A 100,0000 by 100,000 symmetric dense matrix in single precision requires 20 gigabytes of memory (storing only the upper triangle) or 40 gigabytes of memory for double precision. Thus it is too large to fit within the memory of available GPU's.
In order to solve this problem using GPU acceleration you'd have to develop an algorithm that sends smaller chunks of work to the GPU. Copying parts of the matrix into the GPU in each iteration is unlikely to work well. Rather, you could try to use multiple GPU's in parallel, with each GPU storing part of the matrix throughout the algorithm. For example, you could use four of the NVIDIA Tesla K20X's to store your single precision matrix.
The appropriate algorithms to consider for this kind of problem are iterative schemes that use matrix-vector multiplications. Start by looking at the algorithms implemented in ARPACK. Since ARPACK can use BLAS routines for the matrix-vector multiplications, you should be able to simply link ARPACK with some GPU implementation of the BLAS to get a working code.