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On the generalized Ball bases

On the generalized Ball bases. Speaker: Chengming Zhuang Oct.23 Advances in Computational Mathematics (2006) Jorge Delgado ,Juan Manuel Peña. Authors: University of Zaragoza( 萨拉戈萨 ).

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On the generalized Ball bases

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  1. On the generalized Ball bases Speaker: Chengming Zhuang Oct.23 Advances in Computational Mathematics (2006) Jorge Delgado ,Juan Manuel Peña

  2. Authors: University of Zaragoza(萨拉戈萨) • [1] J.Delgado, J.M.Peña, A shape preserving representation with an evaluation algorithm of linear complexity, CAGD 2003, 20, 1-10 • [2] J.Delgado, J.M.Peña, Progressive iterative approximation and bases with the fastest convergence rates, CAGD 2007, 24, 10-18 • [3] J. Delgado and J.M. Peña,Monotonicity preservation of some polynomial and rational representations, in: Information Visualisation (IEEE Computer Society, Los Alamitos, CA, 2002) pp. 57–62. • [4] J.M.Peña, B-splines and optimal stability, Math. Comp. 66 (1997) 1555–1560. • [5] J.M.Peña, Error analysis of algorithms for evaluating Bernstein–Bézier type multivariate polynomials, in: Curves and Surfaces Design, eds. P.J. Laurent, P. Sablonnière and L.L. Schumaker (Vanderbilt Univ. Press, Nashville, TN, 2000) pp. 315–324.

  3. Introduction • Cubic polynomials Ball basis:

  4. Wang Ball System • Wang-Ball[1989]: • In addition, if m is even, and, if m is odd,

  5. Said-Ball basis • Said-Ball[1987]: • If m is even

  6. Outline • Shape preserving properties • Boundary tangent property, • Strictly monotonicity preserving • Backward error analysis of the evaluation algorithms • Conditioned numbers

  7. Shape preserving properties • Control points : • is called the control polygon of curve • is a blending system: • Nonnegative • Convex hull property

  8. Shape preserving properties • Collocation matrix of at is given by: • (u0, . . . , un) is blending if and only if all its collocation matrices are stochastic • A matrix is totally positive (TP) if all its minors are nonnegative. • A system of functions is TP when all its collocation matrices are TP.

  9. Shape preserving properties • Proposition 1:The Wall-Ball basis and the Said-Ball basis satisfy the boundary tangent property.

  10. Shape preserving properties • Proposition 2:The Wang-Ball basis is TP if and only if • Proof : By [6], the basis is TP if and only if the matrix is TP.

  11. Monotonicity preserving • is monotonicity preserving if for any , the function is increasing. • Lemma 1. • (1) is monotonicity preserving if and only if is a constant function and are increasing functions. • (2) is strictly monotonicity preserving if and only if it is monotonicity preserving and is a strictly increasing function.

  12. Monotonicity preserving • Theorem 1. The Wang-Ball basis is strictly monotonicity preserving for all • Proof: By lemma 3.3 of [10], it is sufficient to prove that, If m is odd:

  13. Theorem 2. The Wang-Ball basis is geometrically convexity preserving if and only if • Weak Chebyshev: ‘s square collocation matrices have nonnegative determinant. • A strictly monotonicity preserving system ia called geometrically convexity preserving if for . • :blending strictly monotonicity preserving system. is geometrically convexity preserving if and only if is a weak Chebyshev system(i < j). (by [5]) • For m >=4, the determinant of at 0<0.1<0.5 is -0.0008.

  14. Proposition 3. The Said-Ball basis is NTP. By theorem 1 of [15], the result holds for odd m. Where ,A is TP By 3.1 of [1], it is also TP;

  15. Theorem 3. All the rational Said-Ball basis obtained from the Said-Ball basis as with positive weights are geometrically convexity preserving. • Said-Ball basis is NTP; • By corollary 4.6 of [5], it is sufficient to prove Said-Ball basis is strictly monntonicity preserving. • Since , are increasing, is strictly increasing

  16. Matrix of change of basis • Bernstein basis multiplied by certain nonnegative matrices and :

  17. Matrix of change of basis • Proposition 4: The Wang-Ball basis and the Said-Ball basis are related, for ,by:

  18. If m is odd: By [26], ‘s degree less than or equal to m-1 , use the reduction for Said-Ball curve, we have:

  19. Lemma 2. If , where A is a nonnegative matrix. Then A is stochastic.

  20. Stability properties • Standard notations: • Given the computed element in floating point arithmetic will be denoted by either u: the unit roundoff op: any of the elementary operations • Given define:

  21. Stability properties • Remark 1. • VS basis:

  22. Stability properties • Theorem 4. Consider Wang-ball basis, Said-Ball basis, VS basis’s evaluation algorithms, if the computed value satisfies : If m is odd: If m is even:

  23. Stability properties • Given , where is called a condition number for the evaluation of f (x) with the basis u • By corollary 2.2 of [18] the forward error bound for evaluation algorithms: • by lemma 2.1 of [22], if A is nonnegative:

  24. Example • Consider: sp and dp mean single and double

  25. Conclusions • Wang–Ball and theSaid–Ball bases present lower computational complexity than the de Casteljau algorithm • Shape preserving properties of the Said–Ball basis • Wang–Ball bases are satisfy the boundary tangent property, strictly monotonicity preserving, not satisfy further shape preserving properties for m >= 4 • Backward error analysis of the evaluation algorithms • Said–Ball basis is better conditioned (and so better root conditioned) than the Wang–Ball basis.

  26. References • [1] T. Ando, Totally positive matrices, Linear Algebra Appl. 90 (1987) 165–219. • [2] A.A. Ball, CONSURF, Part I: Introduction to conic lifting title, Comput. Aided Design 6 (1974) 243– • 249. • [3] A.A. Ball, CONSURF, Part II: Description of the algorithms, Comput. Aided Design 7 (1975) 237– • 242. • [4] A.A. Ball, CONSURF, Part III: How the program is used, Comput. Aided Design 9 (1977) 9–12. • [5] J.M. Carnicer, M. Garcia-Esnaola and J.M. Peña, Convexity of rational curves and total positivity, • J. Comput. Appl. Math. 71 (1996) 365–382. • [6] J.M. Carnicer and J.M. Peña, Shape preserving representations and optimality of the Bernstein basis, • Adv. Comput. Math. 1 (1993) 173–196. • [7] J.M. Carnicer and J.M. Peña, Monotonicity preserving representations, in: Curves and Surfaces in • Geometric Design, eds. P.J. Laurent, A. Le Méhauté and L.L. Schumaker (A.K. Peters, Boston, 1994) • pp. 83–90. • [8] N. Dejdumrong and H.N. Phien, Efficient algorithms for Bezier curves, Comput. Aided Geom. Design • 17 (2000) 247–250. • [9] N. Dejdumrong, H.N. Phien, H.L. Tien and K.M. Lay, Rational Wang–Ball curves, Internat. J. Math.

  27. References • Educ. Sci. Technol. 32 (2001) 565–584. • [10] J. Delgado and J.M. Peña,Monotonicity preservation of some polynomial and rational representations, • in: Information Visualisation (IEEE Computer Society, Los Alamitos, CA, 2002) pp. 57–62. • [11] G. Farin, Curves and Surfaces for Computer Aided Geometric Design, 4th edn (Academic Press, San • Diego, CA, 1996). • [12] R.T. Farouki and T.N.T. Goodman, On the optimal stability of the Bernstein basis, Math. Comp. 65 • (1996) 1553–1566. • [13] R.T. Farouki and V.T. Rajan, On the numerical condition of polynomials in Bernstein form, Comput. • Aided Geom. Design 4 (1987) 191–216. • [14] M. Gasca and C.A. Micchelli, Total Positivity and Its Applications (Kluwer Academic Publ., Dordrecht, • 1996). • [15] T.N.T. Goodman and H.B. Said, Shape preserving properties of the generalised Ball basis, Comput. • Aided Geom. Design 8 (1991) 115–121. • [16] W. Guojin and C. Min, New algorithms for evaluating parametric surface, Progress in Natural Science • 11 (2001) 142–148. • [17] N.J. Higham, Accuracy and Stability of Numerical Algorithms (SIAM, Philadelphia, PA, 1996). • [18] E. Mainar and J.M. Peña, Error analysis of corner cutting algorithms, Numer. Algorithms 22 (1999) • 41–52. • [19] J.M. Peña, B-splines and optimal stability, Math. Comp. 66 (1997) 1555–1560.

  28. References • [20] J.M. Peña, Shape Preserving Representations in Computer Aided-Geometric Design (Nova Science • Publishers, Commack, NY, 1999). • [21] J.M. Peña, Error analysis of algorithms for evaluating Bernstein–Bézier type multivariate polynomials, • in: Curves and Surfaces Design, eds. P.J. Laurent, P. Sablonnière and L.L. Schumaker (Vanderbilt • Univ. Press, Nashville, TN, 2000) pp. 315–324. • [22] J.M. Peña, On the optimal stability of bases of univariate functions, Numer.Math. 91 (2002) 305–318. • [23] H.B. Said, Generalized Ball curve and its recursive algorithm, ACM. Trans. Graph. 8 (1989) 360–371. • [24] L.L. Schumaker andW. Volk, Efficient evaluation of multivariate polynomials, Comput. Aided Geom. • Design 3 (1986) 149–154. • [25] H. Shi-Min,W. Guojin and S. Jiaguang, A type of triangular ball surface and its properties, J. Comput. • Sci. Technol. 13 (1998) 63–72. • [26] H. Shi-Min, W. Guo-Zhao and J. Tong-Guang, Properties of two types of generalized Ball curves, • Comput. Aided Design 28 (1996) 125–133. • [27] H.L. Tien, D. Hansuebsai and H.N. Phien, Rational Ball curves, Internat. J. Math. Educ. Sci. Technol. • 30 (1999) 243–257. • [28] G.J. Wang, Ball curve of high degree and its geometric properties, Appl. Math. J. Chinese Univ. 2 • (1987) 126–140. • [29] J.H. Wilkinson, The evaluation of the zeros of ill-conditioned polynomials, Parts I and II, Numer. • Math. 1 (1959) 150–166, 167–180. • [30] J.H. Wilkinson, Rounding Errors in Algebraic Processes, Notes on Applied Science, Vol. 32 (Her • Majesty’s Stationery Office, London, 1963).

  29. The End!Thank you!

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