And Descriptions of Tables and Books

The algorithms and programs in this handy book deal almost exclusively with cubic splines. A short introduction stresses the maximum "smoothness" of "natural" splines. The next chapter reviews the numerical solution of tridiagonal linear systems by Gauss elimination and of block tridiagonal linear systems by relaxation. Chapter 3 offers programs for cubic spline interpolation with various (separated) boundary conditions. The next chapter deals with periodic splines and with the interpolation of curves by interpolation of their component functions with respect to some parametrization. Also, various "smoothing methods" are described, all of which turn out to be interpolation methods based on piecewise cubic Hermite interpolation with the slopes at the interpolation points estimated locally. The author's own construction of an area matching cubic spline approximation to histograms follows. Quintic spline interpolation is the subject of Chapter 5, and block underrelaxation (!) is used for the solution of the appropriately ordered linear system for the determination of the interpolant. The interpolating quintic spline fares badly in the comparison with the cubic spline interpolant as given in five accompanying pictures. The cubic smoothing spline of (Schoenberg and) Reinsch is introduced in the next chapter without any motivation at all as a cubic spline whose deviation at each data point (and knot) is proportional to the jump in its third derivative across the knot, with proportionality factor at each knot to be determined. Block underrelaxation is used in its construction, and the user is expected to choose the proportionality factors, perhaps interactively. Chapter 7 gives a very nice summary of the author's efforts to obtain simple interpolating functions (and curves) which are less wiggly than the cubic spline is at times. Starting with Schweikert's spline in tension, the author discusses a general class of methods in which each cubic piece is replaced by a more flexible function, involving an exponential perhaps or a rational function, and offers programs for the construction of such "generalized cubic spline" interpolants. Again, much is left to interactive user efforts and the simple and attractive alternative of sticking to ordinary cubic splines and combatting extraneous inflection points by the suitable addition of knots instead is not mentioned at all. The last chapter uses tensor products of the various univariate approximation schemes to produce interpolants or approximants to data on a rectangular grid. An English translation, by W. D. Hoskins and H. W. Sager, with the title Spline