On the inversion of block tridiagonals without storage constraints

A strategy was developed to permit trade-offs between the number of floating point operations required and the storage requirements for the solution of certain difference problems, such as block tridiagonal systems of equations. This is done by recomputing some intermediate results instead of storing them. Reducing the storage to the square root of the current requirement roughly doubles the number of computations. Reducing the storage more than this tends to make the number of computations prohibitively large. In theory, if m is the order of each sub-matrix in the block tridiagonal matrix, one can solve any linear system with only 5m(2) + 1 temporary storage cells. In many cases m is a constant and quite small. For example, in solving a factored form of the three-dimensional Navier-Stokes equations, the size m of the block tridiagonals is 5. This method lends itself to efficient use on computers with parallel processing or vector processing architectures. On these computers the larger number of floating point operations is more than offset by the decrease in I/O and the increased percentage of vector operations made possible by this algorithm.