[Rd] Adding a Matrix Exponentiation Operator
Martin Maechler
maechler at stat.math.ethz.ch
Tue Apr 8 15:57:18 CEST 2008
>>>>> "VG" == Vincent Goulet <vincent.goulet at act.ulaval.ca>
>>>>> on Tue, 08 Apr 2008 09:28:00 -0400 writes:
VG> Le dim. 6 avr. à 07:01, Rory Winston a écrit :
>> Hi Martin
>>
>> Thanks for the detailed reply. I had a look at the matrix power
>> implementation in the actuar package and the modified version in the
>> expm
>> package. I have a couple of questions/comments:
>>
>> 1. Firstly, I seem to have trouble loading expm.
>>
>>> install.packages("expm",repos="http://R-Forge.R-project.org")
>> trying URL '
>> http://R-Forge.R-project.org/bin/macosx/universal/contrib/2.6/expm_0.9-1.tgz
>> '
>> Content type 'application/x-gzip' length 149679 bytes (146 Kb)
>> opened URL
>> ==================================================
>> downloaded 146 Kb
>> ...
>>> library("expm")
>> Error in namespaceExport(ns, exports) : undefined exports :matpow
>> Error: package/namespace load failed for 'expm'
VG> [snip]
VG> The current version of the package on R-Forge is 0.9-2 and the
VG> NAMESPACE issue should be fixed there.
Yes, and I told Rory explicitly that he should wait a day before
downloading the windows version of 'expm' ...
{Siiiggh, it's not only the documentation that people are not reading ...}
>> 2. On to the package implementation, I see we actually have very
>> similar
>> implementations. The main differences are:
>>
>> i) For an exponent equal to -1, I call back into R for the solve()
>> function
>> using eval() and CAR/CDR'ing the arguments into place. The actuar
>> package
>> calls dgesv() directly. I suspect that the direct route is more
>> efficient
>> and thus the more preferable one. I see that your implementation
>> doesnt
>> calculate the inverse for an exponent of -1,is there a specific
>> reason for
>> doing that?
VG> The rationale is: you seldom *really* need to inverse a matrix, so we
VG> won't help you go that route. If you *really* need the explicit
VG> inverse, then use solve() directly (as the error message says).
>> ii) Regarding complex matrices: I guess we should have support for
>> this, as
>> its not unreasonable that someone may do this, and it should be
>> pretty easy
>> to implement. My function doesnt have full support yet.
>>
>> iii) A philosophical question - where the the "right" place for the
>> %^%
>> operator? Is it in the operator list at a C level along with %*% and
>> the
>> like? Or is it in an R file as a function definition?
VG> Well... both. The operator %^% is defined at the R level but the
VG> computations are done at the C level by function matpow(). Or perhaps
VG> I didn't understand what you mean, here.
Thank you, Vincent.
I think Rory was asking about how the integration into "R base"
should happen. In that case there's much more choice than just
the .Call() or .External() one: There's also .Internal() and ".Primitive",
and for instance %*% is primitive.
One current advantage of having %^% be primitive would be that
it can automatically also be an S4 and S3 generic function.
However there are plans to make this easier for R 2.8.0++ and we
are talking about that version of R anyway.
>> I dont really have a
>> preference either way...have you an opinion on this?
>>
>> Thanks
>> Rory
VG> HTH
VG> Vincent
>>
>>
>>
>> On Sat, Apr 5, 2008 at 6:52 PM, Martin Maechler <maechler at stat.math.ethz.ch
>> >
>> wrote:
>>
>>>>>>>> "RW" == Rory Winston <rory.winston at gmail.com>
>>>>>>>> on Sat, 5 Apr 2008 14:44:44 +0100 writes:
>>>
RW> Hi all I recently started to write a matrix
RW> exponentiation operator for R (by adding a new operator
RW> definition to names.c, and adding the following code to
RW> arrays.c). It is not finished yet, but I would like to
RW> solicit some comments, as there are a few areas of R's
RW> internals that I am still feeling my way around.
>>>
RW> Firstly:
>>>
RW> 1) Would there be interest in adding a new operator %^%
RW> that performs the matrix equivalent of the scalar ^
RW> operator?
>>>
>>> Yes. A few weeks ago, I had investigated a bit about this and
>>> found several R-help/R-devel postings with code proposals
>>> and then about half dozen CRAN packages with diverse
>>> implementations of the matrix power (I say "power" very much on
>>> purpose, in order to not confuse it with the matrix exponential
>>> which is another much more interesting topic, also recently (+-
>>> two months?) talked about.
>>>
>>> Consequently I made a few timing tests and found that indeed,
>>> the "smart matrix power" {computing m^2, m^4, ... and only those
>>> multiplications needed} as you find it in many good books about
>>> algorithms and e.g. also in *the* standard Golub & Van Loan
>>> "Matrix Computation" is better than "the eigen" method even for
>>> large powers.
>>>
>>> matPower <- function(X,n)
>>> ## Function to calculate the n-th power of a matrix X
>>> {
>>> if(n != round(n)) {
>>> n <- round(n)
>>> warning("rounding exponent `n' to", n)
>>> }
>>> if(n == 0)
>>> return(diag(nrow = nrow(X)))
>>> n <- n - 1
>>> phi <- X
>>> ## pot <- X # the first power of the matrix.
>>> while (n > 0)
>>> {
>>> if (n %% 2)
>>> phi <- phi %*% X
>>> if (n == 1) break
>>> n <- n %/% 2
>>> X <- X %*% X
>>> }
>>> return(phi)
>>> }
>>>
>>> "Simultaneously" people where looking at the matrix exponential
>>> expm() in the Matrix package,
>>> and some of us had consequently started the 'expm' project on
>>> R-forge.
>>> The main goal there has been to investigate several algorithms
>>> for the matrix exponential, notably the one buggy implementation
>>> (in the 'Matrix' package until a couple of weeks ago, the bug
>>> stemming from 'octave' implementation).
>>> The authors of 'actuar', Vincent and Christophe, notably also
>>> had code for the matrix *power* in a C (building on BLAS) and I
>>> had added an R-interface '%^%' there as well.
>>>
>>> Yes, with the goal to move that (not the matrix exponential yet)
>>> into standard R.
>>> Even though it's not used so often (in percentage of all uses of
>>> R), it's simple to *right*, and I have seen very many versions
>>> of the matrix power that were much slower / inaccurate / ...
>>> such that a reference implementation seems to be called for.
>>>
>>> So, please consider
>>>
>>> install.packages("expm",repos="http://R-Forge.R-project.org")
>>>
>>> -- but only from tomorrow for Windows (which installs a
>>> pre-compiled package), since I found that we had accidentally
>>> broken the package trivially by small changes two weeks ago.
>>>
>>> and then
>>>
>>> library(expm)
>>> ?%^%
>>>
>>>
>>> Best regards,
>>> Martin Maechler, ETH Zurich
>>>
>>>
>>>
>>>
RW> operator? I am implicitly assuming that the benefits of
RW> a native exponentiation routine for Markov chain
RW> evaluation or function generation would outstrip that of
RW> an R solution. Based on my tests so far (comparing it to
RW> a couple of different pure R versions) it does, but I
RW> still there is much room for optimization in my method.
RW> 2) Regarding the code below: Is there a better way to do
RW> the matrix multiplication? I am creating quite a few
RW> copies for this implementation of exponentiation by
RW> squaring. Is there a way to cut down on the number of
RW> copies I am making here (I am assuming that the lhs and
RW> rhs of matprod() must be different instances).
>>>
RW> Any feedback appreciated ! Thanks Rory
>>>
RW> <snip>
>>>
RW> /* Convenience function */ static void
RW> copyMatrixData(SEXP a, SEXP b, int nrows, int ncols, int
RW> mode) { for (int i=0; i < ncols; ++i) for (int j=0; j <
RW> nrows; ++j) REAL(b)[i * nrows + j] = REAL(a)[i * nrows +
RW> j]; }
>>>
RW> SEXP do_matexp(SEXP call, SEXP op, SEXP args, SEXP rho)
RW> { int nrows, ncols; SEXP matrix, tmp, dims, dims2; SEXP
RW> x, y, x_, x__; int i,j,e,mode;
>>>
RW> // Still need to fix full complex support mode =
RW> isComplex(CAR(args)) ? CPLXSXP : REALSXP;
>>>
RW> SETCAR(args, coerceVector(CAR(args), mode)); x =
RW> CAR(args); y = CADR(args);
>>>
RW> dims = getAttrib(x, R_DimSymbol); nrows =
RW> INTEGER(dims)[0]; ncols = INTEGER(dims)[1];
>>>
>>>
RW> if (nrows != ncols) error(_("can only raise square
RW> matrix to power"));
>>>
RW> if (!isNumeric(y)) error(_("exponent must be a
RW> scalar integer"));
>>>
RW> e = asInteger(y);
>>>
RW> if (e < -1) error(_("exponent must be >= -1")); else
RW> if (e == 1) return x;
>>>
RW> else if (e == -1) { /* return matrix inverse via
RW> solve() */ SEXP p1, p2, inv; PROTECT(p1 = p2 =
RW> allocList(2)); SET_TYPEOF(p1, LANGSXP); CAR(p2) =
RW> install("solve.default"); p2 = CDR(p2); CAR(p2) = x; inv
RW> = eval(p1, rho); UNPROTECT(1); return inv; }
>>>
RW> PROTECT(matrix = allocVector(mode, nrows * ncols));
RW> PROTECT(tmp = allocVector(mode, nrows * ncols));
RW> PROTECT(x_ = allocVector(mode, nrows * ncols));
RW> PROTECT(x__ = allocVector(mode, nrows * ncols));
>>>
RW> copyMatrixData(x, x_, nrows, ncols, mode);
>>>
RW> // Initialize matrix to identity matrix // Set x[i *
RW> ncols + i] = 1 for (i = 0; i < ncols*nrows; i++)
RW> REAL(matrix)[i] = ((i % (ncols+1) == 0) ? 1 : 0);
>>>
RW> if (e == 0) { ; // return identity matrix } else
RW> while (e > 0) { if (e & 1) { if (mode == REALSXP)
RW> matprod(REAL(matrix), nrows, ncols, REAL(x_), nrows,
RW> ncols, REAL(tmp)); else cmatprod(COMPLEX(tmp), nrows,
RW> ncols, COMPLEX(x_), nrows, ncols, COMPLEX(matrix));
>>>
RW> copyMatrixData(tmp, matrix, nrows, ncols,
RW> mode); e--; }
>>>
RW> if (mode == REALSXP) matprod(REAL(x_), nrows,
RW> ncols, REAL(x_), nrows, ncols, REAL(x__)); else
RW> cmatprod(COMPLEX(x_), nrows, ncols, COMPLEX(x_), nrows,
RW> ncols, COMPLEX(x__));
>>>
RW> copyMatrixData(x__, x_, nrows, ncols, mode); e
RW> /= 2; }
>>>
RW> PROTECT(dims2 = allocVector(INTSXP, 2));
RW> INTEGER(dims2)[0] = nrows; INTEGER(dims2)[1] = ncols;
RW> setAttrib(matrix, R_DimSymbol, dims2);
>>>
RW> UNPROTECT(5); return matrix; }
>>>
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>>>
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