You are probably familiar with this page<p><a href="http://www.fceia.unr.edu.ar/~fisicomp/apuntes/biblios/wnotnr.htm" rel="nofollow">http://www.fceia.unr.edu.ar/~fisicomp/apuntes/biblios/wnotnr...</a><p>and the suggested alternatives<p><a href="http://www.fceia.unr.edu.ar/~fisicomp/apuntes/biblios/altnr.htm" rel="nofollow">http://www.fceia.unr.edu.ar/~fisicomp/apuntes/biblios/altnr....</a><p>In addition to those suggestions, here are a few more relevant to the present discussion.  For dense matrices and how to think about linear algebra algorithms, I highly recommend Trefethen and Bau "Numerical Linear Algebra".  For sparse direct solvers, Tim Davis' book (<a href="http://www.ec-securehost.com/SIAM/FA02.html" rel="nofollow">http://www.ec-securehost.com/SIAM/FA02.html</a>) is good, and you can transition from the "learning"-level implementation to Umfpack which is his production-quality solver (it's what is behind Matlab's backslash).
For iterative solvers, Saad is pretty standard.  For multiphysics solvers, I don't think any book can match the Knoll and Keyes' 2004 review on Jacobian-free Newton-Krylov methods (it's very accessible).<p>The GSL has better implementations for many general-purpose algorithms in NR.  SciPy is great if you work in Python.  For scalable linear and nonlinear solvers, look at PETSc.