A general solution for a special tri-diagonal linear system

Abstract

The tri-diagonal linear system with coefficients of (–1, 2, –1) along the bandwidth and the transposed vector of (1k, 2k, …., nk) as the right hand sides is solved in a general form linking with the classical Faulhaber’s Formula. We derive the general solution for any k and conduct computational experiments compared with using the direct Thomas’s algorithm. The result shows that in the case where k = 1 to 16, the general solution clearly performs more efficiently. For cases where k = 1, the results also indicate that as the problem size continuously expands, there will always be a point at which the general solution processes faster.