A criterion for the positivity of a cubic polynomial on a given interval is derived. By means of this result a necessary and sufficient condition is given under which cubicC
1-spline interpolants are nonnegative. Further, since such interpolants are not uniquely determined, for selecting one of them the geometric curvature is minimized. The arising optimization problem is solved numerically via dualization.

Positivity of cubic polynomials on intervals and positive spline interpolation

The paper presents a damped and perturbed Newton-type method for solving linear complementarity problems with positive-semidefinite matricesM. In particular, the following properties hold: all occurring subproblems are linear equations; each subproblem is uniquely solvable without any assumption; every accumulation point generated by the method solves the linear complementarity problem. The additional property ofM to be an R0-matrix is sufficient, but not necessary, for the boundedness of the iterates. Provided thatM is positive definite on a certain subspace, the method converges Q-quadratically.

A Newton-type method for positive-semidefinite linear complementarity problems

A necessary and sufficient criterion is presented under which the property of positivity carry over from the data set to rational quadratic spline interpolants. The criterion can always be satisfied if the occuring parameters are properly chosen.

Positive interpolation with rational quadratic splines

A survey on univariate data interpolation and approximation by splines of given shape

We consider convex interpolation with
cubic 
$C^2$
 splines on
grids built by adding two knots in each subinterval of neighbouring data
sites. The additional knots have to be variable in order to get a chance
to always retain convexity. By means of the staircase algorithm we
provide computable intervals for the added knots such that all knots
from these intervals allow convexity preserving spline interpolation of
 
$C^2$
 continuity.

An always successful method in univariate convex C 2 interpolation

The problem of finding convex spline interpolants with minimal mean curvature leads to a quadratic optimization problem of special structure. In the present note a corresponding dual problem without constraints is derived. Its objective function is piecewise quadratic and therefore admits an effective numerical treatment.

W. Burmeister

Papers

Convex spline interpolants with minimal curvature