A Practical Framework for Infinite-Dimensional Linear Algebra

We describe a framework for solving a broad class of infinite-dimensional linear equations, consisting of almost banded operators, which can be used to representing linear ordinary differential equations with general boundary conditions. The framework contains a data structure for on which row operations can be performed, allowing for the solution of infinite-dimensional linear equations by the adaptive QR approach. The algorithm achieves O(nopt) complexity, where nopt is the number of degrees of freedom required to achieve a desired accuracy, which is determined adaptively. In addition, special tensor product equations, such as partial differential equations on rectangles, can be solved by truncating the operator in the y-direction with ny degrees of freedom and using a generalized Schur decomposition to upper triangularize, before applying the adaptive QR approach to the x-direction, requiring O(n3y + n2ynoptx) operations. The framework is implemented in the ApproxFun package written in the Julia programming language, which achieves highly competitive computational costs by exploiting unique features of Julia. Using this framework, partial differential equations that require as many as 2.5 million unknowns can be solved in less than 4 seconds.