Karl:

library(Matrix)

## Matrix Exponential

source(system.file("test-tools.R", package = "Matrix"))

## e ^ 0 = 1  - for matrices:
assert.EQ.mat(expm(Matrix(0, 3,3)), diag(3), tol = 0)# exactly
## e ^ diag(.) = diag(e ^ .):
assert.EQ.mat(expm(as(diag(-1:4), "dgeMatrix")), diag(exp(-1:4)))
set.seed(1)
rE <- replicate(100,
            { x <- rlnorm(12)
              relErr(as(expm(as(diag(x), "dgeMatrix")),
                        "matrix"),
                     diag(exp(x))) })
stopifnot(mean(rE) < 1e-15,
          max(rE)  < 1e-14)
summary(rE)

## Some small matrices

m1 <- Matrix(c(1,0,1,1), nc = 2)
e1 <- expm(m1)
assert.EQ.mat(e1, cbind(c(exp(1),0), exp(1)))

m2 <- Matrix(c(-49, -64, 24, 31), nc = 2)
e2 <- expm(m2)
## The true matrix exponential is 'te2':
e_1 <-  exp(-1)
e_17 <- exp(-17)
te2 <- rbind(c(3*e_17 - 2*e_1, -3/2*e_17 + 3/2*e_1),
             c(4*e_17 - 4*e_1, -2  *e_17 + 3  *e_1))
assert.EQ.mat(e2, te2, tol = 1e-13)
## See the (average relative) difference:
all.equal(as(e2,"matrix"), te2, tolerance = 0) # 1.48e-14 on "lynne"

## The ``surprising identity''      det(exp(A)) == exp( tr(A) )
## or                           log det(exp(A)) == tr(A) :
stopifnot(all.equal(c(determinant(e2)$modulus), sum(diag(m2))))

## a very simple nilpotent one:
(m3 <- Matrix(cbind(0,rbind(6*diag(3),0))))# sparse
stopifnot(all(m3 %*% m3 %*% m3 %*% m3 == 0))# <-- m3 "^" 4 == 0
e3 <- expm(m3)
E3 <- expm(Matrix(m3, sparse=FALSE))
s3 <- symmpart(m3) # dsCMatrix
es3 <- expm(s3)
e3. <- rbind(c(1,6,18,36),
	     c(0,1, 6,18),
	     c(0,0, 1, 6),
	     c(0,0, 0, 1))
stopifnot(is(e3, "triangularMatrix"),
	  is(es3, "symmetricMatrix"),
	  identical(e3, E3),
	  identical(as.mat(e3), e3.),
	  all.equal(as(es3,"generalMatrix"),
		    expm(as(s3,"generalMatrix")))
	  )


## This used to be wrong {bug in octave-origin code}:
M6 <- Matrix(c(0, -2, 0, 0, 0, 0,
              10, 0, 0, 0,10,-2,
              0,  0, 0, 0,-2, 0,
              0, 10,-2,-2,-2,10,
              0,  0, 0, 0, 0, 0,
              10, 0, 0, 0, 0, 0), 6, 6)

exp.M6 <- expm(M6)
as(exp.M6, "sparseMatrix")# prints a bit more nicely
stopifnot(all.equal(t(exp.M6),
		    expm(t(M6)), tol = 1e-12),
          all.equal(exp.M6[,3], c(0,0,1,0,-2,0), tolerance = 1e-12),
          all.equal(exp.M6[,5], c(0,0,0,0, 1,0), tolerance = 1e-12),
          all(exp.M6[3:4, c(1:2,5:6)] == 0)
          )

cat('Time elapsed: ', proc.time(),'\n') # for ``statistical reasons''