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.