We present here the algorithms and user interface of a Matlab program, Fie, that solves numerically Fredholm integral equations of the second kind on an interval [ a , b ] to a specified, modest accuracy. The kernel function K ( s , t ) is moderately smooth on [ a , b ] ×[ a , b ] except possibly across the diagonal s = t . If the interval is finite, provides for kernel functions that behave in a variety of ways across the diagonal, that is, K ( s , t ) may be smooth, have a discontinuity in a low-order derivative, have a logarithmic singularity, or have an algebraic singularity. Fie also solves a large class of integral equations with moderately smooth kernel function on [0, ∞ ).

Complex step differentiation (CSD) is a technique for computing very accurate numerical derivatives in languages that support complex arithmetic. We describe here the development of a CSD package in MATLAB called PMAD. We have extended work done in other languages for scalars to the arrays that are fundamental to MATLAB. This extension raises questions that we have been able to resolve in a satisfactory way. Our goal has been to make it as easy as possible to compute approximate Jacobians in MATLAB that are all but exact. Although PMAD has a fast option for the expert that implements CSD as in previous work, the default is an object-oriented implementation that asks very little of the user.

The MATLAB program bvp4c solves two--point boundary value problems (BVPs) of considerable generality. The numerical method requires partial derivatives of several kinds. To make solving BVPs as easy as possible, the default in bvp4c is to approximate these derivatives with finite differences. The solver is more robust and efficient if analytical derivatives are supplied. In this article we investigate how to use automatic differentiation (AD) to obtain the advantages of analytical derivatives without giving up the convenience of finite differences. In bvp4cAD we have approached this ideal by a careful use of the MAD AD tool and some modification of bvp4c.

Our goal was to make it as easy as possible to solve a large class of boundary value problems (BVPs) for ordinary differential equations in the Matlab problem solving environment (PSE). We present here theoretical and software developments resulting in bvp4c, a capable BVP solver that is exceptionally easy to use.

Computing the solution of the initial value problem in ordinary differential equations (ODEs) may be only part of a larger task. One such task is finding where an algebraic function of the solution (an event function) has a root (an event occurs). This is a task which is difficult both in theory and in software practice. For certain useful kinds of event functions, it is possible to avoid two fundamental difficulties. It is described how to achieve the reliable solutions of such problems in a way that allows the capability to be grafted onto popular codes for the initial value problem.

Popular codes for the numerical solution of nonstiff ordinary differential equations (ODEs) are based on a (fixed order) Runge-Kutta method, a variable order Adams method, or an extrapolation method. Extrapolation can be viewed as a variable order Runge-Kutta method. It is plausible that variation of order could lead to a much more efficient Runge-Kutta code, but numerical comparisons have been contradictory. We reconcile previous comparisons by exposing differences in testing methodology and incompatibilities of the implementations tested. An experimental Runge-Kutta code is compared to a state-of-the-art extrapolation code. With some qualifications, the extrapolation code shows no advantage. Extrapolation does not appear to be a particularly effective way to vary the order of Runge-Kutta methods. Although an acceptable way to solve nonstiff problems, our tests raise the question as to whether there is any point in pursuing it as a separate method.

Rosenbrock MethodsRosenbrock formulas have shown promise in research codes for the solution of initial value problems for stiff systems of ordinary differential equations (ODEs).To help assess their practical value, the author wrote an item of mathematical software based on one such formula.This required a varmty of algomthmic and software developments Those of general interest are reported in this paper Among them is a way to select automatically, at every step, an exphcit Runge-Kutta formula or a Rosenbrock formula according to the stiffness of the problem.Solving linear systems is important to methods for stiff ODEs and is rather special for Rosenbrock methods.A cheap, effective estimate of the condition of the linear systems is derived.Some numerical results are presented to illustrate the developments.

In 1975 Enright, Hull, and Lindberg published a set of problems for the testing of codes for stiff ordinary differential equation {ODE) problems.It has been widely used since then.In this paper a critical evaluation of the test set is made, and changes are recommended that make the set a more useful tool.Comments about the difficulties of testing software in this area are made which may lead to more thoughtful use of the test set.

Storage requirements, thetr techniques, and misconceptions are reviewed for Runge-Kutta methods.A new technique provides a substantial reduction of storage for high order methods.

article Free Access Share on Automatic Solution of the Sturm-Liouville Problem Authors: P. B. Bailey Numerical Mathematics Division 5122, Sandia Laboratories, Albuquerque, NM Numerical Mathematics Division 5122, Sandia Laboratories, Albuquerque, NMView Profile , M. K. Gordon Numerical Mathematics Division 5122, Sandia Laboratories, Albuquerque, NM Numerical Mathematics Division 5122, Sandia Laboratories, Albuquerque, NMView Profile , L. F. Shampine Numerical Mathematics Division 5122, Sandia Laboratories, Albuquerque, NM Numerical Mathematics Division 5122, Sandia Laboratories, Albuquerque, NMView Profile Authors Info & Claims ACM Transactions on Mathematical SoftwareVolume 4Issue 3Sept. 1978 pp 193–208https://doi.org/10.1145/355791.355792Published:01 September 1978Publication History 93citation841DownloadsMetricsTotal Citations93Total Downloads841Last 12 Months54Last 6 weeks12 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Although there are a great many variants of Adams methods, only two possibilities are represented among the very best pieces of mathematmal software.The behavior of this software when solwng mildly stiff initial value problems Is studied.Substantial differences between various implementatmns of the same method, as well as those between chfferent methods, are explained.

article Free Access Share on Stiffness and Nonstiff Differential Equation Solvers, II: Detecting Stiffness with Runge-Kutta Methods Author: L. F. Shampine Numerical Mathematics Division 5122, Sandia Laboratories, Albuquerque, NM Numerical Mathematics Division 5122, Sandia Laboratories, Albuquerque, NMView Profile Authors Info & Claims ACM Transactions on Mathematical SoftwareVolume 3Issue 1March 1977 pp 44–53https://doi.org/10.1145/355719.355722Online:01 March 1977Publication History 48citation0DownloadsMetricsTotal Citations48Total Downloads0Last 12 Months0Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

article Global Error Estimates for Ordinary Differential Equations Share on Authors: L. F. Shampine Division 2642, Sandia Laboratories, Albuquerque, NM Division 2642, Sandia Laboratories, Albuquerque, NMView Profile , H. A. Watts Division 2642, Sandia Laboratories, Albuquerque, NM Division 2642, Sandia Laboratories, Albuquerque, NMView Profile Authors Info & Claims ACM Transactions on Mathematical SoftwareVolume 2Issue 2June 1976 pp 172–186https://doi.org/10.1145/355681.355687Published:01 June 1976 62citation814DownloadsMetricsTotal Citations62Total Downloads814Last 12 Months13Last 6 weeks2 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

article Free AccessArtifacts Evaluated & ReusableArtifacts Available Share on Algorithm 504: GERK: Global Error Estimation For Ordinary Differential Equations [D] Authors: L. F. Shampine Division 2642, Sandia Laboratories, Albuquerque, NM Division 2642, Sandia Laboratories, Albuquerque, NMView Profile , H. A. Watts Division 2642, Sandia Laboratories, Albuquerque, NM Division 2642, Sandia Laboratories, Albuquerque, NMView Profile Authors Info & Claims ACM Transactions on Mathematical SoftwareVolume 2Issue 2June 1976 pp 200–203https://doi.org/10.1145/355681.355690Published:01 June 1976Publication History 11citation430DownloadsMetricsTotal Citations11Total Downloads430Last 12 Months16Last 6 weeks3 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

A procedure using the minimax polynomial fit of degree n on the set of extrema of the Chebyshev polynomial T n+1 is examined for its effectiveness in generating near-minimax fits on [-1, 1]. It is compared to Powell's results for interpolation at the zeros of T n .