Current algorithms to simultaneously tridiagonalize a pair of symmetric matrices are impractical for large sparse matrices because they successively update the matrices and so destroy their sparsity. This study describes a new iterative algorithm based on a three term recurrence that can be interrupted midstream much like the well known Lanczos method. While the new algorithm also involves the action of the inverse of a matrix pencil as previous algorithms do, the sparse context means that this step can itself be handled with a classical method based on a Lanczos recurrence for linear systems or other methods meant for sparse problems. Our study is enabling in that it allows researchers to explore using the new algorithm in applications where a partial simultaneously tridiagonalization may be useful but not considered so far because no practical algorithm was available.