NUMA11/FMNN01 - Course library

• Netlib contains routines for Numerical Linear Algebra that are used in professional contexts, for example in MATLAB. Of special importance is the sublibrary LAPACK. If you look at the documentation of a certain built-in linear algebra function in MATLAB's "helpdesk", you often find a reference to a LAPACK. Once you have found the LAPACK routine in Netlib you can browse it and find out what methods and programming tricks are used. Note that different MATLAB releases may use different routines for the same built-in function.
• Templates for the Solution of Linear Systems: Building Blocks for Iterative methods is a classic text targeted towards those who want to experiment with own implementations.
• NAG is a commercial library with well documented routines for linear algebra. The codes cost, but reading the documentation is for free. This means that you can get free advice on what algorithms are efficient in a particular situation, for example when solving Large Scale Linear Systems .
• On deriving the inverse of a sum of matrices is an interesting review paper which gives useful expressions for inverses to partitioned matrices.
• The computation of elementary unitary matrices clarifies some of the obscurities on pages 71-72 in the course book.
• An analysis of sparse approximate inverse preconditioners... is a paper from the rich literature on how to construct efficient special-purpose preconditioners.
• Eigenvalues in combinatorial optimization gives a fascinating survey of the use of eigenvalues and eigenvectors of graphs in combinatorial optimization.
• Solution of the Sylvester Matrix Equation ... is a paper on how to solve certain matrix equations. It is written by, among others, Cleve Moler -- the MATLAB inventor and co-founder of the MathWorks Inc.
• The LAPACK pages Eigenvalue problems and Symmetric Eigenproblems give useful information about professional eigenvalue algorithms. The pages What's New in Version 3.3? and What's New in Version 3.3.1? presents enthusiastically the very latest progress within this research field.
• The German Wikipedia is a forum where the QR Algorithmus is discussed.
• Methods of conjugate gradients for solving linear systems is a nice historical review.
• The idea behind Krylov methods is an easy-to-read text. It ties together several topics discussed in class, such as generalized pseudo inverses and GMRES. The paper GMRES on (nearly) singular systems deals with similar issues, but is more technical.
• Implementation of the GMRES Method Using Householder Transformations discusses how Housholder reflections can replace the modified column-oriented Gram-Schmidt procedure in the Arnoldi process in a GMRES setting.
• Kbxref.mat . This is the system matrix K, the right hand side b, and the reference solution xref for most problems in Assignment 8.
• The function fredholm.m generates system matrix, right hand side, and reference solution for the last problem in Assignment 8. Also available as text version .
• The function m1Y.m enables fast matrix-free multiplication which could be used by those interested for the last problem in Assignment 8. Also available as text version m1Y.txt . More codes of this type is available from the Fast Multipole Method project .
• Pseudospectra of Linear Operators gives a review of a favourite topic of one of the textbook authors.
• The papers A fast direct solver for a class of elliptic differential equations and An adaptive fast direct solver for boundary integral equations in two dimensions present very recent development in the area of fast direct solvers for linear systems.