A Practical Guide to Splines

by Carl R. de Boor

This book is based on the author's experience with calculations involving polynomial splines. It presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines. After two chapters summarizing polynomial approximation, a rigorous discussion of elementary spline theory is given involving linear, cubic and parabolic splines. The computational handling of piecewise polynomial functions (of one variable) of arbitrary order is the subject of chapters VII and VIII, while chapters IX, X, and XI are devoted to B-splines. The distances from splines with fixed and with variable knots is discussed in chapter XII. The remaining five chapters concern specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting. The present text version differs from the original in several respects. The book is now typeset (in plain TeX), the Fortran programs now make use of Fortran 77 features. The figures have been redrawn with the aid of Matlab, various errors have been corrected, and many more formal statements have been provided with proofs. Further, all formal statements and equations have been numbered by the same numbering system, to make it easier to find any particular item. A major change has occured in Chapters IX-XI where the B-spline theory is now developed directly from the recurrence relations without recourse to divided differences. This has brought in knot insertion as a powerful tool for providing simple proofs concerning the shape-preserving properties of the B-spline series.

372 pages, Paperback

First published August 26, 1994

About the author

Carl-Wilhelm Reinhold de Boor is a German-American mathematician and professor emeritus at the University of Wisconsin–Madison.

Having earned only a high school diploma after 3 and a half years of study at Hamburg University, de Boor entered Harvard as a graduate student of mathematics. After working for a year as a research assistant to Garrett Birkhoff, he went to work for General Motors Research in Warren, Michigan, where he met splines. He received his first postgraduate degree, a Ph.D. from the University of Michigan, in 1966, and then became an assistant professor at Purdue University. In 1972, he accepted a position as professor of mathematics and computer science at the University of Wisconsin–Madison, working out of the UW's Army Math Research Center, which had recently been bombed in opposition to the Vietnam War.