Spline Matrices and Their Applications to Some Higher Order Methods for Boundary Value Problems

Gregory, John; Zeman, Marvin

doi:10.1137/0725027

Cited by 7 publications

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“…This is a linear system AhEh = /i 3 Q + 0(/i 5 ), where Ah is a block tridiagonal matrix, Eh is the m(N -1) error vector, and Q(a k ) is the fcth component of Q, described above. Extensions of the author's quadratic form theory [1] and generalizations of results for ordinary differential equations by the author and Zeman [3] lead to an error of the form Hü^l^ < C/i 3 / 2 . Using these results it may…”

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Numerical methods for extremal problems in the calculus of variations and optimal control theory

Gregory¹

1988

Bull. Amer. Math. Soc.

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Introduction. New, general methods are given to find numerical solutions for extremal problems in the calculus of variations and optimal control theory. Theoretical methods are derived and used to establish pointwise a priori error estimates with maximum error at the node point, ||e||oo, equal to 0(h 2 ) and a Richardson error of 0(h 4 ). This is done under the weak assumption that there are no conjugate points on the interval and not under the usual convexity assumptions.Of practical interest is that these methods (i) are very easy to implement, (ii) hold for well-defined mixtures of initial value and boundary value problems, (iii) use multipliers, and not ill-conditioned penalty methods, for both equality and inequality constraints, in a natural, efficient manner, and (iv) are applicable to transversality, type-minimal time problems.The heart of these methods is the algorithm (4) and the a priori estimates in Theorem 2 for the m-dependent variable problem in the calculus of variations given below. Once this is established we quote Hestenes [5] and show that very general optimal control problems can be easily reformulated and solved as calculus of variations problems.

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