Direct methods are proposed to solve Qv = g where Q is a quasi-tridiagonal matrix. Such matrices arise in solving finite difference equations of problems for pure elliptic equations and for the symmetric positive systems of K. 0. Friedrichs. An example is given of the latter type for solving a boundary problem for the Tricomi equation. The methods employ a partitioned decomposition of Q into a product of lower and upper triangular matrices and a criterion is given for the methods to apply. Direct methods seem especially useful where iterative methods are found not feasible. Other methods for special types of Q are proposed which involve the inversion of relatively small matrices.
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