In 1991 I was invited by Philippe Toint to give a presentation, on Conjugate Gradients and related iterative methods, at the university of Namur (Belgium). I had prepared a few-hand-written notes to guide myself through an old-fashioned presentation with blackboard and chalk. Some listeners asked for a copy of the notes and afterwards I heard from Philippe that they had been quite instructive for his students. This motivated me to work them out in LATEX and that led to the first seven or so pages of my lecture notes. I took the habit of expanding them before and after new lectures and after I had read new interesting aspects of iterative methods. Around 1995 I put the then about thirty pages on my website. They turned out to be quite popular and I received many suggestions for improvement and expansion, most of them by e-mail from various people: novices in the area, students, experts in this field, and users from other fields and industry.

For instance, research groups at Philips Eindhoven used the text for their understanding of iterative methods and they sometimes asked me to comment on certain novel ideas that they had heard of at conferences or picked up from literature. This led, amongst others, to sections on GPBi-CG, and symmetric complex systems. Discussions with colleagues about new developments inspired me to comment on these in my Lecture Notes and so I wrote sections on Simple GMRES and on the superlinear convergence of Conjugate Gradients.

A couple of years ago, I started to use these Lecture Notes as material for undergraduate teaching in Utrecht and I found it helpful to include some exercises in the text.