Assuming Peter's equation applies, I think a direct for loop with multiplication would be a more efficient way to obtain this answer than repeated use of a power operator.
On June 18, 2019 8:01:09 AM CDT, Martin Maechler <maech...@stat.math.ethz.ch> wrote: >>>>>> 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.dun...@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.dun...@gmail.com > >>> <mailto:murdoch.dun...@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@r-project.org <mailto:R-help@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. > >>> > >> > >> ______________________________________________ > >> R-help@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....@cbs.dk Priv: pda...@gmail.com > > > ______________________________________________ > > R-help@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. > >______________________________________________ >R-help@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. -- Sent from my phone. Please excuse my brevity. ______________________________________________ R-help@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.
