Numerical Analysis

Below is a list of reviews of books on numerical analysis.

A First Course in the Numerical Analysis of Differential Equations
Iserles, A. (2nd Ed. 2008). ISBN 9780521734905.
Prereqs: Familiarity with ODEs, basic PDEs, and linear algebra

A standard introduction to numerical techniques for solving differential equations. It starts off with discussing numerical techniques for ODEs including multistep and Runge-Kutta methods, with a section of stiffness, before moving on to the Poisson equation as a model problem for numerical techniques for PDEs. It discusses a variety of techniques, with some content on numerical linear algebra as well. It concludes by discussing diffusion and advection, two important time-dependent PDEs that also serve as important model problems in their own right. Pretty standard material, roughly comparable to a two course sequence on numerical methods for differential equations.

Applied Numerical Linear Algebra
Demmel, J. W. (1997). ISBN 9780898713893.\ Prereqs: Familiarity with basic linear algebra

Demmel’s book is one of the gold standards for this area. It presents a very good perspective towards NLA and is very comprehensive. It covers both theoretical aspects and practical considerations in considerable detail, for each algorithm it discusses. It also has good problems. All in all, an excellent book to both learn from and refer to. Additionally, Demmel has notes for his NLA class that he teaches with his book, so there are additional online resources if one so desires.

Numerical Methods for Conservation Laws
LeVeque, R. J. (2nd Ed. 1992). ISBN 9783764327231.
Prereqs: Familiarity with linear algebra

LeVeque has many books; this I would consider to be one of the more noteworthy ones. This book is divided into two parts: the first part discusses the mathematical theory of PDEs, in specific, advection equations, and the second part discusses numerical techniques for advection equations. It is fairly comprehensive in its treatment of both, but what makes this book stand out is how it treats mathematical theory and numerical considerations together. This perspective is entirely important in actual applied math as it is practiced, and is entirely underemphasized in books.