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I've had a glimpse of Numerical Analysis (majorly, Numerical Methods like root finding, quadratic equations and other preliminary stuff) in my Calculus class but now, I find myself wanting more sophistication in my work.

Is there a good book which will help me understand concepts such as stability of algorithms, designing stable algorithms, error propogation, convergence analysis etc. from a more general point of view?

Essentially, I want to be able to understand and analyze Krylov Subspace Methods (QMR, GMRES and CG) and a few Nonlinear Optimization algorithms better. Especially, how floating point approximation makes a difference to the algorithms.

The problem with most books I've seen is that they start off assuming that the reader knows nothing about Linear Algebra and go on into basics of LU, Gaussian Elimination, QR etc. which I don't need. What I want is more of a "bird's eye view" of Numerical Analysis without going into the details of specific methods. Brevity would be highly appreciated.

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My favorite book on this topic is Accuracy and Stability of Numerical Algorithms by Nick Higham. The first few chapters are on general principles of stability, floating point arithmetic etc. Then starting from simple problems (summation, polynomial evaluation), Higham proceeds to the stability analysis of more elaborate numerical methods. I would highly recommend this book, even for the first few chapters.

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    $\begingroup$ This is a very nice book indeed, sort of The Standard in error analysis. It assumes a background in numerical analysis, and proceeds directly to detailed error analysis. $\endgroup$ – Arnold Neumaier Apr 25 '12 at 18:59
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    $\begingroup$ I had gone through this book in my Library some time last month but surprisingly, I can't seem to buy it in my country. Is there a good alternative to this book? (With an International Edition maybe) $\endgroup$ – Inquest Apr 25 '12 at 19:09
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    $\begingroup$ SIAM publishes an e-book version of Accuracy and Stability of Numerical Algorithms. It's in PDF form, and it's DRM-free. I don't know the price of the e-book, though; SIAM offers a few hundred e-books free to participating member institutions, and Accuracy and Stability of Numerical Algorithms happens to be one of them. It's a good book, so I downloaded it. It can also be ordered online through SIAM's bookstore (and through Cambridge University Press if you're in Europe). The book is 56 USD for SIAM members, and 80 USD otherwise (plus shipping). $\endgroup$ – Geoff Oxberry Apr 25 '12 at 23:34
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Quite recently I've discovered Trefethen and Bau's Numerical Linear Algebra. I really like the style and it seems to me this book satisfies almost all your criteria.

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  • $\begingroup$ I have Bau and it's really good for NLA but what I would prefer is more of a general approach. I don't necessarily need to be confined to Linear Algebra. $\endgroup$ – Inquest Apr 25 '12 at 19:10
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With respect to floating point arithmetic, I think a good starting point is the paper of D. Golberg "What every computer scientist should know about floating-point arithmetic".

Some other fun books to read, besides the already suggested, are:

  • "Matrix computations" by Golub, and van Loan.
  • "Numerical methods that usually work" by Acton.
  • "The art of computer programming" by Knuth.
  • "Domain decomposition: parallel multilevel methods for elliptic partial differential equations" by Smith, Bjørstad, Gropp.

Every book has remarkable chapters but how good is a book at helping to develop the reader's understanding of a topic depends on the reader background and interests. I found these books useful for my work and I recommend you to have a look at them at the library.

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  • $\begingroup$ The book by Acton looks really good but similar to the case above, I can't seem to buy it in my country. Any alternative to the book (probably with an International Edition)? $\endgroup$ – Inquest Apr 25 '12 at 19:18
  • $\begingroup$ You can buy it through amazon, they have international delivery. $\endgroup$ – fcruz Apr 26 '12 at 8:55
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An introductory book which explains the basics very well is Gander, Gander, Kwok: Scientific Computing.

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  • $\begingroup$ As you commented on another recent question I am an Executive Editor for CSE and Mathematics at Springer. That is public information. Personally I think you should add that to your user profile here on SciComp. Again personally I have no problem with you recommending your own (as it were) books but I think this is a community which values openness on such matters. $\endgroup$ – High Performance Mark Oct 22 '14 at 18:36

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