Very-high-precision solutions of a class of Schrödinger type equations

We investigate a method to solve a class of Schrödinger type equation eigenvalue problems numerically to very high precision P (from thousands to a million of decimals). The memory requirement, and the number of high-precision algebraic operations, of the method scale essentially linearly with P   when only eigenvalues are computed. However, since the algorithms for multiplying high-precision numbers scale at a rate between P1.6P1.6 and PlogPloglogP, the time requirement of our method increases somewhat faster than P2P2.