Is there a way to calculate only first few eigen vectors, say first 1o instead of all, which may speed up the computation

If you are dealing with positive definite matrix, then you can try doing, in scilab:

R=chol(X); // so X=R'*R
[U,S,V,rk] = svd(R,tol); // so R=U*S*V' and V*(S^2)*V' aproximates X

This way S^2 is diagonal, with rk nonzero elements, which are the eigenvalues of X greater than tol^2.

I'm not sure if this is fast or not, but it is worth a try. You can't control the number of eigenvalues calculated, but you are getting the bigger ones.

I'm right now programming in c++ using eigen library, I want to calculate first few eigen vectors for a matrix of size 2500x2500 and is pretting slow
is there a way doing this using this library in c++.Thanks in advance!

I don't think this is possible. There is only an option to calculate eigenvectors or not.
I used MKL/LAPACK for a similar case where I needed only the first (minimum) eigenvalue.

With a little programming, you might implement the power method: repeatedly multiplying
an arbitrary vector by your matrix A and normalizing the result will get you the largest eigenvalue
lambda1 (and associated eigenvector v1). Replacing A by A - lambda1*v1*v1' and using the
same procedure will get you the second, etc.

Easy to program and will be much faster if you only need a few eigenvectors/eigenvalues.

Have a look at this thread for a starting code for the power method: viewtopic.php?f=74&t=108033

You can use the ARPACK wrapper in Eigen/unsupported. It works with dense
matrices as well, and lets you compute the first k eigenvalues/vectors.

You'll need ARPACK installed on your system.