Matrix Analysis and Algorithms
The book contains an introduction to matrix analysis, and to the basic algorithms of numerical linear algebra. Further results can be found in many text books. The book of Horn and Johnson is an excellent reference for theoretical results about matrix analysis. The subject of linear algebra, and matrix analysis in particular, is treated in an original and illuminating fashion in [Lax97]. For a general introduction to the subject of numerical linear algebra we recommend the book by Trefethen and Bau; more theoretical treatments of the subject can be found in Demmel, Golub and Van Loan and in Stoer and Bulirsch. Higham's book contains a wealth of information about stability and the effect of rounding errors in numerical algorithms; it is this source that we used for almost all theorems we state concerning backward error analysis. The book of Saad covers the subject of iterative methods for linear systems. The symmetric eigenvalue problem is analysed in Parlett.