I would like to calculate YTM for every top of the book update on the 10-year note traded on Brokertec. There is no closed form solution so have to use a root finding method like Newton-Rhapson. It will obviously need to be fast. Is NR method the fastest? Are there any other better methods?
Brent https://en.wikipedia.org/wiki/Brent's_method may converge a little faster than NR for price-yield.
Before electronic computers, a "yield book" was a massive paper book where one could find nearest dirty price, days to maturity, and coupon, and then interpolate. Some caching might help you if you keep varying only the price, and look at the same bonds and settlement date.