2. Linear Equations

TL;DR: The LAPACK project as discussed by the authors developed a linear algebra package for modern high-performance computers, and the functions developed within that project are being incorporated into the Library as Chapters f07 and f08.

Abstract: 1 Introduction The f Chapters of the Library are concerned with linear algebra and cover a large area. This general introduction is intended to help you decide which particular f Chapter is relevant to your problem. The following Chapters are currently available: Chapter f01 – Matrix Factorizations Chapter f02 – Eigenvalues and Eigenvectors Chapter f03 – Determinants Chapter f04 – Simultaneous Linear Equations Chapter f06 – Linear Algebra Support Functions Chapter f07 – Linear Equations (LAPACK) Chapter f08 – Least-squares and Eigenvalue Problems (LAPACK) Chapter f11 – Large Scale Linear Systems Chapter f12 – Large Scale Eigenproblems Chapter f16 – NAG Interface to BLAS The principal problem areas addressed by the above Chapters are Systems of linear equations Linear least-squares problems Eigenvalue and singular value problems The solution of these problems usually involves several matrix operations, such as a matrix factorization followed by the solution of the factorized form, and the functions for these operations themselves utilize lower level support functions, typically from Chapter f16. You will not normally need to be concerned with these support functions. NAG has been involved in a project, called LAPACK (see Anderson et al. (1999)), to develop a linear algebra package for modern high-performance computers, and the functions developed within that project are being incorporated into the Library as Chapters f07 and f08. It should be emphasised that, while the LAPACK project has been concerned with high-performance computers, the functions do not compromise efficiency on conventional machines. Chapters f11 and f12 contain functions for solving large scale problems, but a few earlier functions are still located in Chapters f01, f02 and f04. For background information on numerical algorithms for the solution of linear algebra problems see Golub and Van Loan (1996). In some problem areas you have the choice of selecting a single function to solve the problem, a so-called Black Box function, or selecting more than one function to solve the problem, such as a factorization function followed by a solve function, so-called General Purpose functions. The following sections indicate which chapters are relevant to particular problem areas. The Black Box functions for solving linear equations of the form Ax ¼ b and AX ¼ B, where A is an n by n real or complex nonsingular matrix, are to be found in Chapters f04 and f07. Such equations can also be solved by selecting a general purpose factorization function from Chapter f01 …