Comments on: Higher-dimensional, higher-order derivatives, functionally

By: John Salvatier

John Salvatier — Tue, 13 Jul 2010 04:09:48 +0000

Your blog posts/article on automatic differentiation are proving very interesting and I think useful to me. I am writing a very general automatic differentiation package in python (which can fake a lot of functional programming), and this has been the most interesting resource I have found. It's been slow going since I knew no Haskell before coming across your stuff, but this is all really elegant.

Thanks!

By: Conal Elliott » Blog Archive » Thoughts on semantics for 3D graphics

Conal Elliott » Blog Archive » Thoughts on semantics for 3D graphics — Mon, 23 Nov 2009 07:41:35 +0000

[...] a :> b contains all derivatives (including zeroth) at a point of a function of type a->b. See Higher-dimensional, higher-order derivatives, functionally. We could perhaps also include derivatives of material [...]

By: Conal Elliott » Blog Archive » Comparing formulations of higher-dimensional, higher-order derivatives

Sun, 25 Jan 2009 01:40:11 +0000

[...] Jason’s IntMap-based implementation and my LinearMap-based implementation described in Higher-dimensional, higher-order derivatives, functionally and in Simpler, more efficient, functional linear maps. For the case of Rn, my formulation uses a [...]

By: Conal Elliott » Blog Archive » Simpler, more efficient, functional linear maps

Tue, 21 Oct 2008 16:54:04 +0000

[...] Higher-dimensional, higher-order derivatives, functionally [...]

By: conal

conal — Wed, 17 Sep 2008 10:06:25 +0000

Tracy Harms brought my attention to a discussion of this post on Jchat

The initial response from John Randall was consistent with what I was describing, as was the follow-up from Tracy. The rest of the many responses seemed to miss the essence of the idea (the perspective of calculus on manifolds) by continuing to identifying derivatives with representations of linear maps (such as numbers, vectors, and matrices) rather than linear maps themselves. Correspondingly, they used various forms of multiplication of representations (e.g., scalar, dot, or matrix product) rather than the simpler and more general operation of composition of linear maps/functions.

I suspect that some of the confusion came from the dual role of linear maps in differentiation:

The derivative functional is itself a linear map between functions.
A single derivative value (an element of the range of the derivative of a function) is also a linear map.

The latter use of linear maps is what I described.

About Raul Miller's post, 0 represents the (only) constant linear map, and 1 (or an identity matrix, etc) represents the identity linear map.

By: Mark Wassell

Mark Wassell — Mon, 30 Jun 2008 08:43:07 +0000

Hi,

I am trying to write a Haskell function that will create a surface by extruding a (closed) curve f along a curve g. So given f: u -> (x,y) and g : v -> (x,y,z), I end up with h : (u,v) -> (x,y,z).

As I understand it the calculation for a given (u,v) is to calculate the slope of g at v and then to translate and tilt f u accordingly.

I think I need to have Double :~> (Double,Double) and Double :~> (Double,Double,Double) as types for f and g. Firstly I need the slope of g at v and secondly I need to pass on the means of obtaining the derivatives of the resulting surface (although thinking about the torus example, this can come for free?). I will be returning (Double,Double) :~> (Double,Double,Double).

Does this sound as though I am on the right track?

By: Functional Thinking » Performance problems with functional representation of derivatives

Thu, 05 Jun 2008 06:22:02 +0000

[...] some Haskell code for infinite Higher-dimensional, higher-order derivatives, discussed in his blog. Interesting stuff! Conal and I tried using the implementation of these ideas inside an animation [...]

By: Conal Elliott » Blog Archive » Functional linear maps

Conal Elliott » Blog Archive » Functional linear maps — Wed, 04 Jun 2008 06:10:03 +0000

[...] About « Higher-dimensional, higher-order derivatives, functionally [...]

By: conal

conal — Fri, 23 May 2008 15:43:52 +0000

Yes, the notion of derivative towers in Beautiful differentiation seems to correspond to the Taylor series expanded at that point, with coefficients multiplied by progressive factorials. See The Music of Streams by Doug McIlroy, especially the conversions between Horner and Maclaurin form.

The towers in this current post ("Higher dimensional, ...") are much more general, and I imagine have an analogous relationship to a generalized notion of Taylor series and probably to section 2.4 in The Music of Streams.

By: Lemming

Lemming — Fri, 23 May 2008 14:45:07 +0000

I guess the derivative tower is a power series when neglecting factorial factors.