How expensive is it to compute the eigenvalues of

Question

How expensive is it to compute the eigenvalues of a matrix?

What is the complexity of the best algorithms?

How long might it take in practice if I have a 1000 x 1000 matrix? I assume it helps if the matrix is sparse?

Are there any cases where the eigenvalue computation would not terminate?

In R, I can compute the eigenvalues as in the following toy example:

m<-matrix( c(13,2, 5,4), ncol=2, nrow=2 )
eigen(m, only.values=1)
$values
[1] 14  3

Does anyone know what algorithm it uses?

Are there any other (open-source) packages that compute the eigenvalue?

Answer 1

Most of the algorithms for eigen value computations scale to big-Oh(n^3), where n is the row/col dimension of the (symmetric and square) matrix.

For knowing the time complexity of the best algorithm till date you would have to refer to the latest research papers in Scientific Computing/Numerical Methods.

But even if you assume the worse case, you would still need at least 1000^3 operations for a 1000x1000 matrix.

R uses the LAPACK routine's (DSYEVR, DGEEV, ZHEEV and ZGEEV) implementation by default. However you could specify the EISPACK=TRUE as a parameter to use a EISPACK's RS, RG, CH and CG routines.

The most popular and good open source packages for eigenvalue computation are LAPACK and EISPACK.