Recurrences and Triangular Systems

Abstract

A recurrence relation is a rule that defines each element of a sequence in terms of the preceding elements; it forms one of the most basic tools in discrete and computational mathematics with myriad applications in scientific computing, ranging from numerical linear algebra, the numerical solution of ordinary and partial differential equations and orthogonal polynomials, to dependence analysis in restructuring compilers and logic design. In fact, almost every computational task relies on recursive techniques and hence involves recurrence relations. Therefore, recurrence solvers are computational kernels and need to be implemented as efficient primitives on various architectures. Throughout this book we will encounter many linear and nonlinear recurrences that can be solved using techniques from this chapter. Linear recurrences can usually be expressed in the form of mostly banded lower triangular linear systems, so much of the discussion is devoted to parallel algorithms for solving \(\textit{Lx} = f\), where L is a lower triangular matrix.