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A well known theorem by Bolzano states that, whenever a continuous real function changes sign in some interval, it must have a zero in this interval. Then-dimensional generalisation of this theorem, due to C. Miranda, is used for the construction of an error estimation procedure for approximate solutions of nonlinear systems of equations. Ein bekannter Satz(More)
Segmented approximations with free knots for a given continuous function are discussed in a general form. This form includes any kind of continuous approximating segments, and a great variety of error criteria, such as anyL s -norm, or the maximum norm of the pointwise error. It is shown that a solution to the problem exists under very general conditions,(More)
The L1 approximation of strictly convex functions by means of first degree splines with a fixed number of knots is studied. The main theoretical results are a system of equations for the knots, which solves the problem, and an estimate of the approximation error. The error estimation allows the determination of bounds for the number of knots needed so that(More)
New a posteriori (computable) upper bounds for theL 2-norms, both ofD(u−v) and ofu−v are proposed, whereu is the exact solution of the boundary value problem $$Au: = - D(pDu) + qu = f, x \in G and u = 0,x \in \partial G$$ andv any approximation of it (D is here the vector of partial derivatives with respect to the components ofx). It is shown that the new(More)
A new error bound for any approximate solutionu of the two-point boundary value problemAy:=−(py′)′+qy=f,y(0)=0, y(1)=0, is proposed. This error bound depends on the deviationAu−fjust like the one which is proportional to ‖Au−f‖2, but in the case of Ritz-Galerkin approximations by cubic splines it behaves asymptotically likeh 3, whereh is the knot distance,(More)
Optimal segmented approximations with free knots for a continuous function have (with the exception of trivial cases) strictly monotonically decreasing error with increasing number of knots, if the error functional is strictly isotonic and the approximation segments belong to the linear space of solutions of a linear homogeneous differential equation. More(More)
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