Sorry, I meant "Symbolic differentiation is not automatic differentiation",
see the link to the wikipedia article
<https://en.wikipedia.org/wiki/Automatic_differentiation>.
Thanks for the links to the libraries they do similar things to what I want
to do, but not quiet since they don't seem to support interval arithmetic.
Also, I was thinking about this in terms of types, e.g., if I define a
function such as
def f(x, y):
return (x/y).cos()
in sage and I give it elements of RDF, it will return me numbers, but if I
give it elements in RIF, I get intervals. And I was hoping there would be a
"AutomaticDifferentiationField" so that if I pass suitable elements as
inputs to f, I automatically obtain the derivatives.
It wouldn't be fancy: an element in an AutomaticDifferentiationField for
one variable would be just a pair (value, derivative) with addition as (v,
d) + (v', d') = (v+v', d+d'), multiplication as (v, d) * (v', d') = (v *
v', v * d' + v' * d), division as (v, d) / (v', d') = (v/v', (d * v' - v *
d')/v'^2), and so on.
An AutomaticDifferentiationField should probably be in sage's category of
fields and have a base_field() (similar to PolynomialRing has a
base_ring()), probably subclass from Algebra, elements allow coercion from
elements in the base_field(). It is essentially like PolynomialRing where
the base ring needs to be a field and where you discard higher monomials
when multiplying. But it also would support division, sin, cos, ...
according to how you would ordinarily differentiation using, e.g., chain
rule.
For several variables, it would be a tuple (value,
derivativeWithRespectToFirstDim, ..., derivativeWithRespectToLastDim).
Does something like that exist in SageMath?
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