Black-Box Toolbox: Further Reading

Further Reading

This page contain pointers to some of the most important aspects of BBTools.

SVD

An exceptionally readable introduction to the SVD can be found in [20], but many other textbooks include it.

The standard reference for dense decompositions, including the SVD, is [7], although some followers insist that the 2nd edition [6] was better. A good introduction to a newer algorithm, divide-and-conquer, is given in [4], although it is presented as an eigenproblem.

Lanczos Bidiagonalization

Both the SVD computations and equation solving relies heavily on this procedure. A good reference in the classical tradition is [15], although it is put in the context of the LSQR-algorithm.

Self-contained treatments of Krylov-methods and Lanczos algorithms can be found in [20] and [4].

Regularization

Most litterature, that is not specific to a single topic, treat regularization in the context of Hilbert spaces. Although this concept is powerful, and more general than matrices, it is also heavy to read.

One standard reference, which consistently use the matrix formalism, is [9], which also appear to become a standard reference.

This book also contains a good description of the L-curve method. Although Mr. Hansen denies to have invented this method, this appears to be a common conception. To avoid confusion, he is usually credited for "popularizing" the L-curve method.

References

[4] James Weldon Demmel. Applied Numerical Linear Algebra. SIAM, Philadelphia, PA, 1997. ISBN: 0-89871-389-7. (Book)

[6] Gene Howard Golub and Charles Francis Van Loan. Matrix Computations. John Hopkins University Press, 2nd edition, 1989. ISBN: 0-8018-3739-1. (Book)

[7] Gene Howard Golub and Charles Francis Van Loan. Matrix Computations. John Hopkins University Press, 3rd edition, 1996. ISBN: 0-8018-5414-8. (Book)

[9] Per Christian Hansen. Rank-Deficient and Discrete Ill-Posed Problems. SIAM, 1998. ISBN: 0-89871-403-6. (Book)

[15] Christopher Conway Paige and Michael A. Saunders. LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares. ACM Transactions on Mathematical Software (TOMS), 8(1):43-71, 1982. ACM Press. ISSN: 0098-3500. (Paper)

[20] Lloyd Nicholas Trefethen and David Bau III. Numerical Linear Algebra. SIAM, Philadelphia, PA, 1997. ISBN: 0-89871-487-7. (Book)