[R] eigenvalues for a sparse matrix

Tamas Papp tpapp at axelero.hu
Wed Apr 7 11:44:20 CEST 2004


I have the following problem.  It has two parts.

1. I need to calculate the stationary probabilities of a Markov chain,
eg if the transition matrix is P, I need x such that

xP = x

in other words, the left eigenvectors of P which have an eigenvalue of

Currently I am using eigen(t(P)) and then pick out the vectors I need.
However, this seems to be an overkill (I only need a single vector!)
and takes a lot of time -- P is 1176 x 1176!  Is there a faster way?

2. In fact, P has a structure: it comes from the solution of a
discrete dynamic optimzation problem.  There are exogenous (X) and
endogenous (N) states, and I have a policy function X x N -> N, which
gives the choice of the agent for any (x,n) in (X,N).  X has an
exogenous transition matrix.  I use the following function to build
the "global" transition matrix:

globalTransition <- function(U, modelenv) {
  G <- matrix(0, modelenv$Nn*modelenv$Xn, modelenv$Nn*modelenv$Xn)
                                        # this matrix is very sparse
  for (i in 1:modelenv$Nn) {
    r <- (i - 1)*modelenv$Xn            # start at this row
    for (j in 1:modelenv$Xn) {
      G[r+j, 1:modelenv$Xn+modelenv$Xn*(U[j,i]-1)] <-
        modelenv$Xtrans[j, 1:modelenv$Xn]

Nn and Xn are the number of engo- and exogenous states, U is the said
policy function in a matrix form, and Xtrans is the transition matrix
of exogenous states.

The row (and column) indexes of G run like this: 

index Nn Xn
1     1  1
2     1  2
3     1  3
Xn    1  Xn
Xn+1  2  1
Nn*Xn Nn Xn

As you can see from the above function, each row contains just a few
nonzero elements in columns Xn*(U[j,i]-1)+1, ...; this means that the
agent chose the endogenous state n = U[j,i].



Tamás K. Papp
E-mail: tpapp at axelero.hu
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