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I am interested in clean algorithms for approximating solutions and so I am interested in numerical analysis, but most of the books I have seen get bogged down in error analysis or they spend a lot of time and effort in squeezing an additional 2% efficiency from a classical algorithm. What are some works which just have the fundamental ideas and methods?

flag	asked May 11 2010 at 9:17 teil 2,313●7●38

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What sort of solutions are you approximating? The amount of error control you have to do to get a meaningful answer depends strongly on the problem domain. – S. Carnahan♦ May 11 2010 at 13:40

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I always feel a bit awkward recommending a book with which I am heavily associated, but Nick Trefethen wrote a beautiful article on the general ideas of numerical analysis for the Princeton Companion to Mathematics. It's even available online: comlab.ox.ac.uk/nick.trefethen/NAessay.pdf – gowers Jun 8 2010 at 15:37

It has been 4 weeks now. Voting to close... – S. Carnahan♦ Jun 8 2010 at 15:50

This is a very broad question. Are you seeking approximations of solutions of PDE? Large linear systems? Eigenvalue problems? Optimization problems? Integral equations? Any book which professes to do all of these will not do any in depth. A collection of 'just the methods' would, for example, be 'Numerical Recipes in C', but is this the level that you want? – Nilima Nigam Jun 1 2011 at 4:59

Ed Gorcenski · Accepted Answer · 2010-05-11 15:06:59Z UTC

Are you looking for a reference that links the field of numerical analysis to mathematical concepts moreso than algorithmic concepts? Matrix Computations by Golub and Van Loan is a fairly important book that studies the algebraic structures of matrices and derives algorithms from those properties. If you're looking for an entry-level work, I keep a copy of Michael Heath's book Scientific Computing on my desk. It covers fundamental concepts and algorithms fairly well, in my opinion.

Do you have a specific problem domain in mind?

Nilima Nigam · Answer 2 · 2011-06-01 05:07:07Z UTC

To augment Timur's answer:

Claes Johnson's introductory book on FEM
Braess's book on FEM
Iserles's book on numerical analysis of DE
Gottlieb and Orzsag's book on spectral methods
Nick Trefethen's book on spectral methods for spectral collocation ideas.
Quarteroni, Sacco, Saleri on numerical methods.
From 'the horse's mouth', the Cleve Moler book on numerical computing using Matlab.

I've picked these books for their balance of important algorithms and key insights, delivered with clear prose.

I also like Strikwerda's book on finite difference methods, and the Hairer-Wanner books on numerical methods for ODE. But these focus a lot on error analysis, which may not be what you wish.

John D. Cook · Answer 3 · 2010-06-08 18:50:41Z UTC

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One of my favorites is "Numerical Methods that Work" by Forman S. Acton. It's an old book; some of the examples seem quaint now that we have far more compute power. But the principles haven't changed.

Acton's book is good about illustrating technique, learning how to think like a numerical analyst.

answered Jun 8 2010 at 18:50

John D. Cook
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The sequel "Real Computing Made Real" was written in the same spirit; just think of Acton here as a Dutch uncle: "You did what? What the hell?!" :) – J. M. Sep 20 2010 at 22:03

timur · Answer 4 · 2011-06-01 03:45:12Z UTC

My favorites:

Lloyd Nicholas Trefethen, David Bau. Numerical linear algebra
Randall J. LeVeque. Finite difference methods for ordinary and partial differential equations
Randall J. LeVeque. Numerical methods for conservation laws
Susanne C. Brenner and L. Ridgway Scott. Mathematical theory of finite element methods

Reference request for conceptual numerical analysis

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