This paper describes mathematical and software developments for a suite of programs for solving ordinary differential equations in MATLAB.

The MATLAB ODE Suite

A family of embedded Runge-Kutta formulae

Survey of Numerical Methods for Trajectory Optimization

In principle, the exponential of a matrix could be computed in many ways. Methods involving approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polyn...

/pdf/nineteen-dubious-ways-to-compute-the-exponential-of-a-matrix-49qnou6xyl.pdf

Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later ⁄

Boltzmann codes are used extensively by several groups for constraining cosmo- logical parameters with Cosmic Microwave Background and Large Scale Structure data. This activity is computationally expensive, since a typical project requires from 10 4 to 10 5 Boltz- mann code executions. The newly released code CLASS (Cosmic Linear Anisotropy Solving System) incorporates improved approximation schemes leading to a simultaneous gain in speed and precision. We describe here the three approximations used by CLASS for basic �CDM models, namely: a baryon-photon tight-coupling approximation which can be set to first order, second order or to a compromise between the two; an ultra-relativistic fluid approximation which had not been implemented in public distributions before; and finally a radiation streaming approximation taking reionisation into account.

/pdf/the-cosmic-linear-anisotropy-solving-system-class-part-ii-3p3bg6yhsk.pdf

The Cosmic Linear Anisotropy Solving System (CLASS). Part II: Approximation schemes

Computer solution of ordinary differential equations : the initial value problem

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.

/pdf/solving-odes-with-matlab-1yxxtyf9zz.pdf

Solving ODEs with MATLAB

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.

A BVP solver based on residual control and the Maltab PSE

http://www.radford.edu/%7Ethompson/webddes/manuscript.pdf

Lawrence F. Shampine

Papers

The MATLAB ODE Suite

Computer solution of ordinary differential equations : the initial value problem

Solving ODEs with MATLAB

A BVP solver based on residual control and the Maltab PSE

Solving DDEs in MATLAB