Constrained approximation by splines with free knots

In this paper, a method that combines shape preservation and least squares approximation by splines with free knots is developed. Besides the coefficients of the spline a subset of the knot sequence, the so-calledfree knots, is included in the optimization process resulting in a nonlinear least squares problem in both the coefficients and the knots. The original problem, a special case of aconstrained semi-linear least squares problem, is reduced to a problem that has only the knots of the spline as variables. The reduced problem is solved by a generalized Gauss-Newton method. Special emphasise is given to the efficient computation of the residual function and its Jacobian. Dedicated to our colleague and teacher Prof. Dr. J. W. Schmidt on the occasion of his 65th birthday Research of the first author was supported by Deutsche Forschungsgemeinschaft under grant Schm 968/2-1,2-2.