[R] matrix of eigenvalues

[Douglas Bates]

Spencer Graves wrote:
>      I think you need the Schur decomposition, which seems currently not 
> to be available in R.  The documentation for the the Matrix package 
> describes a "Schur" function, but it's not available in the current 
> Matrix package, as mentioned in my post on this issue yesterday (subj:  
> Schur decomposition).
>      Moreover, Lindsey's mexp in rmutils won't work either:
>  > A = matrix(cbind(c(-1,1),c(-4,3)),nrow=2)
>  > mexp(A)
> Error in solve.default(z$vectors) : system is computationally singular: 
> reciprocal condition number = 4.13756e-017
>  > mexp(A, "series")
> Error in t * x : non-numeric argument to binary operator
>  >
>      Have you considered adding a little noise: > mexp(A+1e-6*rnorm(4))
>          [,1]       [,2]
> [1,] -2.718284 -10.873139
> [2,]  2.718284   8.154859
>  > mexp(A+1e-6*rnorm(4))
>                        [,1]                     [,2]
> [1,] -2.718284-1.060041e-12i -10.873126-1.575184e-12i
> [2,]  2.718284+7.492895e-13i   8.154846+1.578515e-12i
>  > mexp(A+1e-6*rnorm(4))
>                        [,1]                     [,2]
> [1,] -2.718284+6.146195e-13i -10.873127+1.586731e-12i
> [2,]  2.718284-4.004574e-13i   8.154847-8.515411e-13i
>  > mexp(A+1e-6*rnorm(4))
>                        [,1]                     [,2]
> [1,] -2.718283+3.077538e-13i -10.873130+2.433609e-13i
> [2,]  2.718283-3.197442e-14i   8.154847-2.782219e-13i
>  >
>      hope this helps.  spencer graves
> (I believe this is described in one of Richard Bellman's matrix analysis 
> book.)

I rather suspected that that original question was about the matrix 
exponential and linear systems of differential equations.  The solution 
to such a system can only be written using the matrix exponential for 
diagonalizable systems and this is the classic example of a 
nondiagonalizable system.

Calculation of the matrix exponential is an operation that seems 
straightforward in theory and can be very difficult in practice.  Moler 
and van Loan have a classic paper on "Nineteen Dubious Ways to Calculate 
the Matrix Exponential" which I would recommend reading.

In the proposal for updated LAPACK/ScaLAPACK libraries, Demmel and 
Dongarra state that they will include code for the matrix exponential in 
their proposed package.