Numerical Linear Algebra

This is a concise, insightful introduction to the field of numerical linear algebra. The clarity and eloquence of the presentation make it popular with teachers and students alike. The text aims to expand the readers view of the field and to present standard material in a novel way. All of the most important topics in the field are covered with a fresh perspective, including iterative methods for systems of equations and eigenvalue problems and the underlying principles of conditioning and stability. Presentation is in the form of 40 lectures, which each focus on one or two central ideas. The unity between topics is emphasized throughout, with no risk of getting lost in details and technicalities. The book breaks with tradition by beginning with the QR factorization - an important and fresh idea for students, and the thread that connects most of the algorithms of numerical linear algebra.

Spis treści:
Preface
Part I. Fundamental
1. Matrix-vector multiplication
2. Orthogonal vectors and matrices
3. Norms
4. The singular value decomposition
5. More on the SVD
Part II. QR Factorization and Least Squares
6. Projectors
7. QR factorization
8. Gram-Schmidt orthogonalization
9. MATLAB
10. Householder triangularization
11. Least squares problems
Part III. Conditioning and Stability
12. Conditioning and condition numbers
13. Floating point arithmetic
14. Stability
15. More on stability
16. Stability of householder triangularization
17. Stability of back substitution
18. Conditioning of least squares problems
19. Stability of least squares algorithms
Part IV. Systems of Equations
20. Gaussian elimination
21. Pivoting
22. Stability of Gaussian elimination
23. Cholesky factorization
Part V. Eigenvalues
24. Eigenvalue problems
25. Overview of Eigenvalue algorithms
26. Reduction to Hessenberg or tridiagonal form
27. Rayleigh quotient, inverse iteration
28. QR algorithm without shifts
29. QR algorithm with shifts
30. Other Eigenvalue algorithms
31. Computing the SVD
Part VI. Iterative Methods
32. Overview of iterative methods
33. The Arnoldi iteration
34. How Arnoldi locates Eigenvalues
35. GMRES
36. The Lanczos iteration
37. From Lanczos to Gauss quadrature
38. Conjugate gradients
39. Biorthogonalization methods
40. Preconditioning
Appendix
Notes
Bibliography
Index.