Hi Waldek, This thread is almost one year old, any opinions on this?
- Qian On 4/17/24 10:29 PM, Qian Yun wrote: > 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. >>> -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to fricas-devel+unsubscr...@googlegroups.com. To view this discussion visit https://groups.google.com/d/msgid/fricas-devel/5eb9d215-03ad-4541-b6fc-e57cf9a3548d%40gmail.com.
