Algorithms for Approximation II. J. C. Mason Professor of Computational Mathematics, Royal Military College of Science, Shrivenham, UK and

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1 Algorithms for Approximation II Based on the proceedings of the Second International Conference on Algorithms for Approximation, held at Royal Military College of Science, Shrivenham, July 1988 Edited by J. C. Mason Professor of Computational Mathematics, Royal Military College of Science, Shrivenham, UK and M. G. Cox Senior Principal Scientific Officer, National Physical Laboratory, Teddington, UK London New York CHAPMAN AND HALL

2 Contents Contributors Preface Part One DEVELOPMENT OF ALGORITHMS 1 1. Spline Approximation 3 Constrained spline approximation of functions and 4 data based on constrained knot removal E. Arge, M. Dcehlen, T. Lyche* and K. Morken Near real-time spline fitting of long sequences of 21 uniformly-spaced data G. T. Anthony^ and M. G. Cox An algorithm for knot location in bivariate least 30 Squares spline approximation M. Bozzinii and F. de Tisi A knot placement strategy for least Squares spline 37 fitting based on the use of local polynomial approximations M. G. Cox, P. M. Harris t and H. M. Jones An algorithm for nonlinear splines with non- 46 negativity constraints G. Opfert Spline curve fitting of digitized contours 54 C. Potieri and C. Vercken A B-spline approximation algorithm for quasi- 62 interpolation or filtering C. Rabuti On knots and nodes for spline interpolation 72 P. W. Smith t

3 Polynomial and Piecewise Polynomial Approximation 79 A basis for certain Spaces of multivariate polynomials 80 and exponentials W. Dahmen* Monotone piecewise cubic data fitting 99 F. N. Fritschi Direct and converse results on simultaneous 107 approximation by the method of Bernstein- Durrmeyer Operators M. Heilmann and M. W. Müller] Orthogonality and approximation in a Sobolev space 117 A. Iserlesi, P. E. Koch, S. P. N0rsett and J. M. Sanz-Serna Piecewise polynomial approximation of polynomial 5 curves M. A. Lachance-\ Calculation of the energy of a piecewise polynomial 4 surface E. Quaki and L. L. Schumaker Interpolation 145 Radial basis function interpolation on an infinite 146 regulär grid M. D. Buhmanni and M. J. D. Powell* The Fourier Operator of even order and its 170 application to an extremum problem in interpolation L. Brutmani On multivariate polynomial interpolation 177 N. Dyni and A. Ron Algorithms for the construction of data dcpcndent 185 triangulations N. Dyn, D. Levin and S. Rippen Algorithms for Computing best parametric cubic 193 interpolation C. Rademacher and K. Schereri 4. Smoothing and Constraint Methods 209 Data fitting by penalized least Squares 210 M. Von Golitschek and L. L. Schumaker*

4 A semiinfinite programming algorithm for 228 constrained best approximation K. W. Bosworthi Inference region for a method of local approximation 236 by using the residuals M. Bozzini and L. Lenarduzzij Complex Approximation 245 Numerical methods for Chebyshev approximation of 246 complex-valued functions G. A. Watson* A fast algorithm for linear complex Chebyshev 265 approximation P. T. P. Tangi Part Two APPLICATIONS Computer Aided Design and Geometrie Modelling 277 Uniform subdivision algorithms for curves and 278 surfaces N. Dyn, J. A. Gregory* and D. Levin Approximation by spheres 296 T. B. Boffeyi, M. G. Cox, L. M. Delves and C. J. Pursglove Interpolation of scattered data on a spherical domain 303 T. A. Foleyi Least Squares best fit geometric elements 311 A. B. Forbes^ Uniform pieeewise approximation on the sphere 320 W. Freedeni and /. C.Mason 7. Applications in Numerical Analysis 335 Approximation theory and numerical linear algebra 336 L. N. Trefethen* An algorithm for Computing minimum norm Solutions 361 of the finite moment problem M. Frontini']', G. Rodriguez and S. Seatzu Numerical Solution of the biharmonic equation using 369 different types of bivariate spline functions R. H. J. Gmelig Meylingt

5 Quadrature Solution of integral equations: a uniform 377 treatment of Fredholm and Volterra equations G. O. Olaofei Increasing the convergence modulus of an asymptotic 387 expansion: an algorithm for numerical differentiation G. Walzt Approximation and parameter estimation in ordinary 395 differential equations J. Williams t 8. Applications in Other Disciplines 405 Applications of discrete Li methods in science and 406 engineering C. Zala and /. Barrodale* Constrained complex approximation algorithms in 424 communication engineering J. C. Mason*, A. E. Trefethen and S. J. Wilde Integration of absolute amplitude from a decibel 449 B-spline fit R. W. Alleni and /. G. Metcalfe A nonlinear least Squares data fitting problem arising 458 in microwave measurement M. G. Cox and H. M. Jonest A complex minimax algorithm for phase-only 466 adaptation in antenna arrays /. C. Mason and S. J. Wilder Part Three CATALOGUE OF ALGORITHMS 477 A catalogue of algorithms for approximation 479 E. Grosse* *Invited Speaker ^Speaker