In an ideal world this second edition should have been written at least three years ago but, needless to say, this is not an ideal world. Annoyingly, there are just 24 hours per day, rather less annoyingly I have joyously surrendered myself to the excitements of my own research and, being rather good at finding excuses, delayed the second edition again and again.

Yet, once I braced myself, banished my writer's block and started to compose in my head the new chapters, I was taken over by the sheer pleasure of writing. Repeatedly I have found myself, as I often do, thanking my good fortune for working in this particular corner of the mathematical garden, the numerical analysis of differential equations, and striving in a small way to communicate its oft-unappreciated beauty.

The last sentence is bound to startle anybody experienced enough in the fashions and prejudices of the mathematical world. Numerical analysis is often considered neither beautiful nor, indeed, profound. Pure mathematics is beautiful if your heart goes after the joy of abstraction, applied mathematics is beautiful if you are excited by mathematics as a means to explain the mystery of the world around us. But numerical analysis? Surely, we compute only when everything else fails, when mathematical theory cannot deliver an answer in a comprehensive, pristine form and thus we are compelled to throw a problem onto a number-crunching computer and produce boring numbers by boring calculations. This, I believe, is nonsense.