Numerical Calculus

Abstract

In this chapter, we review the standard numerical methods to perform differential and integral calculus using finite precision mathematics. Indeed, the price to pay to invoke the powerful help of numerical methods is to introduce a finite precision cutoff which shall be taken carefully into account as it enters as a numerical error. Nevertheless, whenever this error is kept under control and correctly bounded, e.g., to machine precision (see Appendix A), we say that the final result is numerically exact. Hereafter, we will review the most successful numerical approaches to solve integrals of functions and partial differential equations, and specify them to solve the Schrödinger equation for few-body systems, that is, whenever it is possible to write explicitly the system wave function and the operators acting on them. These methods form the basis for the numerical approaches based on tensor network methods that we will introduce in the next parts of the book.