The recursive doubling algorithm as developed by Stone can be used to solve a tridiagonal linear system of size n on a parallel computer with n processors using O(log n) parallel arithmetic steps. In this paper, we give a limited processor version of the recursive doubling algorithm for the solution of tridiagonal linear systems using O(n / p + log p) parallel arithmetic steps on a parallel computer with p < n processors. The main technique relies on fast parallel prefix algorithms, which can be efficiently mapped on the hypercube architecture using the binary-reflected Gray code. For p ≪ n this algorithm achieves linear speedup and constant efficiency over its sequential implementation as well as over the sequential LU decomposition algorithm. These results are confirmed by numerical experiments obtained on an Intel iPSC/d5 hypercube multiprocessor.