On the Reliability of Newton’s Method

I recently re-read my thesis, “On the Reliability of Newton’s Method in the Presence of Singularity” as a way to get back into thinking about numerical analysis.  On the one hand, it was interesting reading something that I wrote seven years ago, on the other hand…  …Yeah, I’m pretty rusty.

My thesis work basically hangs on two main theorems that, over the course of four years, I pretty well seared into my brain.  There was a point, during my last year, that I could walk up to a chalk board and write down those theorems and their proofs from memory.  After reading them this time around, I can tell you, there’s no chance of that right now.  That’s not to say I had trouble understanding it – in fact, the reason I chose my own thesis to dust off my numerical analysis memory was because I knew I could at read my own proofs.

I found, as I was doing my research, that proof technique varies quite a bit by author, in terms of how much stuff they do a in single step and how much they leave the reader hanging.  I find that, comparatively, I tend to be a bit more explicit in my proofs – I will call out more things and work through the algebra with a few more steps with the goal of (hopefully) making it easier to read.  As I try to ease myself back into more numerical analysis work, I found that reading my own proofs, made the jump to other works a bit less painful.

The odd thing about reading something that I wrote seven years ago, was that I spotted a couple of typos.  Granted, it was only a couple, but they were definitely more jarring to me now than they were when I wrote them (especially for them to have gone unnoticed.)  Another thing that I noticed in reading my work again is just how tricky inequality algebra is to read.  As a writer, you have to worry about getting it right, but as a reader it’s much more challenging to keep ahead of the writer, as there is just so much they could possibly do.

There were a couple of really neat things about reading my thesis though.  First, I was able to remember what I was able to do.  Sure, my results weren’t ground-breaking, but, to me, they had some interesting consequences that was fun to be able to noodle through again.  Also, I found the little section at the end about future work – that was extremely inspiring!  While some of it would certainly take a lot of work to spin back up to, it was really interesting to think about again.

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