On the generalized Ball bases

Speaker: Chengming Zhuang

Oct.23Advances 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, 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.

Introduction

Cubic polynomials Ball basis:

2 2 2 2(1 ) , 2 (1 ) , 2 (1 ),t t t t t t

Wang Ball System

Wang-Ball[1989]:

In addition, if m is even,

and, if m is odd,

2

2

( ) 2 (1 ) 0 / 2 1

( ) 2 (1 ) ( 1) / 2 1

m i i ii

m m i m i m ii

a t t t i m

a t t t m i m

/ 2 / 2 / 2/ 2 ( ) 2 (1 )m m m m

ma t t t

( 1) / 2 ( 1) / 2 ( 1) / 2( 1) / 2

( 1) / 2 ( 1) / 2 ( 1) / 2( 1) / 2

( ) 2 (1 )

( ) 2 (1 )

m m m mm

m m m mm

a t t t

a t t t

Said-Ball basis Said-Ball[1987]:

If m is even

/ 2 1

/ 2 1

/ 2( ) (1 ) 0 ( 1) / 2

/ 2( ) (1 ) / 2 1

m i mi

m m ii

m is t t t i m

i

m m is t t t m i m

m i

/ 2 / 2/ 2 ( ) (1 )

/ 2m m mm

ms t t t

m

Outline

Shape preserving properties Boundary tangent property, Strictly monotonicity preserving Backward error analysis of the evaluation al

gorithms Conditioned numbers

Shape preserving properties

Control points : is called the control polygon of curve

is a blending system: Nonnegative

Convex hull property

0 , , nP P

0 nP P

0

( ) ( )n

i ii

t Pu t

0 1( , , , )nu u u

0

( ) 1n

ii

u t

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.

0 1( ( ), ( ), , ( ))nu t u t u t

0 mt t

00 ; 0

0

, ,: ( ( ))

, ,n

j i i m j nm

u uM u t

t t

Shape preserving properties Proposition 1:The Wall-Ball basis and the Said-

Ball basis satisfy the boundary tangent property.0( , , )m m

ma a

0( , , )m mms s

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.

0( , , )m mma a

3m

0( , , )m mma a mA

0 0( , , ) ( , , )m m m mm m ma a b b A

Monotonicity preserving is monotonicity preserving if for any , the function is incre

asing. 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.

0 1( , , , )nu u u

0 1 nin R 0

n

i ii

u

:m

i jj i

v u

0 1( , , , )nu u u1, , mv v

0 1( , , , )nu u u

1

m

ii

v

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:

0( , , )m mma a

2m

/ 2 / 2 1

( 1) / 2

sin ( )

sin ( )

m mm m

mm

v and v are increa g functions m is even

v is increa g function m is odd

1 1 1 1 1( 1) / 2 ( 1) / 2 ( 1) / 2 ( 1) / 2 ( 1) / 2 ( 1) / 2

1 1( 1) / 2 ( 1) / 2

' ( ) ' ' '

'

m m m m m mm m m m m m

m mm m

v ta v a ta v

a tv

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.

0( , , )m mma a

2 3m

0( , , )mu u0( , , )mu u

0( , , )mu u

0( , , )mu u

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;

0( , , )m mms s

1 10 0 1( , , ) ( , , )m m m m

m ms s s s A

( 1)( )ij m mA a

1, 0 / 2 / 2 1 1 1

0,ij

if i j m or m i j ma

otherwise

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 mon

ntonicity preserving. Since , are increasing,

is strictly increasing

1 1, , mv v

1

mmi

i

v

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

and :

( )TjiB b

( )TjiA a

0 0( , , ) ( , , )m m m m Tm ms s b b B

0 0( , , ) ( , , )m m m m Tm ma a b b A

0

0

0

( , , )

( , , )

( , , )

m mm

m mm

m mm

s s Said Ball basis

a a Wang Ball basis

b b Bernstein basis

Matrix of change of basis Proposition 4: The Wang-Ball basis and the Said-

Ball basis are related, for ,by:0( , , )m m

ma a0( , , )m m

ms s 4m

0 0( , , ) ( , , )m m m mm m ma a s s F

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 comp

lexity than the de Casteljau algorithm Shape preserving properties of the Said–Ball basis Wang–Ball bases are satisfy the boundary tangent property, strictly m

onotonicity 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) t

han the Wang–Ball basis.

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 basi

s, 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.

References Educ. Sci. Technol. 32 (2001) 565–584. [10] J. Delgado and J.M. Peña,Monotonicity preservation of some polynomial and rational representat

ions, 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, Sa

n 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., Dordrec

ht, 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 Scien

ce 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.

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).

The End! Thank you!

0 1 00 0

00 ; 0 0 0

0

0 0 0 0

0 0

( ) ( )( , , , ) ( ) 1

, ,: ( ( )) ( , , )( , , ) 32 3

, ,

( , , ) ( , , ) ( , , ) ( , , ) ( )

( , , ) ( , ,

n n

i i n i m mi i

n m m m mj i i m j n m m

m

m m m m m m m m Tm m m i m m ji

m m mm

t Pu t u u u u t t t A

u uM u t a a s s m m

t t

a a b b A r s s b b B b

a a s

01

1 1

1 10 0 1

1

1 1 1 1 1( 1) / 2 ( 1) / 2 ( 1) / 2 ( 1) / 2 ( 1) / 2 ( 1) / 2

)

( , , )

, ,

( , , ) ( , , )

' ( ) ' ' '

1, 0 / 2 / 2 1 1 1

0,

mm m

mmi m

i

m

mm m m m

i m mi

m m m m m mm m m m m m

ij

s F

v v u u

v v

v s s s s A

v ta v a ta v

if i j m or m i j ma

otherwise