A Survey of Matrix Computations
Charles F. Van Loan · 1990
This chapter is a three-level introduction to the field of matrix computations. The first section is very informal and is designed to acquaint the reader with the basic tools of the trade. The tools range from the algorithmic (What is a Householder matrix, and how they can be used to solve the least square problem?) to the analytic (What happens to the solution of a least square problem if we perturb the data?). In the second section, we discuss the matrix factorizations that figure heavily in numerical linear algebra. For each factorization, we survey algorithms, associated mathematical properties, and applications. Sprinkled throughout this section are special topics that illustrate the power of the factorization paradigm. In the final section, we use one factorization (Cholesky) to illustrate various aspects of high performance matrix computations. Successful computing nowadays requires the design of codes that pay careful attention to the flow of data during execution. Our goal is to make the reader more aware of these data movement concerns.