[Haskell-cafe] matrix computations based on the GSL
lemming at henning-thielemann.de
Thu Jun 30 04:39:02 EDT 2005
Let me a bit elaborate on what I wrote yesterday.
On Wed, 29 Jun 2005, Henning Thielemann wrote:
> I think matrices and derivatives are very different issues. I have often
> seen that the first derivative is considered as vector, and the second
> derivative is considered as matrix. In this spirit it is used like
> x^T * (D2 f)(x) * x
It should be h^T * D2 f x * h (if at all the transposition notation is
used) if the curvature of f at x is meant.
> but this is only abuse of the common multiplication definitions. A good
> interpretation and notation should seamless extend to higher derivatives.
> But the interpretation above does not work in higher dimensions.
> I like the following type for derivation.
> derive :: ((i -> a) -> b) -> ((i -> a) -> (i -> b))
> Here i is the index type, (i -> a) is the vector type, b is the type the
> vector function maps to. Its derivative has the same type of argument, but
> the result is a vector with indices of type i. You see that it is easy to
> repeat the application of 'derive', just replace b by say i->b. The second
> derivative yields vectors of type (i -> i -> b). This can be interpreted
> as matrix because it has two indices. But this is certainly not a matrix
> which represents a linear mapping as usual, but it is a matrix
> representing a bilinear form. The only thing we need is a multiplication
> to reduce one level of indices.
> mul :: (i -> c) -> (i -> b) -> b
> Though, what we still need is a general (overloaded?) definition of the
> scaling of b by c and a sum of b.
That is b must be a vector space with respect to c.
'derive' in this form is a bundled partial derivative, but it can be
identified with the total derivative. That is
derive f x i is the partial derivative of f at x with respect to the i-th component
derive f x is the total derivative of f at x.
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