Approximation Theory and Approximation Practice
by Lloyd N. Trefethen · first published 2012
15mentions
9recommended by
10HN commenters
2015–2025first – last mentioned
90% recommend10% neutral
Why HN recommends it
Clear, intuitive treatment of approximation theory, especially Chebyshev polynomials, mixing mathematical ideas with practical computation; suits engineers and numerical-methods learners seeking approachable depth.
Summarised by Claude from recommending comments
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Folks interested in this topic should look at Chebfun, http://www.chebfun.org Also at Lloyd Trefethen’s book Approximation Theory and Approximation Practice, of which the first 6 chapters are available online, http://www.chebfun.org/ATAP/ The author of this blog post might as well have used Clenshaw’s algorithm as he mentioned was suggested by Numerical…
on “Chebyshev Approximation” ↗…overload them to work on their continuous function approximations. This makes for very clean and readable code for their users, though the internals get pretty complex. Also recommended is Trefethen’s book Approximation Theory and Approximation Practice, which explains the mathematical ideas underlying the library. First 6 chapters are freely available online: http://www.chebfun.org/ATAP/
on “Mathematical software wish list” ↗This page crashes my chrome tab!! Anyway, anyone interested enough in this to be reading this comment thread should go look up approximation theory and approximation practice by trefethen [0], which is a very very good book, that covers chebyshev polynomials in a very clear way. The real party trick you get out of it, though, is that by taking samples of functions at the roots…
on “Approximating sin(x) to 5 ULP with Chebyshev polynomials” ↗…polynomial. Operations like numerical differentiation are much more accurate when you make the right “global” (i.e. not just at one single point) approximation. Also check out Nick Trefethen’s book Approximation Theory and Approximation Practice, the first 6 chapters of which are available online: http://www.chebfun.org/ATAP/atap-first6chapters.pdf
on “Hack the derivative” ↗Nice article, thanks. For anyone interested in the topic, I highly recommend Trefethen's Approximation Theory and Approximation Practice. It's approachable, intuitive, and fun while still covering a lot of technical detail.
on “Implementing the Exponential Function” ↗…average case relative error. It's not uncommon for the Remez exchange to produce polynomials which actually have worse average case relative error. This is also covered in considerable depth in Trefethen's Approximation Theory and Approximation Practice.
on “How Many Decimals of Pi Do We Really Need? (2016)” ↗Worth looking at also is Approximation Theory and Approximation Practice, which is centered around Chebfun. Almost everything by Trefethen is unusually well written, Spectra and Psuedospectra is one of my favorites.
on “Best polynomial approximation equal ripple error” ↗I wrote this intro for a friend and since then more people have found it useful: https://xn--2-umb.com/22/approximation If you want a proper textbook, I recommend L.N. Trefethen (2019). “Approximation Theory and Approximation Practice.” Which is available online here: https://epubs.siam.org/doi/book/10.1137/1.9781611975949
on “Implementing Cosine in C from Scratch” ↗A great book on this subject is Approximation Theory and Approximation Practice: https://people.maths.ox.ac.uk/trefethen/ATAP/ Also chebfun! https://www.chebfun.org/
on “Chebyshev approximation and how it can help (2012)” ↗I'd suggest: Trefethen, Lloyd N., Approximation theory and approximation practice (Extended edition), SIAM, Philadelphia, PA (2020), ISBN 978-1-611975-93-2.
on “Gaussian integration is cool” ↗I would check out "An Introduction to Numerical Analysis" by Suli and Mayers or "Approximation Theory and Approximation Practice" by Trefethen. The former covers all the major intro numerical analysis topics in a format that is suitable for someone with ~undergrad math or engineering backgrounds. The latter goes deep into Chebyshev…
on “Gaussian integration is cool” ↗…much by these, namely sparse direct solvers. Tim Davis has a good intro book. Martinsson has a new book on fast direct solvers---unclear how useful such solvers are in general. Approximation theory: "Approximation Theory and Approximation Practice" by Trefethen. Extremely important book. Many numerical methods for solving 1D problems (rootfinding, approximation, optimization, solving differential equations, nonlinear boundary value problems, representing…
on “Matters Computational (2010) [pdf]” ↗Check out https://www.chebfun.org/ if you want to see some crazy stuff you can do with polynomials! Also the book Approximation Theory and Approximation Practice https://epubs.siam.org/doi/book/10.1137/1.9781611975949 is really great, by the author of Chebfun.
on “Polynomial Interpolation” ↗…that means the whole network is just a polynomial. And sure, ```technically''' that's true but there are good reasons that numerics aren't just all polynomials all the time (check out Lloyd Trefethen's 'Approximation Theory and Approximation Practice', for example). tanh is nice and smooth but its Taylor series doesn't converge outside a small region, and the whole reason you'd use tanh in the first place is because you'll have things outside that…
on “Complexity is a source of income in open source ecosystems (2019)” ↗…that I'm quite clear on the definition of these. If you would like to read what I'm saying but from a more authoritative reference that you feel you can trust, you can just take a look at Trefethen's "Approximation Theory and Approximation Practice". I'm just quoting contents of Chapter 4 at you. Again, like I said in my first response to you, what you're saying isn't wrong, it just misses the mark a bit. If you want to compute the L2 projection…
on “Gaussian integration is cool” ↗