$A$ is a $n \times n$ symmetric matrix. I would like to know the time complexity of calculating $r$-largest eigenvalues and vectors.
When we need all eigenvalues and eigenvectors, it means $r=n$, I think the time complexity is known as $O(n^3)$. I would like to know the case of $r<n$.
Restricting the number of eigenvalues doesn't buy you much. You're really looking at slightly-worse than $O(m^{\omega})$, where $\omega$ is the exponent of Matrix Multiplication.
Note that what you get is an approximation of the eigenvalues, due to the nature of floating-point arithmetic (i.e., writing down the eigenvalues in binary on a Turing Machine).
[1] https://cstheory.stackexchange.com/questions/2611/complexity-of-finding-the-eigendecomposition-of-a-matrix
[2] https://arxiv.org/abs/1912.08805