[R] Add transitivity to a matrix?

[Martin Maechler]

>>>>> peter dalgaard 
>>>>>     on Tue, 18 Jun 2019 11:45:39 +0200 writes:

    > Sounds like this is isomorphic to reachability in graph
    > theory. I wonder if

    >   (sum_1^n M^i) > 0

    > would suffice?

neat! (and I guess correct)

    > -pd

Which reminds me that in the relatively distant past as
maintainer of the 'expm' package I had introduced "%^%" (to
compute matrix *integer* powers) with this first part of help() :

--------------------------------------------------------------------------
Matrix Power

Description:

     Compute the k-th power of a matrix. Whereas ‘x^k’ computes
     _element wise_ powers, ‘x %^% k’ corresponds to k - 1 matrix
     multiplications, ‘x %*% x %*% ... %*% x’.

Usage:

     x %^% k
     
Arguments:

       x: a square matrix.

       k: an integer, k >= 0.

Details:

     Argument k is coerced to integer using as.integer.

     The algorithm uses O(log2(k)) matrix multiplications.

Value:

     A matrix of the same dimension as ‘x’.

Note:

     If you think you need ‘x^k’ for k < 0, then consider instead
     ‘solve(x %^% (-k))’.

........
........

--------------------------------------------------------------------------

and I had thought / wondered to myself if this should not be
brought into base R [or then at least 'Matrix' which is
installed with R (almost surely)] but I think never got around
to propose that.

Opinions?


    >> On 18 Jun 2019, at 02:08 , Duncan Murdoch
    >> <murdoch.duncan using gmail.com> wrote:
    >> 
    >> On 17/06/2019 7:34 p.m., Bert Gunter wrote:
    >>> Depends on what you mean by "simple" of course, but
    >>> suppose that: M[i,j] & M[j,k] & M[k,n] are TRUE and
    >>> M[i,k] and M[i,n] are FALSE.  Then the procedure would
    >>> see that M[i,k] needs to change to TRUE, but not that
    >>> M[i,n] needs to also become TRUE *after* M[i,k] changes.
    >>> This seems to imply that an iterative solution is
    >>> necessary.
    >> 
    >> Right, that's a good point.
    >> 
    >> Duncan Murdoch
    >> 
    >>> One such procedure, via repeated matrix multiplication
    >>> to check for and impose transitivity, appears to be
    >>> suggested by this discussion:
    >>> https://math.stackexchange.com/questions/228898/how-to-check-whether-a-relation-is-transitive-from-the-matrix-representation
    >>> Cheers, Bert On Mon, Jun 17, 2019 at 10:29 AM Duncan
    >>> Murdoch <murdoch.duncan using gmail.com
    >>> <mailto:murdoch.duncan using gmail.com>> wrote: On 17/06/2019
    >>> 1:19 p.m., Duncan Murdoch wrote: > Suppose I have a
    >>> square logical matrix M which I'm thinking of as a >
    >>> relation between the row/column numbers.
    >>> >
    >>> > I can make it into a symmetric relation (i.e. M[i,j]
    >>> being TRUE implies > M[j,i] is TRUE) by the calculation
    >>> >
    >>> > M <- M | t(M)
    >>> >
    >>> > Is there a simple way to ensure transitivity,
    >>> i.e. M[i,j] & M[j,k] both > being TRUE implies M[i,k] is
    >>> TRUE?
    >>> >
    >>> > The operation should only change FALSE or NA values to
    >>> TRUE values; TRUE > values should never be changed.  I
    >>> also want the changes to be minimal; changing everything
    >>> to TRUE would satisfy transitivity, but isn't useful to
    >>> me.  Duncan Murdoch
    >>> ______________________________________________
    >>> R-help using r-project.org <mailto:R-help using r-project.org>
    >>> mailing list -- To UNSUBSCRIBE and more, see
    >>> https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do
    >>> read the posting guide
    >>> http://www.R-project.org/posting-guide.html and provide
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    >>> 
    >> 
    >> ______________________________________________
    >> R-help using r-project.org mailing list -- To UNSUBSCRIBE and
    >> more, see https://stat.ethz.ch/mailman/listinfo/r-help
    >> PLEASE do read the posting guide
    >> http://www.R-project.org/posting-guide.html and provide
    >> commented, minimal, self-contained, reproducible code.

    > -- 
    > Peter Dalgaard, Professor, Center for Statistics,
    > Copenhagen Business School Solbjerg Plads 3, 2000
    > Frederiksberg, Denmark Phone: (+45)38153501 Office: A 4.23
    > Email: pd.mes using cbs.dk Priv: PDalgd using gmail.com

    > ______________________________________________
    > R-help using r-project.org mailing list -- To UNSUBSCRIBE and
    > more, see https://stat.ethz.ch/mailman/listinfo/r-help
    > PLEASE do read the posting guide
    > http://www.R-project.org/posting-guide.html and provide
    > commented, minimal, self-contained, reproducible code.