Properties and applications of the constrained dual Bernstein - PowerPoint PPT Presentation

International Conference on Scientific Computing Properties and applications of the constrained dual Bernstein polynomials Stanisław Lewanowicz & Paweł Woźny Institute of Computer Science University of Wrocław, POLAND e-mail: Pawel.Wozny@ii.uni.wroc.pl S. Margherita di Pula, Sardinia, Italy, October 10–14, 2011

Part I. Definitions and properties

2/21 Dual Bernstein basis polynomials • Bernstein basis polynomials of degree n � n � B n x i ( 1 − x ) n − i ( 0 ≤ i ≤ n ) . i ( x ) = i SC 2011 Paweł Woźny, University of Wrocław, Poland

2/21 Dual Bernstein basis polynomials • Bernstein basis polynomials of degree n � n � B n x i ( 1 − x ) n − i ( 0 ≤ i ≤ n ) . i ( x ) = i • Associated with the Bernstein basis, there is a unique dual basis D n 0 ( x ; α, β ) , D n 1 ( x ; α, β ) , . . . , D n n ( x ; α, β ) ∈ Π n defined so that � D n i , B n j � J = δ ij ( i, j = 0, 1, . . . , n ) , where � 1 ( 1 − x ) α x β f ( x ) g ( x ) d x � f, g � J := ( α, β > − 1 ) 0 SC 2011 Paweł Woźny, University of Wrocław, Poland

2/21 Dual Bernstein basis polynomials • Bernstein basis polynomials of degree n � n � B n x i ( 1 − x ) n − i ( 0 ≤ i ≤ n ) . i ( x ) = i • Associated with the Bernstein basis, there is a unique dual basis D n k ( x ; α, β ) ∈ Π n ( 0 ≤ k ≤ n ) defined so that D n i , B n � � J = δ ij ( i, j = 0, 1, . . . , n ) , j where � 1 ( 1 − x ) α x β f ( x ) g ( x ) d x � f, g � J := ( α, β > − 1 ) 0 • Shifted Jacobi polynomials R ( α,β ) ( x ) are orthogonal wrt the inner product � f, g � J , i.e., k � � R ( α,β ) , R ( α,β ) J = δ kl h k ( k, l = 0, 1, . . . ; h k > 0 ) . k l SC 2011 Paweł Woźny, University of Wrocław, Poland

3/21 Constrained dual Bernstein basis polynomials • Let us define � � Π ( k,l ) P ∈ Π n : P ( i ) ( 0 ) = 0 P ( j ) ( 1 ) = 0 ( 0 ≤ i < k ) , ( 0 ≤ j < l ) := , n � � where k + l ≤ n . Certainly, Π ( k,l ) B n k , B n k + 1 , . . . , B n = lin . n n − l SC 2011 Paweł Woźny, University of Wrocław, Poland

3/21 Constrained dual Bernstein basis polynomials • Let us define � � Π ( k,l ) P ∈ Π n : P ( i ) ( 0 ) = 0 P ( j ) ( 1 ) = 0 ( 0 ≤ i < k ) , ( 0 ≤ j < l ) := , n � � where k + l ≤ n . Certainly, Π ( k,l ) B n k , B n k + 1 , . . . , B n = lin . n n − l • There is a unique constrained dual Bernstein basis of degree n D ( n,k,l ) ( x ; α, β ) , D ( n,k,l ) k + 1 ( x ; α, β ) , . . . , D ( n,k,l ) ( x ; α, β ) ∈ Π ( k,l ) , n − l n k satisfying the relation � � D ( n,k,l ) , B n J = δ ij ( i, j = k, k + 1, . . . , n − l ) , j i where � 1 ( 1 − x ) α x β f ( x ) g ( x ) d x � f, g � J := ( α, β > − 1 ) 0 SC 2011 Paweł Woźny, University of Wrocław, Poland

4/21 Constrained and unconstrained dual Bernstein polynomials • Constrained dual Bernstein polynomials D ( n,k,l ) ( x ; α, β ) can be expressed in terms of i the unconstrained dual Bernstein polynomials of degree n − k − l , with parameters α + 2l and β + 2k in the following way: � − 1 � n − k − l �� n x k ( 1 − x ) l D n − k − l D ( n,k,l ) ( x ; α, β ) = ( x ; α + 2l, β + 2k ) i i − k i − k i SC 2011 Paweł Woźny, University of Wrocław, Poland

5/21 Constrained dual Bernstein basis polynomials. Previous results • Ciesielski, 1987 ( α = β = 0, k = l = 0 ) : definition, recurrence relation SC 2011 Paweł Woźny, University of Wrocław, Poland

5/21 Constrained dual Bernstein basis polynomials. Previous results • Ciesielski, 1987 ( α = β = 0, k = l = 0 ) : definition, recurrence relation. • Jüttler, 1998 ( α = β = 0, k = l ) : Bernstein-Bézier representation SC 2011 Paweł Woźny, University of Wrocław, Poland

5/21 Constrained dual Bernstein basis polynomials. Previous results • Ciesielski, 1987 ( α = β = 0, k = l = 0 ) : definition, recurrence relation. • Jüttler, 1998 ( α = β = 0, k = l ) : Bernstein-Bézier representation. • L&W, 2006 ( α, β > − 1, k = l = 0 ) : recurrence relation, orthogonal expansion, ”short” representation SC 2011 Paweł Woźny, University of Wrocław, Poland

5/21 Constrained dual Bernstein basis polynomials. Previous results • Ciesielski, 1987 ( α = β = 0, k = l = 0 ) : definition, recurrence relation. • Jüttler, 1998 ( α = β = 0, k = l ) : Bernstein-Bézier representation. • L&W, 2006 ( α, β > − 1, k = l = 0 ) : recurrence relation, orthogonal expansion, ”short” representation. • Rababah and Al-Natour, 2007: extented Jüttler’s results to the case of arbitrary α, β > − 1 SC 2011 Paweł Woźny, University of Wrocław, Poland

5/21 Constrained dual Bernstein basis polynomials. Previous results • Ciesielski, 1987 ( α = β = 0, k = l = 0 ) : definition, recurrence relation. • Jüttler, 1998 ( α = β = 0, k = l ) : Bernstein-Bézier representation. • L&W, 2006 ( α, β > − 1, k = l = 0 ) : recurrence relation, orthogonal expansion, ”short” representation. • Rababah and Al-Natour, 2007: extented Jüttler’s results to the case of arbitrary α, β > − 1 . • W&L, 2009 ( α, β > − 1 , and k, l ∈ N ): recurrence relation, orthogonal expansion, ”short” representation, relation beetwen constrained and unconstrained dual Bernstein polynomials SC 2011 Paweł Woźny, University of Wrocław, Poland

6/21 Constrained dual Bernstein basis polynomials. Applications • Least-squares approximation by Bézier curves (Jüttler, 1998) SC 2011 Paweł Woźny, University of Wrocław, Poland

6/21 Constrained dual Bernstein basis polynomials. Applications • Least-squares approximation by Bézier curves (Jüttler, 1998). • Computing roots of polynomials (Bartoň and Jüttler, 2007; Liu et al. , 2009) SC 2011 Paweł Woźny, University of Wrocław, Poland

6/21 Constrained dual Bernstein basis polynomials. Applications • Least-squares approximation by Bézier curves (Jüttler, 1998). • Computing roots of polynomials (Bartoň and Jüttler, 2007; Liu et al. , 2009). • Degree reduction of Bézier curves and surfaces (L&W) SC 2011 Paweł Woźny, University of Wrocław, Poland

6/21 Constrained dual Bernstein basis polynomials. Applications • Least-squares approximation by Bézier curves (Jüttler, 1998). • Computing roots of polynomials (Bartoň and Jüttler, 2007; Liu et al. , 2009). • Degree reduction of Bézier curves and surfaces (L&W). • Polynomial approximation of rational Bézier curves (L&W) SC 2011 Paweł Woźny, University of Wrocław, Poland

6/21 Constrained dual Bernstein basis polynomials. Applications • Least-squares approximation by Bézier curves (Jüttler, 1998). • Computing roots of polynomials (Bartoň and Jüttler, 2007; Liu et al. , 2009). • Degree reduction of Bézier curves and surfaces (L&W). • Polynomial approximation of rational Bézier curves (L&W). ⇓ • Problem. Given a function f . Find the Bézier form of the polynomial P n ∈ Π ( k,l ) n which gives the minimum value of the norm � || f − P n || L 2 := � f − P n , f − P n � J . SC 2011 Paweł Woźny, University of Wrocław, Poland

6/21 Constrained dual Bernstein basis polynomials. Applications • Least-squares approximation by Bézier curves (Jüttler, 1998). • Computing roots of polynomials (Bartoň and Jüttler, 2007; Liu et al. , 2009). • Degree reduction of Bézier curves and surfaces (L&W). • Polynomial approximation of rational Bézier curves (L&W). ⇓ • Problem. Given a function f . Find the Bézier form of the polynomial P n ∈ Π ( k,l ) n which gives the minimum value of the norm � || f − P n || L 2 := � f − P n , f − P n � J . • Solution: l � � � f, D ( n,k,l ) a j B n P n ( t ) = j ( t ) , a j := ( · ; α, β ) j J j = k SC 2011 Paweł Woźny, University of Wrocław, Poland

6/21 Constrained dual Bernstein basis polynomials. Applications • Least-squares approximation by Bézier curves (Jüttler, 1998). • Computing roots of polynomials (Bartoň and Jüttler, 2007; Liu et al. , 2009). • Degree reduction of Bézier curves and surfaces (L&W). • Polynomial approximation of rational Bézier curves (L&W) ⇓ • Problem. Given a function f . Find the Bézier form of the polynomial P n ∈ Π ( k,l ) n which gives the minimum value of the norm � || f − P n || L 2 := � f − P n , f − P n � J . • Solution: � 1 l � ( 1 − x ) α x β f ( x ) D ( n,k,l ) a j B n P n ( t ) = j ( t ) , a j = ( x ; α, β ) d x j 0 j = k SC 2011 Paweł Woźny, University of Wrocław, Poland

7/21 Dual Bernstein polynomials: explicit formulae (L&W, 2006) • Recurrence relation � � i i D n + 1 D n n + 1D n ( x ; α, β ) = 1 − i ( x ; α, β ) + i − 1 ( x ; α, β ) + i n + 1 i R ( α,β ) ϑ n n + 1 ( x ) , where Γ ( α + β + 1 ) ( 2n + α + β + 3 )( α + β + 2 ) n ϑ n i := (− 1 ) n − i + 1 Γ ( α + 1 ) Γ ( β + 1 ) ( β + 1 ) i ( α + 1 ) n + 1 − i SC 2011 Paweł Woźny, University of Wrocław, Poland