G. M. Phillips & Peter J. Taylor 
Theory and Applications of Numerical Analysis [PDF ebook] 

Theory and Applications of Numerical Analysis is a self-contained Second Edition, providing an introductory account of the main topics in numerical analysis. The book emphasizes both the theorems which show the underlying rigorous mathematics andthe algorithms which define precisely how to program the numerical methods. Both theoretical and practical examples are included. – a unique blend of theory and applications- two brand new chapters on eigenvalues and splines- inclusion of formal algorithms- numerous fully worked examples- a large number of problems, many with solutions