https://github.com/oldk1331/fricas/commit/642ce7d42e4168313816ba2dd703bfa04efa4cf0
Here is a sketch implementation of this idea,
now dummyvars are gone. The new implementation
is much shorter than the old one. The nesting kernels
of high order derivatives are gone as well.
- Qian
On 4/16/24 20:47, Qian Yun wrote:
(I have not tried to implement the following idea in FriCAS.)
I have always liked the idea presented in SICM (sister book
of SICP) Structure and Interpretation of Classical Mechanics:
https://mitp-content-server.mit.edu/books/content/sectbyfn/books_pres_0/9579/sicm_edition_2.zip/chapter001.html
No dummy var involved. Instead view "diff" as a higher order
operator: it acts on other operator and returns a new operator.
So instead of the having new kernels (and new dummy var)
for each new argument,
(1) -> f := operator 'f
(1) f
(2) -> argument first kernels D(f(x),x)
(2) [f(%A), %A, x]
(3) -> argument first kernels D(f(x^2),x)
2
(3) [f(%B), %B, x ]
(4) -> argument first kernels D(f(x^3),x)
3
(4) [f(%C), %C, x ]
The new plan should be generating a "new operator" "%diff(f)"
and its argument are x,x^2,x^3 respectively.
This "new operator" "%diff(f)" can store the argument "f"
into its properties, I guess. And add special "%equal" for it.
- Qian
On 4/16/24 09:43, Waldek Hebisch wrote:
If you look at 'src/input/derham.input' you will notice comment
saying that it is quite slow for large n. I looked a bit
at this and first observation is that slowness is due to
kernel manipulations. Profile shows that almost all time goes
into kernel maniputation, mainly 'SCACHE;linearSearch'.
This is called from 'FS-;diffdiff' which is responsible for
differentiating formal derivatives. Linear search is necessary
due to the way we implement formal derivatives. But it is
actually worse: repeating the same calculation shows growing
time. Just to see scale, one of calculation in
'src/input/derham.input' is to calculate second exterior
derivative of general form in n variables. General form
involves n formal function in n variables. First exterior
derivative needs to compute n^2 partial derivatives (n functions
times n variables). Second exterior derivative needs
n(n-1)n partial derivatives. So togeter about n^3 derivatives.
For n=8 this is about 512. Even squared (due to linear search)
this does not look like very large number of operations.
However, formal derivative of order 2 contains derivatives of
order 1. And it seems that due to use of dummies we create
new kernels, so when we repeat calculation, then our cache keep
growing.
This suggest that we should re-think representation of formal
derivatives and possibly also use of dummies.
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