[R] deriv for psigamma

[Martin Maechler]

>>>>> "UweL" == Uwe Ligges <ligges at statistik.uni-dortmund.de>
>>>>>     on Tue, 31 Jul 2007 12:13:45 +0200 writes:

    UweL> francogrex wrote:
    >> Hi, 2 questions:

[....]

    >> Question 2:
    >> 
    >>> deriv(~gamma(x),"x")
    >> 
    >> expression({
    >> .expr1 <- gamma(x)
    >> .value <- .expr1
    >> .grad <- array(0, c(length(.value), 1), list(NULL, c("x")))
    >> .grad[, "x"] <- .expr1 * psigamma(x)
    >> attr(.value, "gradient") <- .grad
    >> .value
    >> })
    >> 
    >> BUT
    >> 
    >>> deriv3(~gamma(x),"x")
    >> Error in deriv3.formula(~gamma(x), "x") : Function 'psigamma' is not in the
    >> derivatives table
    >> 

    >> What I want is the expression for the second derivative (which I believe is
    >> trigamma(x), or psigamma(x,1)), how can I obtain that?


    UweL> By using some algebraic software (rather than a numeric one) or 
    UweL> contributing complete derivatives tables for the next R release.

Yes, but for the present case, one could argue that
the R internal code which "knows"  
    d/dx lgamma(x) = psi(x) = digamma(x) = psigamma(x,0)
should easily be enhanced to also "know"

    d/dx psigamma(x, n) = psigamma(x, n+1)

and consequently (but maybe with an extra clause)

    d/dx psigamma(x) = psigamma(x, 1)

The code is in  R*/src/main/deriv.c
and patches which implement the above and (there are few more
'FIXME's there ... ;-) against
    https://svn.r-project.org/R/trunk/src/main/deriv.c
are welcome - after useR!2007

Martin Maechler, ETH Zurich