##
**Bivariate natural spline smoothing.**
*(English)*
Zbl 0561.41011

Delay equations, approximation and application, Int. Symp. Mannheim/Ger. 1984, ISNM 74, 165-179 (1985).

[For the entire collection see Zbl 0554.00008.]

This paper deals with the problem of smoothing data given on a rectangular grid. More specifically, the authors discuss an efficient algorithm for calculating the well-known bivariate natural splines which minimize a combination of smoothness and goodness of fit, or which minimize smoothness subject to some prescribed goodness of fit. The algorithms are based on a representation using tensor-products of univariate natural B-splines. It is shown that that first problem leads to a convenient linear system of equations of a special tensor form which can be very efficiently solved. The second problem is solved by computing a sequence of solutions of the first problem in an iterative process which adjusts the weight between the measure of smoothness and the measure of goodness of fit.

This paper deals with the problem of smoothing data given on a rectangular grid. More specifically, the authors discuss an efficient algorithm for calculating the well-known bivariate natural splines which minimize a combination of smoothness and goodness of fit, or which minimize smoothness subject to some prescribed goodness of fit. The algorithms are based on a representation using tensor-products of univariate natural B-splines. It is shown that that first problem leads to a convenient linear system of equations of a special tensor form which can be very efficiently solved. The second problem is solved by computing a sequence of solutions of the first problem in an iterative process which adjusts the weight between the measure of smoothness and the measure of goodness of fit.

### MSC:

41A15 | Spline approximation |