Numerical Linear Algebra by Lloyd N Trefethen (PDF)

Description

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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is the clearest, most to the point, and most understandable book I’ve seen on this subject. It’s like a distillation of aha moments laid out for the reader. The other reviews already here cover plenty, so I’ll only add a little. (FYI: I thought the book by Demmel looked very good too, but this one seemed to resonate just a bit more with me.)I’ve seen too many books on numerical linear algebra where reading them is like hacking your way through a dense jungle – cumbersome, cluttered notation, lots of theorem-proof stuff, and too little in the way of calculational guidance. The author demonstrates their mastery, and we are left to sometimes feel a bit sluggish and dim as we try to keep up. This book is the antidote.It still contains all the key topics as you can see in the table of contents, and it still does pay attention to clearly proving important theorems. But it somehow keeps you engaged at every step rather than just tiring you out. Perhaps it’s a combination of factors, like the short, digestible chapters (which are really lectures), each covering a limited amount of material within just a few pages. And the text is laid out comfortably instead of cramming material onto the page. Also, a generous number of schematic diagrams and sample data plots are given to aid the exposition. The notation is clear and as uncluttered as possible. In summary, it is possible to sit down and enjoy reading and learning from this book.

⭐This is without a doubt the most gentle treatment of not just algorithms and efficiency in numerical computations, but also the theory that allows these algorithms to work, and I really wish other books would follow suit. Too many books are more concerned with jamming ideas in your face over hundreds of topics because they fear becoming outdated in an ever expanding field (ESPECIALLY where linear algebra, or just vectors/data, is concerned).But this book has a clear purpose in mind. “This is a vector. A bunch of sorted data is a vector. This is how we need to handle it. This is how we’re going to improve it. You can manipulate data like this, but not like that.” And with that, you’re left not just wanting more, but you’re also fully prepared to tackle a more advanced text (like Iterative Methods for Sparse Linear Systems, an ideal “next step” from here. Or perhaps you’re ready for linear programming/optimization methods? Or maybe even branching into nonlinear programming? Or convex analysis?). And you have the best foundation for being able to read through those jargon-heavy papers with giant summation signs and vector notation strewn everywhere without feeling “Oh, geez, another one of THESE papers.”And this book, with all of its chapters and contents, is in no hurry to catch up with modern methods. It takes time to explain “Hey, look, we get that there are better ways of doing it. But the reasons those methods exist is because of what we’re trying to show you.”And the problems, wow! They’re sorted into every possible category. You have your typical Simple – Challenging range, but you also have problems clearly designed for engineers with little abstract mathematical analysis ideas, problems clearly designed for computer scientists who have some knowledge but want something specifically applicable to them, problems clearly designed for the mathematician who can prove, and problems clearly designed for the mathematician who has more interest in application but needs more meat than the average engineer.And the presentation, wow! So much care is given to how much white space is needed between theorems, sections, equations, and algorithms. Trefethen and Bau know that math books, particularly numerical ones, tend to cram information too close together which can hurt the eyes.I’m absolutely gah-gah for this book. And you would be, too, if you sat down, read it, and worked through it. This is the second book I’ve ever had any desire to just sit down and try and work every problem through, and I’m almost done with it! And while there’s only one book out there I frequently reread, this is one of the very books I keep coming back to when I need a quick reference while trying to sift through something like Bender’s Decomposition or trying to construct a genetic algorithm.

⭐Face it, most math textbooks are awful: boring to read, not much insight, little more than a compendium of definitions, theorems, proofs, and examples. Trefethen and Bau is an exception to that rule. Indeed, the field of numerical linear algebra is unusual in having available several top-notch textbooks: Golub and Van Loan, Stewart’s two volumes, Saad’s books on iterative methods, Demmel’s introduction, Watkins’ undergraduate level treatment, and T&B. All of these are excellent (and any student in numerical analysis should delve into all of them) but to my tastes T&B and Stewart are the standouts for insight and simply being fun to read.If you’re a student using T&B in a course, to use it effectively you need to understand that T&B is a book to be read carefully for understanding; it’s not a typical textbook suited only for “mining” for examples and solutions to homework problems. My students have sometimes complained — accurately — that T&B is short on details and worked examples, and many of the proofs are just sketches. But that’s a feature, not a bug: you can learn much by filling in the missing steps. This is book for reading, so take the time to read it, to think about what you’ve read, and to fill in the gaps; it’s worth it. If you need some worked examples, Watkins has them in great detail and would be a good supplement to T&B (though see the caveat below).The only minor gripe I have about T&B is that the order of topics (QR before LU before Cholesky) is unusual, which makes it a little awkward to coordinate with other books such as Watkins which do Cholesky before LU before QR.

⭐Good book on the topic but horribly overpriced at £50 for a thin paperback copy with standard 70 paper!!!

⭐very easy layout to understand the process of working out. i would recommend to those has very little knowledge of computing or numerical methods

⭐This book is a treasure among all the books in the domain of linear algebra. The author presents everything from a really practical point of view, which would help you understand how these algorithms running in scientific computing softwares.Compare with the book named matrix algebra, this book outline the fundamental idea of numerical linear algebra, such as SVD, orthogonality, QR decomposition, which makes the material really easy to catch up with and every theorem in this book looks so natural, even you need a bit derivation to get the same result.So overall I would rate this book 5 stars. It’s a must have book if you are an engineering student or even you are a math major student.

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