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Summary. This paper describes a fast and numerically stable method for solving the discrete Dirichlet problem for Poisson's equation in case of a rectangle (and mainly, a square). By using a special calculus for difference operators, the system of linear equations is reduced to a block-triangular system such that the diagonal blocks are heavily diagonally dominant. For a standard version of the algorithm, the number of operations and the computing time are proportional to h -~ (h =mesh width). The method is one of total reduction compared with the method of block-cyclic reduction (odd-even reduction) [2], which we describe as a method of partial reduction.-Due to the developed calculus, many generalizations are possible.-In a following part II of the paper, the algorithm and numerical results will be described in detail.