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 ) [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, [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– [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.

Introduction Cubic polynomials Ball basis:

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

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

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

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

Shape preserving properties Collocation matrix of at is given by: (u 0,..., u n ) 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.

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

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.

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.

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:

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

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;

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

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

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

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

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

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:

Stability properties Remark 1. VS basis:

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:

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:

Example Consider: sp and dp mean single and double

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.

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