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**Curve and surface fitting with splines.**
*(English)*
Zbl 0782.41016

Oxford: Clarendon Press. xvii, 285 p. (1993).

The problem of smoothing of an observational data occurs frequently in many problems that arise in science and engineering. The work by Dierckx concentrates on numerical algorithms for curve and surface fitting with the aid of univariate and tensor product splines. The outline of this book is as follows.

Part I (Chapters 1-2) is devoted to the discussion of basic properties of spline functions with special emphasis on \(B\)-splines and tensor product splines. Algorithms for calculating with these functions are also included.

Part II (Chapters 3-7) deals with curve fitting algorithms. An introduction to curve fitting is given in Chapter 3. The least-squares and smoothing techniques are discussed in Chapters 4 and 5, respectively. Therein, the knot placing strategy and the choice of a smoothing parameter are presented as well. Smoothing with periodic and parametric splines is the main topic of Chapter 6. Shape preserving approximation, with special emphasis on locally convex/concave spline approximants, is described in Chapter 7. Using the least-squares criterion of optimality of the spline fitting function the author demonstrates how the Theil-Van de Panne algorithm can be utilized to compute the spline function in question. Various techniques for surface fitting are discussed in Part III (Chapters 8-12). In Chapter 8 the author gives a brief review of the existing methods which are either constructive or variational in their nature. The surface fitting algorithms for the data on a rectangular grids are presented in the next two chapters. The fitting surfaces are the bivariate tensor product splines. These methods can be regarded as generalizations of those designed for the univariate fitting functions. Problems involving the incomplete grids or extremely large data sets are addressed briefly. Algorithms for the surface fitting for the data on certain nonrectangular meshes are presented in Chapters 11 and 12. Therein the interested reader will find a method for the surface reconstruction from planar contours. Finally, Part IV (Chapter 13) contains a short description of the FITPACK software package for computing smoothing splines curves and surfaces. Among the main features of this library are portability, transparency and modularity.

A distinguished feature of this book is its lucid presentation. Dierckx’s choice of examples is excellent. The discussion is clear and complete enough to serve as an introduction for those beginning a study of spline fitting algorithms. The work is self-contained except that the author presuppose a familiarity with numerical analysis and some linear algebra.

Part I (Chapters 1-2) is devoted to the discussion of basic properties of spline functions with special emphasis on \(B\)-splines and tensor product splines. Algorithms for calculating with these functions are also included.

Part II (Chapters 3-7) deals with curve fitting algorithms. An introduction to curve fitting is given in Chapter 3. The least-squares and smoothing techniques are discussed in Chapters 4 and 5, respectively. Therein, the knot placing strategy and the choice of a smoothing parameter are presented as well. Smoothing with periodic and parametric splines is the main topic of Chapter 6. Shape preserving approximation, with special emphasis on locally convex/concave spline approximants, is described in Chapter 7. Using the least-squares criterion of optimality of the spline fitting function the author demonstrates how the Theil-Van de Panne algorithm can be utilized to compute the spline function in question. Various techniques for surface fitting are discussed in Part III (Chapters 8-12). In Chapter 8 the author gives a brief review of the existing methods which are either constructive or variational in their nature. The surface fitting algorithms for the data on a rectangular grids are presented in the next two chapters. The fitting surfaces are the bivariate tensor product splines. These methods can be regarded as generalizations of those designed for the univariate fitting functions. Problems involving the incomplete grids or extremely large data sets are addressed briefly. Algorithms for the surface fitting for the data on certain nonrectangular meshes are presented in Chapters 11 and 12. Therein the interested reader will find a method for the surface reconstruction from planar contours. Finally, Part IV (Chapter 13) contains a short description of the FITPACK software package for computing smoothing splines curves and surfaces. Among the main features of this library are portability, transparency and modularity.

A distinguished feature of this book is its lucid presentation. Dierckx’s choice of examples is excellent. The discussion is clear and complete enough to serve as an introduction for those beginning a study of spline fitting algorithms. The work is self-contained except that the author presuppose a familiarity with numerical analysis and some linear algebra.

Reviewer: E.Neuman (Carbondale)

### MSC:

41A15 | Spline approximation |

65D10 | Numerical smoothing, curve fitting |

41-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to approximations and expansions |