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Theorems of the alternative for linear algebraic equations and inequalities are considered in this paper. Classical theorems of the alternative, such as Gordan’s theorem, Farkas’ theorem, S...

Boundary-value ordinary differential equations discretized by collocation generally give rise to systems of non-linear algebraic equations. Algorithms containing Newton-like iterations without the known disadvantages of Newton’s method have led to quasi-Newton methods for solving these systems. This paper attempts to provide more efficient approximations to the diagonal matrices which form part of the Jacobian, and presents a quasi-Newton method where the problem is discretized by collocation using cubic splines. The method is applied to several problems and the results are compared with those obtained by some of the other well-known methods.