To put the world of linear algebra to advanced use, it is not enough to merely understand the theory; there is a significant gap between the theory of linear algebra and its myriad expressions in nearly every computational domain. To bridge this gap, it is essential to process the theory by solving many exercises, thus obtaining a firmer grasp of its diverse applications. Similarly, from a theoretical perspective, diving into the literature on advanced linear algebra often reveals more and more topics that are deferred to exercises instead of being treated in the main text. As exercises grow more complex and numerous, it becomes increasingly important to provide supporting material and guidelines on how to solve them, supporting students’ learning process.
This book provides precisely this type of supporting material for the textbook “Numerical Linear Algebra and Matrix Factorizations,” published as Vol. 22 of Springer’s Texts in Computational Science and Engineering series. Instead of omitting details or merely providing rough outlines, this book offers detailed proofs, and connects the solutions to the corresponding results in the textbook. For the algorithmic exercises the utmost level of detail is provided in the form of MATLAB implementations. Both the textbook and solutions are self-contained. This book and the textbook are of similar length, demonstrating that solutions should not be considered a minor aspect when learning at advanced levels.
Arvustused
This is a companion volume to Lyches textbook on numerical linear algebra and matrix factorization. This combination of this book of exercises and solutions with the original book makes the text more approachable, and the additional teaching present in the solutions is quite valuable. This book would also be useful to instructors teaching a linear algebra course at a comparable level. There are a lot of good exercises here that would supplement those in other texts. (Bill Satzer, MAA Reviews, July 5, 2021)
A Short Review of Linear Algebra.- Diagonally Dominant Tridiagonal
Matrices; Three Examples.- Gaussian Eliminationa nd LU Factorizations.- LDL*
Factorization and Positive Definite Matrices.- Orthonormal and Unitary
Transformations.- Eigenpairs and Similarity Transformations.- The Singular
Value Decomposition.- Matrix Norms and Perturbation Theory for Linear
Systems.- Least Squares.- The Kronecker Product .- Fast Direct Solution of a
Large Linear System.- The Classical Iterative Methods.- The Conjugate
Gradient Method.- Numerical Eigenvalue Problems.- The QR Algorithm.
Tom Lyche is professor emeritus at the University of Oslo. His research interests are in numerical analysis, and splines in approximation theory. He is the author with Jean-Louis Merrien of the book "Exercises in Computational Mathematics with MATLAB" and editor on numerous books on spline methods, computer aided geometric design and approximation theory. He has been a co-organizer of many conferences, in particular the two conference series "Mathematical Methods for Curves and Surfaces" in Norway and "Curves and Surfaces" in France.
Georg Muntingh is a research scientist at the Department of Mathematics and Cybernetics at SINTEF Digital. Jointly with Dr. Tor Dokken, he edited the book SAGA --- Advances in Shapes, Geometry, and Algebra, published by Springer in 2014. His research has an interdisciplinary nature and spans a wide range of research fields, with papers published on algebraic geometry, approximation theory, combinatorics, geometric modelling, physics and machine learning.
Øyvind Ryan is associate professor at the Department of Mathematics at the University of Oslo. His main interest lies in book projects on topics building heavily on linear algebra, and he has published the book Linear Algebra, Signal Processing, and Wavelets A Unified Approach, published by Springer in 2019.
