Differentiation of freealg objects

Differentiation of freealg objects

# S3 method for class 'freealg'
deriv(expr, r, ...)

Arguments

expr: Object of class freealg
r: Integer vector. Elements denote variables to differentiate with respect to. If r is a character vector, it is interpreted as a=1,b=2,...,z=26; if of length 1, “aab” is interpreted as c("a","a","b")
...: Further arguments, currently ignored

Author

Robin K. S. Hankin

Details

Experimental function deriv(S,v) returns $\frac{\partial^r S}{\partial v_1\partial v_2\ldots\partial v_r}$. The Leibniz product rule

$$\left(u\cdot v\right)'=uv'+u'v$$

operates even if (as here) $u,v$ do not commute. For example, if we wish to differentiate $aaba$ with respect to $a$, we would write $f(a) = aaba$ and then

$$f(a+\delta a) = (a+\delta a)(a+\delta a)b(a+\delta a)$$

and working to first order we have

$$f(a+\delta a) -f(a)= (\delta a)aba + a(\delta a)ba + aab(\delta a).$$ In the package:



    > deriv(as.freealg("aaba"),"a")
    free algebra element algebraically equal to
    + 1*aab(da) + 1*a(da)ba + 1*(da)aba

A term of a freealg object can include negative values which correspond to negative powers of variables. Thus:


    > deriv(as.freealg("AAAA"),"a")
    free algebra element algebraically equal to
    - 1*AAAA(da)A - 1*AAA(da)AA - 1*AA(da)AAA - 1*A(da)AAAA

(see also the examples). Vector r may include negative integers which mean to differentiate with respect to the inverse of the variable:


    > deriv(as.freealg("3abcbCC"),"C")
    free algebra element algebraically equal to
    + 3*abcbC(dC) + 3*abcb(dC)C - 3*abc(dC)cbCC

It is possible to perform repeated differentiation by passing a suitable value of r. For $\frac{\partial^2}{\partial a\partial c}$:


    > deriv(as.freealg("aaabAcx"),"ac")
    free algebra element algebraically equal to
    - 1*aaabA(da)A(dc)x + 1*aa(da)bA(dc)x + 1*a(da)abA(dc)x + 1*(da)aabA(dc)x

The infinitesimal indeterminates (“da” etc) are represented by SHRT_MAX+r, where r is the integer for the symbol, and SHRT_MAX is the maximum short integer. This includes negative r. So the maximum number for any symbol is SHRT_MAX. Inverse elements such as A, being represented by negative integers, have differentials that are SHRT_MAX-r.

Function deriv() calls helper function lowlevel_diffn() which is documented at Ops.freealg.Rd.

A vignette illustrating this concept and furnishing numerical verification of the code in the context of matrix algebra is given at inst/freealg_matrix.Rmd.

Examples


deriv(as.freealg("4*aaaabaacAc"),1)
#> free algebra element algebraically equal to
#> - 4aaaabaacA(da)Ac + 4aaaaba(da)cAc + 4aaaab(da)acAc + 4aaa(da)baacAc +
#> 4aa(da)abaacAc + 4a(da)aabaacAc + 4(da)aaabaacAc

x <- rfalg()
deriv(x,1:3)
#> free algebra element algebraically equal to
#> 0

y <- rfalg(7,7,17,TRUE)

deriv(y,1:5)-deriv(y,sample(1:5)) # should be zero
#> free algebra element algebraically equal to
#> 0