the fifth order and extended regions of absolute stability.
About: This article is published in Journal of Computational and Applied Mathematics.The article was published on 1980-03-01 and is currently open access. It has received 3106 citations till now.
Citations
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Journal ArticleDOI
TL;DR: This paper describes mathematical and software developments for a suite of programs for solving ordinary differential equations in MATLAB.
Abstract: This paper describes mathematical and software developments for a suite of programs for solving ordinary differential equations in MATLAB.

3,330 citations


Cites methods from "A family of embedded Runge-Kutta fo..."

  • ...The new ode23 is based on the Bogacki-Shampine (2, 3) pair [3] (see also [37]) and the new ode45 is based on the Dormand-Prince (4, 5) pair [12]....

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Book
06 Sep 2007
TL;DR: This book discusses infinite difference approximations, Iterative methods for sparse linear systems, and zero-stability and convergence for initial value problems for ordinary differential equations.
Abstract: Finite difference approximations -- Steady states and boundary value problems -- Elliptic equations -- Iterative methods for sparse linear systems -- The initial value problem for ordinary differential equations -- Zero-stability and convergence for initial value problems -- Absolute stability for ordinary differential equations -- Stiff ordinary differential equations -- Diffusion equations and parabolic problems -- Addiction equations and hyperbolic systems -- Mixed equations -- Appendixes: A. Measuring errors -- B. Polynomial interpolation and orthogonal polynomials -- C. Eigenvalues and inner-product norms -- D. Matrix powers and exponentials -- E. Partial differential equations.

1,349 citations


Cites methods from "A family of embedded Runge-Kutta fo..."

  • ...For example, the ode45 routine in MATLAB uses a pair of embedded Runge-Kutta methods of order 4 and 5 due to Dormand and Prince [25]....

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Posted Content
TL;DR: This work presents a stochastic differential equation (SDE) that smoothly transforms a complex data distribution to a known prior distribution by slowly injecting noise, and a corresponding reverse-time SDE that transforms the prior distribution back into the data distribution by Slowly removing the noise.
Abstract: Creating noise from data is easy; creating data from noise is generative modeling. We present a stochastic differential equation (SDE) that smoothly transforms a complex data distribution to a known prior distribution by slowly injecting noise, and a corresponding reverse-time SDE that transforms the prior distribution back into the data distribution by slowly removing the noise. Crucially, the reverse-time SDE depends only on the time-dependent gradient field (\aka, score) of the perturbed data distribution. By leveraging advances in score-based generative modeling, we can accurately estimate these scores with neural networks, and use numerical SDE solvers to generate samples. We show that this framework encapsulates previous approaches in score-based generative modeling and diffusion probabilistic modeling, allowing for new sampling procedures and new modeling capabilities. In particular, we introduce a predictor-corrector framework to correct errors in the evolution of the discretized reverse-time SDE. We also derive an equivalent neural ODE that samples from the same distribution as the SDE, but additionally enables exact likelihood computation, and improved sampling efficiency. In addition, we provide a new way to solve inverse problems with score-based models, as demonstrated with experiments on class-conditional generation, image inpainting, and colorization. Combined with multiple architectural improvements, we achieve record-breaking performance for unconditional image generation on CIFAR-10 with an Inception score of 9.89 and FID of 2.20, a competitive likelihood of 2.99 bits/dim, and demonstrate high fidelity generation of 1024 x 1024 images for the first time from a score-based generative model.

1,174 citations


Cites methods from "A family of embedded Runge-Kutta fo..."

  • ...Using a black-box ODE solver (Dormand & Prince, 1980) not only produces high quality samples (Table 2, details in Appendix D.4), but also allows us to explicitly trade-off accuracy for efficiency....

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  • ...In our experiments, we use the RK45 ODE solver (Dormand & Prince, 1980) provided by scipy.integrate.solve_ivp in all cases....

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MonographDOI
01 Apr 2003
TL;DR: In this article, the authors provide a sound treatment of ODEs with Matlab in about 250 pages, with a discussion of "the facts of life" for the problem, mainly by means of examples.
Abstract: From the Publisher: This book is for people who need to solve ordinary differential equations (ODEs), both initial value problems (IVPs) and boundary value problems (BVPs) as well as delay differential equations (DDEs). These topics are usually taught in separate courses of length one semester each, but solving ODEs with Matlab provides a sound treatment of all three in about 250 pages. The chapters on each of these topics begin with a discussion of "the facts of life" for the problem, mainly by means of examples. Numerical methods for the problem are then developed - but only the methods most widely used. Although the treatment of each method is brief and technical issues are minimized, the issues important in practice and for understanding the codes are discussed. Often solving a real problem is much more than just learning how to call a code. The last part of each chapter is a tutorial that shows how to solve problems by means of small but realistic examples.

685 citations

Journal ArticleDOI
TL;DR: A family of explicit Runge-Kutta formulas that contains imbedded formulas of all orders 1 through 4 is derived, which is very efficient for problems with smooth solution as well as problems having rapidly varying solutions.
Abstract: Explicit Runge-Kutta methods (RKMs) are among the most popular classes of formulas for the approximate numerical integration of nonstiff, initial value problems. However, high-order Runge-Kutta methods require more function evaluations per integration step than, for example, Adams methods used in PECE mode, and so, with RKMs, it is expecially important to avoid rejected steps. Steps are often rejected when certain derivatives of the solutions are very large for part of the region of integration. This corresponds, for example, to regions where the solution has a sharp front or, in the limit, some derivative of the solution is discontinuous. In these circumstances the assumption that the local truncation error is changing slowly is invalid, and so any step-choosing algorithm is likely to produce an unacceptable step. In this paper we derive a family of explicit Runge-Kutta formulas. Each formula is very efficient for problems with smooth solution as well as problems having rapidly varying solutions. Each member of this family consists of a fifty-order formula that contains imbedded formulas of all orders 1 through 4. By computing solutions at several different orders, it is possible to detect sharp fronts or discontinuities before all the function evaluations defining the full Runge-Kutta step have been computed. We can then either accpet a lower order solution or abort the step, depending on which course of action seems appropriate. The efficiency of the new algorithm is demonstrated on the DETEST test set as well as on some difficult test problems with sharp fronts or discontinuities.

673 citations


Cites background or methods from "A family of embedded Runge-Kutta fo..."

  • ...It has been suggested by Dormand and Prince [5] that every elementary differential should contribute to both the local error and its estimate....

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  • ...With this in mind, we chose the coefficients of our CSIRK formulas so that their local truncation errors have the form suggested by Dormand and Prince [5] and Shampine [13]....

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References
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Journal ArticleDOI
TL;DR: According to criteria involving the number of function evaluations, overhead cost, and reliability, the best general-purpose method, if function evaluations are not very costly, is one due to Bulirsch and Stoer, however, when function evaluated methods are relatively expensive, variable-order methods based on Adams formulas are best.
Abstract: Numerical methods for systems of first order ordinary differential equations are tested on a variety of initial value problems. The methods are compared primarily as to how well they can handle relatively routine integration steps under a variety of accuracy requirements, rather than how well they handle difficulties caused by discontinuities, stiffness, roundoff or getting started. According to criteria involving the number of function evaluations, overhead cost, and reliability, the best general-purpose method, if function evaluations are not very costly, is one due to Bulirsch and Stoer. However, when function evaluations are relatively expensive, variable-order methods based on Adams formulas are best. The overhead costs are lower for the method of Bulirsch and Stoer, but the Adams methods require considerably fewer function evaluations. Krogh’s implementation of a variable-order Adams method is the best of those tested, but one due to Gear is also very good. In general, Runge–Kutta methods are not co...

735 citations

01 Jul 1969

537 citations


"A family of embedded Runge-Kutta fo..." refers background or methods in this paper

  • ...the two "minimum truncation error" formulae RK5(4)7M, RK5 (4)6M are more efficient than the RKF4 (Fehlberg [8])....

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  • ...Fehlberg ([7], [8]) has developed embedded RK formulae which have a 'small' principal truncation term in the lower order formula....

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  • ...Practical tests [8], however, indicate that, in spite of our comments in section 1, good results for the lower order mode are obtained by attempting to make the a i as small as possible....

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  • ...For RKF4 [8], which practical results ([5], [6]) indicate is preferable in local extrapolation mode, 11&(6) 112 = 3....

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Journal ArticleDOI
TL;DR: In this paper, a set of η first order simultaneous differential equations in the dependent variables y 1, y 2, y 3, y 4, y 5, y 6 and the independent variable x is considered.
Abstract: We consider a set of η first order simultaneous differential equations in the dependent variables y1, y2, …, yn and the independent variable x ⋮ No loss of gernerality results from taking the functions f1, f2, …, fn to be independent of x, for if this were not so an additional dependent variable yn+1, anc be introduced which always equals x and thus satisfies the differential equation

394 citations


"A family of embedded Runge-Kutta fo..." refers background in this paper

  • ...nr + 1' being the elementary differentials [1] of order r + i offNote that if the formula is of order p, then -Or --- 0, r = 1, 2 ....

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  • ...Butcher [1] has listed the expressions for the elementary differentials for up to order 8 and Harris [3] has considered the computer derivation of the equations of condition....

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  • ...Without loss of generality [1] the first order autonomous system y_'(x) = f [y (x)], (1....

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01 Oct 1968
TL;DR: Runge-Kutta formulas of high order with stepsize control through leading truncation error term through leading parallelogram error term.
Abstract: Runge-Kutta formulas of high order with stepsize control through leading truncation error term

346 citations


"A family of embedded Runge-Kutta fo..." refers methods in this paper

  • ...Fehlberg ([7], [8]) has developed embedded RK formulae which have a 'small' principal truncation term in the lower order formula....

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  • ...Shampine [15] has developed a modification to the RKF7 formula of Fehlberg [7] which overcomes this difficulty but it is preferable to develop formulae which are free from this deficiency....

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  • ...Concerning the quadrature problem, it has been noted previously [ 14] that many embedded formulae of high order, such as those of Fehlberg [7] with p ;~ 5, and Dormand & Prince [11] fail because, in this case, the two formulae yield the same numerical approximation for Y(Xn + 1)....

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