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 reader's 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. Contents: Preface; Acknowledgments; Part I: Fundamentals. Lecture 1: Matrix-Vector Multiplication; Lecture 2: Orthogonal Vectors and Matrices; Lecture 3: Norms; Lecture 4: The Singular Value Decomposition; Lecture 5: More on the SVD; Part II: QR Factorization and Least Squares. Lecture 6: Projectors; Lecture 7: QR Factorization; Lecture 8: Gram-Schmidt Orthogonalization; Lecture 9: MATLAB; Lecture 10: Householder Triangularization; Lecture 11: Least Squares Problems; Part III: Conditioning and Stability. Lecture 12: Conditioning and Condition Numbers; Lecture 13: Floating Point Arithmetic; Lecture 14: Stability; Lecture 15: More on Stability; Lecture 16: Stability of Householder Triangularization; Lecture 17: Stability of Back Substitution; Lecture 18: Conditioning of Least Squares Problems; Lecture 19: Stability of Least Squares Algorithms; Part IV: Systems of Equations. Lecture 20: Gaussian Elimination; Lecture 21: Pivoting; Lecture 22: Stability of Gaussian Elimination; Lecture 23: Cholesky Factorization; Part V: Eigenvalues. Lecture 24: Eigenvalue Problems; Lecture 25: Overview of Eigenvalue Algorithms; Lecture 26: Reduction to Hessenberg or Tridiagonal Form; Lecture 27: Rayleigh Quotient, Inverse Iteration; Lecture 28: QR Algorithm without Shifts; Lecture 29: QR Algorithm with Shifts; Lecture 30: Other Eigenvalue Algorithms; Lecture 31: Computing the SVD; Part VI: Iterative Methods. Lecture 32: Overview of Iterative Methods; Lecture 33: The Arnoldi Iteration; Lecture 34: How Arnoldi Locates Eigenvalues; Lecture 35: GMRES; Lecture 36: The Lanczos Iteration; Lecture 37: From Lanczos to Gauss Quadrature; Lecture 38: Conjugate Gradients; Lecture 39: Biorthogonalization Methods; Lecture 40: Preconditioning; Appendix: The Definition of Numerical Analysis; Notes; Bibliography; Index. Audience: Written on the graduate or advanced undergraduate level, this book can be used widely for teaching. Professors looking for an elegant presentation of the topic will find it an excellent teaching tool for a one-semester graduate or advanced undergraduate course. A major contribution to the applied mathematics literature, most researchers in the field will consider it a necessary addition to their personal collections. NUMERICAL LINEAR ALGEBRA......Page 1 Contents......Page 9 Preface......Page 11 Acknowledgments......Page 13 Part I Fundament als......Page 15 Lecture 1. Matrix-Vector Multiplication......Page 17 Lecture 2. Orthogonal Vectors and Matrices......Page 25 Lecture 3. Norms......Page 31 Lecture 4. The Singular Value Decomposition......Page 39 Lecture 5. More on the SVD......Page 46 Part II QR Factorization and Least Squares......Page 53 Lecture 6. Projectors......Page 55 Lecture 7. QR Factorization......Page 62 Lecture 8. Gram-Schmidt Orthogonalization......Page 70 Lecture 9. MATLAB......Page 77 Lecture 10. Householder Triangularization......Page 83 Lecture 11. Least Squares Problems......Page 91 Part III Conditioning and Stability......Page 101 Lecture 12. Conditioning and Condition Numbers......Page 103 Lecture 13. Floating Point Arithmetic......Page 111 Lecture 14. Stability......Page 116 Lecture 15. More on Stability......Page 122 Lecture 16. Stability of Householder Triangularization......Page 128 Lecture 17. Stability of Back Substitution......Page 135 Lecture 18. Conditioning of Least Squares Problems......Page 143 Lecture 19. Stability of Least Squares Algorithms......Page 151 Part IV Systems of Equations......Page 159 Lecture 20. Gaussian Elimination......Page 161 Lecture 21. Pivoting......Page 169 Lecture 22. Stability of Gaussian Elimination......Page 177 Lecture 23. Cholesky Factorization......Page 186 Part V Eigenvalues......Page 193 Lecture 24. Eigenvalue Problems......Page 195 Lecture 25. Overview of Eigenvalue Algorithms......Page 204 Lecture 26. Reduction to Hessenberg or Tridiagonal Form......Page 210 Lecture 27. Rayleigh Quotient, Inverse Iteration......Page 216 Lecture 28. QR Algorithm without Shifts......Page 225 Lecture 30. Other Eigenvalue Algorithms......Page 239 Lecture 31. Computing the SVD......Page 248 Part VI Iterative Methods......Page 255 Lecture 32. Overview of Iterative Methods......Page 257 Lecture 33. The Arnold! Iteration......Page 264 Lecture 34. How Arnold! Locates Eigenvalues......Page 271 Lecture 35. GMRES......Page 280 Lecture 36. The Lanczos Iteration......Page 290 Lecture 37. From Lanczos to Gauss Quadrature......Page 299 Lecture 38. Conjugate Gradients......Page 307 Lecture 39. Biorthogonalization Methods......Page 317 Lecture 40. Preconditioning......Page 327 Appendix. The Definition of Numerical Analysis......Page 335 Notes......Page 343 Bibliography......Page 357 Index......Page 367 Beautiful! Very simply, if you want to have an insight on linear algebraic procedures, and why this and that happens so and so, this is the book. Topic-wise, it is almost complete for a first treatment. Each chapter starts with a gentle introduction, building intuition and then gets into the formal material. The style is solid. Although talking about procedures, it also attempts to give some geometric intuition here and there. It helps. This is not a reference book though. You cannot find every important procedure. An Introduction To The Field Of Numerical Linear Algebra. It Aims To Present The Core, Standard Material In A Novel Way. Topics Include Iterative Methods For Systems Of Equations And Eigenvalue Problems And The Underlying Principles Of Conditioning And Stability. Fundamentals -- Qr Factorization And Least Squares -- Conditioning And Stability -- Systems Of Equations -- Eigenvalues -- Iterative Methods. Lloyd N. Trefethen, David Bau Iii. Includes Bibliographical References (p. 343-352) And Index.