UMCS. Fast multidimensional Bernstein-Lagrange algorithms. The John Paul II Catholic University of Lublin, ul. Konstantynow 1H, Lublin, Poland

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1 Annales Informatica AI XII, ) 7 18 DOI: /v Fast multidimensional Bernstein-Lagrange algorithms Joanna Kapusta 1, Ryszard Smarzewski 1 1 Institute of Mathematics and Computer Science, The John Paul II Catholic University of Lublin, ul. Konstantynow 1H, Lublin, Poland Abstract In this paper we present two fast algorithms for the Bézier curves and surfaces of an arbitrary dimension. The first algorithm evaluates the Bernstein-Bézier curves and surfaces at a set of specific points by using the fast Bernstein-Lagrange transformation. The second algorithm is an inversion of the first one. Both algorithms reduce the initial problem to computation of some discrete Fourier transformations in the case of geometrical subdivisions of the d-dimensional cube. Their orders of computational complexity are proportional to those of corresponding d-dimensional FFT-algorithm, i.e. to ON logn)+odn), where N denotes the order of the Bernstein-Bézier curves. 1 Introduction Let n =n 1,n 2,...,n d ) be a d-tuple of positive integers and K be a field. Moreover, let Q n be a lattice of all N = n 1 n 2...n d multi-indices α =α 1,α 2,...,α d ) with the integer coordinates satisfying inequalities 0 α i <n i for i =1,2,...,d. 1) Using the multi-index notation, we write Bernstein-Bézier vector polynomials of the variable x =x 1,x 2,...,x d ) K d in the form p n x)= α Q n f α B α x), 2) jkapusta@kul.lublin.pl rsmax@kul.lublin.pl

2 8 Fast multidimensional Bernstein-Lagrange algorithms where f α K s are the control points, the summation extends over all n 1 n 2 n d multi-indices α from the lattice Q n, and ) n 1 d ) B α x)= x α 1 x) n α 1 nj 1 = x αj j 1 x j ) nj 1 αj, 3) α where n 1=n 1 1,n 2 1,...,n d 1). Note that p n x) is a Bézier curve or surface in the case when d =1or d =2, respectively. Additionally, suppose that j=1 x α =x 1,α1,x 2,α2,...,x d,αd ),α=α 1,α 2,...,α d ) Q n, 4) are the points in K d such that coordinates α j x i,0,x i,1,...,x i,ni 1 i =1,2,...,d) 5) are pairwise distinct, i.e. x i,j x i,k, whenever j k. Then the Bernstein-Bézier vector polynomial p n x) can be written in the Lagrange form p n x)= α Q n y α L α x), 6) where y α = p n x α ) K s and L α x)= d L αi x i ),L αi x i )= n i 1 j=0 j α i In this paper we present two fast algorithms of the order x i x i,j x i,αi x i,j. 7) ON logn)+odn),n= Q n = n 1 n 2...n d, 8) for the Bernstein-Lagrange transformation T :f β ) β Qn y β ) β Qn, and its inverse, which is defined by T : y β = f α B α x β ),β Q n, 9) α Q n where the coordinates of points x β are such that x i,j = λ i γ j i i =1,2,...,d, j=0,1,...,n i 1) 10) with the scalarsλ i 0, γ i 0andγ i 1i =1,2,...,d) fixed ink. For the simplicity, these algorithms will be established under the additional assumption that s =1,which does not restrict the generality of our considerations. Since the coordinates x i,j j =0,1,...,n i 1) form geometrical progression, it follows that the points x β can be used e.g. in extrapolation problems [1]. It is not clear if it is possible to extend our fast algorithms to the case of arithmetic progression, or more generally to the case when x i,j = λ i x i,j 1 +δ i i =1,2,...,d, j=1,2,...,n i 1), 11) where λ i 0, δ i and x i,0 = κ i belong to the field K [2]. Of course, in order to evaluate the transformation T for the last coordinates one can use multidimensional algorithms

3 Joanna Kapusta, Ryszard Smarzewski 9 based on the de Casteljau algorithm, which have the computational complexity of the order greater than our algorithms, cf. [3], [4], [5] and [6]. Following [7] and [8], our algorithms will use the discrete Fourier transformation and its inverse, which are defined by F m : K m a b K m 12) m 1 b i = a k ψm ik and a i = 1 m 1 b k ψm ik m i =0,1,...,m 1), where ψ m is supposed to be a primitive root of the unity of order m in the field K. It is well known that discrete Fourier transformations can be computed by the famous FFT-algorithm, which has a running time of order Omlogm) [9]. 2 Fast multidimensional convolutions and deconvolutions In order to present fast algorithms for computation of the Bernstein-Lagrange transformations T and T 1, we need fast algorithms for multidimensional convolutions and deconvolutions. For this purpose, suppose that a =a 0,a 1,...) and b =b 0,b 1,...) are two finite or infinite sequences. Then the wrapped convolution is defined by while its deconvolution c i = c =c 0,c 1,...,c m 1 )=a m b 13) i a k b i k i =0,1,...,m 1), 14) is supposed to be the solution a = c m b = c m b 1 b 0 0) a 0 = c 0 /b 0, i 1 a i = c i a k b i k )/b 0 i =1,2,...,m 1) of the lower triangular system of equations 14). Moreover, the convolutionary inverse is such that where bx)= 15) d =d 0,d 1,...d r 1 )=b 1 16) r 1 1/bx)= d k x k +Ox r ), 17) m 1 b k x k and d k = dk dx k 1. 18) bx)) x=0

4 10 Fast multidimensional Bernstein-Lagrange algorithms The wrapped convolution satisfies the formula a m b = { F 1 m [F m a) F m b)]+f 1 m [F m Ψ a) F m Ψ b)]/ψ } /2, 19) where a =a 0,a 1,...,a m 1 ), b =b 0,b 1,...,b m 1 ), Ψ=1,ψ 1 2m,...,ψ m 1 2m ), ψ 2m is the primitive root of order2m of the unity ink, and vector operations of multiplication and division are defined coordinatewise. Formula 19) gives an extremely effective and fast algorithm of the order Om logm) to evaluate wrapped convolutions, which is observed implicitly in [7], see also [8]. Note that it can be also applied to evaluate m 1 b j = a i γ ij j =1,2,...,m 1). 20) Indeed, we have m 1 j m 1 b j = a i γ ij = r j p i q j i + p i q j i+1) j =0,1,...,m 1), 21) where Hence with and r j = i=j+1 j j 1 γ k,p j = a j γ k,q j = 1 j =0,1,...,m 1). j γ k j b j m 1) = r j m 1) d i c j i j = m 1,m,...,2m 2) c i = { pi for i =0,1,...,m 1, d i = 0 for i = m,m+1,...,2m 2 { qm 2 i for i =0,1,...,m 2, q i m 1) for i = m 1,m,...,2m 2. Consequently, if d =d i ) 2m 2, c =c i ) 2m 2 and r =r i ) m 1, then we get where b = d m c ) r, 22) d m c = P m d 2m 1 c) and the projection P m : K 2m 1 K m is defined by P m e)=e m 1,e m,...,e 2m 2 ),e=e 0,e 1,...,e 2m 2 ). 23) It is clear that the order of algorithm 22) is equal to Omlogm). Note, that another algorithm for computing 20), which has the same order of complexity, was presented in [10].

5 Joanna Kapusta, Ryszard Smarzewski 11 The wrapped convolution can be also applied to evaluate the multidimensional convolution u = a b, 24) of a hypermatrix a =a α ) α Qn and vector b =b i ) d, with b i =b i,0,b i,1,...,b i,ni 1). Here coordinates of u are equal to u α = β Q α a β b α β,α=α 1,α 2,...,α d ) Q n. 25) Definition 1 [2]). A hypermatrix w =w α ) α Qn = a i) b i K n1 n2 n d, 1 i d, 26) is said to be the i-th partial hypermatrix convolution of a hypermatrix a =a α ) α Qn and a vector b i =b i,0,b i,1,...,b i,ni 1), whenever each column w β1,...,β i 1,,β i+1,...,β d = a β1,...,β i 1,,β i+1,...,β d ni b i, 0 β j <n j 1, j =1,2,...,i 1,i+1,...,d, of the hypermatrix w is equal to the wrapped convolution of the column and vector b i. a β1,...,β i 1,,β i+1,...,β d = a β1,...,β i 1,j,β i+1,...,β d ) ni 1 j=0. 27) The notation of the partial hypermatrix convolutions enables to rewrite the multidimensional convolution u =u α ) α Qn in the following hypermatrix form ) ) u = a b =... a 1) b 1 ) 2) b 2 3)... d) b d 28) with b i =b i,0,b i,1,...,b i,ni 1) and a =a α ) α Qn [2]. Hence it is clear that the fast algorithm for computing an i-th partial hypermatrix convolution should evaluate N/n i one-dimensional convolutions for vectors of size n i. Therefore, algorithm 28) for computing the multidimensional convolution is of the order N/n 1 )On 1 logn 1 )+...+N/n d )On d logn d )=ON logn),n= n 1 n 2...n d. 29) The same order is in the algorithm ) ) v = a b =... a 1) b 1 2) b 2...) 3) d) b d, 30) where the i-th extended convolution a i) b i i =1,2,...,dis defined as in Definition 1 with a β1,...,β i 1,,β i+1,...,β d ni b i 31) replaced by P ni aβ1,...,β i 1,,β i+1,...,β d 2ni 1 b i ) and a α =0for α/ Q n. In a similar way one can define the hypermatrix deconvolution 32) a = u b, 33)

6 12 Fast multidimensional Bernstein-Lagrange algorithms whenever b i,0 0for i =0,1,...,n i 1. The only difference consists in replacing the operator i) of the i -th partial hypermatrix convolution in Definition 1 by the corresponding operator i) of the i-th partial hypermatrix deconvolution. In other words, each column of the hypermatrix should be equal to a =a α ) α Qn = w i) b i K n1 n2 n d, 1 i d, 34) a β1,...,β i 1,,β i+1,...,β d = w β1,...,β i 1,,β i+1,...,β d ni b i = w β1,...,β i 1,,β i+1,...,β d ni b 1 where 0 β j <n j. Then we have ) ) a =... u d) b d... ) 2) b 2 1) b 1 ) ) ) 35) =... u d) b ) b 1 2 1) b 1 1. d One can prove that the last algorithm for hypermatrix deconvolution is of the order i, ON logn),n= n 1 n 2...n d. 36) For this purpose, it is sufficient to observe that the convolutionary inverse of a vector b =b 0,b 1,...,b m 1 ) K m with b 0 0can be computed by the Newton method of the order Omlogm) [11]. More precisely, let x i+1 =2x i x 2 ib, i =0,1,..., 37) be the Newton iterative formula for the function f x)=x 1 b x 0). Moreover, suppose that the coefficients d 0,d 1,...,d 2 i 1 i 1) 38) of the inverse b0 +b 1 x+...b m 1 x m 1) 1 = d0 +d 1 x+...d 2 i 1x 2i 1 +Ox 2i ) 39) are already computed and that d k =0for all k 2 i. Then the single Newton iteration d =2 d d 2 i+1 d 2 i+1 b. doubles the number of evaluated coefficients d k k =0,1,...,2 i+1 1 ) of the convolutionary inverse. Hence we finally conclude that the iterative Newton formula d =2 d d 2 i d 2 i b, i =2,3,..., log 2 m, 40) with the starting vector d of the form 1 d =, b ) 1 b 0 b 2,0,0..., 0 generates the required convolutionary inverse d =d 0,d 1,...,d m 1 ) 41)

7 Joanna Kapusta, Ryszard Smarzewski 13 ofb =b 0,b 1,...,b m 1 ), b 0 0. Since the computational complexity of the convolution 2 i is equal to Oi2 i ), it is clear that the computational complexity of algorithm 40) is of the order O mlog 2 m+ m 2 log m ) log 22 = Omlogm). 42) This completes the proof that the algorithm 35) is of the order ON logn). 3 Fast Bernstein-Lagrange transformation Now we establish a fast algorithm for evaluating the multivariate polynomial p n x)= ) n 1 x α 1 x) n α 1 43) α α Q n f α at the points x β =x 1,β1,x 2,β2,...,x d,βd ) with the coordinates of the form For this purpose, note that x i,j = λ i γ j i i =1,2,...,d, j=0,1,...,n i 1). 44) p n x)= f α α Q n β Q n α = β Q n x β = β Q n a β x β, α Q β+1 f α where the coefficients a β are given by the formula a β = n 1) n β 1) ) ) n 1 n α 1 x α+β 1) β α β ) ) n 1 n α 1 1) β α α β α β α f α 1) αβ α). α Q β+1 Hence one can use the multidimensional convolution in order to get the algorithm ) ) f f a = r p t =... r 1) p 1 ) 2) p 2 ) 3) d) p d t, 46) where t =t α ) α Qn,r=r α ) α Qn and p i =p i,0,p i,1,...,p i,ni 1) are defined by 45) and r α = α, t α = n 1) n α 1), α Q n, 47) p i,l = 1)l l) i =1,2,...,d, l =0,1,...,n i 1). 48) Therefore, it follows from 28) and 29) that the coefficients a β can be computed by the algorithm 46) of the order ON logn). Furthermore, by inserting formula 44)

8 14 Fast multidimensional Bernstein-Lagrange algorithms into 45), we get y α = p n x α )= β Q n a β λ β γ αβ. Hence, we obtain n d 1 n 2 1 n 1 1 y α =... a β b β w 1,α1 β 1 w 2,α2 β 2...w d,αd β d q α, 49) β d =0 whenever we set and q β = β 2=0 β 1=0 β d j γj, k b β = j=1 w j,l = 1 l γj k d β j 1 λ βj j j=1 Finally, formula 49) yields the following theorem. γ k j, β Q n, 50) j =1,2,...,d, l=0,1,...,n j 1). 51) Theorem 1. If T :f α ) α Qn y α ) α Qn denotes the d-dimensional Bernstein- Lagrange transformation with the points x α =x 1,α1,x 2,α2,...,x d,αd ) defined as in 44), then T can be evaluated by the algorithm ) ) ) a b) 1) w 1 2) w 2 3)... d) w d T : y = a =... f... r 1) p 1 q, 52) ) ) 2) p 2 ) 3) d) p d t, 53) where elements of b =b α ) α Qn,q=q α ) α Qn, w i =w i,ni 2,...,w i,0,w i,0,w i,1,..., w i,ni 1),r=r α ) α Qn,t =t α ) α Qn andp i =p i,ni 2,...,p i,0,p i,0,w i,1,..., p i,ni 1) are defined as in formulae 47), 48), 50) and 51). Moreover, this algorithm has the running time of ON logn))+odn), where N = n 1 n 2 n d. We note that the term OdN) in the running time is an estimate of all auxiliary computations 47), 48), 50) and 51), which do not use convolutions. For the completeness of consideration, we summarize the algorithm for computing the multidimensional Bernstein-Lagrange transformation in more detail. Algorithm 1. The d - dimensional Bernstein-Lagrange transformation T with respect to the points x α =x 1,α1,x 2,α2,...,x d,αd ), where α =α 1,α 2,...,α d ) Q n, n =n 1,n 2,..., n d ) and x i,j = λ i γ j i i =1,2,...,d, j=0,1,2,...,n i 1). Input: A hypermatrix f =f α ) α Qn, scalar vectors λ =λ 1,λ 2,...,λ d ) and γ = γ 1,γ 2,...,γ d ) in K d, and the vector n =n 1,n 2,...,n d ) of positive integers. Output: A hypermatrix y =y α ) α Qn of values y α = p n x α ). 1. Use 47) to evaluate r α,t α for each α Q n. 2. Use 48) to evaluate p j,l for j =1,2,...,d,l=0,1,...,n j 1.

9 Joanna Kapusta, Ryszard Smarzewski Perform the componentwise division v = f/r. 4. For i from 1 to d do the following: 4.1. Compute the partial hypermatrix convolution v = v i) p i. 5. Perform the componentwise multiplication a = v t. 6. Use 50) to evaluate b β,q β for each β Q n. 7. Use 51) to evaluate w j,l for j =1,2,...,d,l=0,1,...,n j Perform the componentwise multiplication u = a b. 9. For i from 1 to d do the following: 9.1. Compute the extended partial hypermatrix convolution u = u i) w i. 10. Perform the componentwise multiplication y = u q. 11. Return y). 4 Inverse multidimensional Bernstein-Lagrange transformation Now we consider the inversion of the multidimensional Bernstein-Lagrange transformation T 1 :y α ) α Qn f α ) α Qn. 54) If we know coefficients y α = px α ) of the Lagrange polynomial 6) at the knots of the form x α =x 1,α1,x 2,α2,...,x d,αd ) 55) x i,j = λ i γ j i,,2,..,d, j =0,1,...,n i 1, 56) then we can find the multivariate divided differences = p n [x 1,0,...,x 1,α1 ;...;x d,0,...,x d,αd ]= c α of the Newton polynomial p n x)= α 1=0α 2=0 d α i 1 c α α Q n j=0 α d =0 β Q α+1 y β 57) d α i x i,βi x i,j ) j=0,j β i x i x i,j ), 58) by using an algorithm of the order ON logn))+odn) presented in [12]. Moreover, by using equality 56) the formula 58) can be rewritten in the following form ) n 1 n 2 n d d α i 1 p n x)=... c α1,α 2,...,α d x αi i 1 λ iγ j i. 59) x i Since we have see [13]) n 1 1 xq k ) = n [ n m] m=0 q j=0 1) m x m q mm 1) 2 60)

10 16 Fast multidimensional Bernstein-Lagrange algorithms with [ n m] q = n ) 1 q i [n] q = m 1 q i ) n m 1 q i [n m] ) q [m] q, it follows from 59) and 60) that where a β = 1 [β] q Hence we get p n x)= n 1 β 1 1n 2 β 2 1 α 1=0 α 2=0 a α = 1 [β] q n 1 1n 2 1 β 1=0 β 2=0... n d 1 d... a β1,β 2,...,β d n d β d 1 α d =0 α Q n β c α β d =0 x βi i, d [α i +β i ] q α i α i 1) 2 c α1,α 2,...,α d γi λ i ) αi. [α i ] q [α+β] q [α] q γ αα 1) 2 λ) α,α Q n, or equivalently ) ) ) a =... c v) d) d 1) d 2) 1) z d zd 1 z1 g, 62) where the elements of v =v α ), g =g α ) and z i =z i,0,z i,1,...,z i,ni 1) are defined by v α = z i,l = d αiαi 1) 1 [α i ] q γ 2 i λ i ) αi, g α = 1 [α] q, α Q n, d [n i l] q i =1,2,...,d,l=0,1,...,n i 1), and the reversed i-th partial hypermatrix convolution 61) 63) w i) z i = ŵ i) z i 64) is defined as the i-th partial hypermatrix convolution with its elements written in the reverse order, where ŵ is the hypermatrix with i-th column written in the reverse order, too. Finally, one can apply 46) to get the following theorem. Theorem 2. Let T 1 :y α ) α Qn f α ) α Qn be the inverse multidimensional Bernstein-Lagrange transformation with respect to the points x α =x 1,α1,x 2,α2,...,x d,αd ) with the coordinates of the form x i,j = λ i γ j i,,2,..,d, j =0,1,2,...,n i 1, 65)

11 Joanna Kapusta, Ryszard Smarzewski 17 where λ i 0, γ i 1andγ i 0. Then it can be evaluated by the algorithm a ) ) T 1 : f =... t d) p d ) d 1) p d ) p 1 r, 66) ) ) ) a =... c v) d) d 1) d 2) 1) z d zd 1 z1 g, 67) where the elements of t =t α ) α Qn,r =r α ) α Qn,g=g α ) α Qn,c=c α ) α Qn, v =v α ) α Qn,z i =z i,0,z i,1,...,z i,ni 1) and p i =p i,0,p i,1,...,p i,ni 1) are defined as in formulae 47), 48), 57) and 63). Moreover, this algorithm has the running time of ON logn)) +OdN), where N = n 1 n 2 n d. Algorithm 2. The inverse d-dimensional Bernstein-Lagrange transformation T 1 with respect to the points x α =x 1,α1,x 2,α2,...,x d,αd ), where α =α 1,α 2,...,α d ) Q n,n=n 1,n 2,..., n d ) and x i,j = λ i γ j i i =1,2,...,d, j=0,1,2,...,n i 1). Input: A hypermatrix y =y α ) α Qn, scalar vectors λ =λ 1,λ 2,...,λ d ), γ =γ 1,γ 2,...,γ d ) in K d, and the vector n =n 1,n 2,...,n d ) of positive integers. Output: A hypermatrix f =f α ) α Qn. 1. Use Algorithm 8 from [12] to evaluate c α for each α Q n. 2. Use 63) to evaluate v α,g α for each α Q n. 3. Use 63) to evaluate z j,l for j =1,2,...,d,l=0,1,...,n j Perform the componentwise multiplication v = c v. 5. For i from d down to 1 do the following: 5.1. Compute the reversed partial hypermatrix convolution v = v i) z i. 6. Perform the componentwise multiplication a = v g. 7. Use 47) to evaluate t α,r α for each α Q n. 8. Use 48) to evaluate p j,l for j =1,2,...,d,l=0,1,...,n j Perform the componentwise division a = a/t. 10. For i from d down to 1 do the following: Compute the partial hypermatrix deconvolution a = a i) p i. 11. Perform the componentwise multiplication f = a r. 12. Return f). 5 Conclusions and remarks In this paper, we present two new algorithms for the d-dimensional Bernstein- Lagrange transformation and its inverse for the points with the coordinates defined by the formulae x α =x 1,α1,x 2,α2,...,x d,αd ),α Q n 68) x i,j = λ i γ j i,,2,...,d, j=0,1,...,n i 1, where γ i 0, γ i 1and λ i 0are fixed.

12 18 Fast multidimensional Bernstein-Lagrange algorithms Roughly speaking, the main feature of these algorithms consists in splitting the computations into two steps. In the first step we compute only quantities, which require to perform only OdN) operations. The second step includes computations of d-dimensional convolutions or deconvolutions of the order ON logn). Thus, the computational complexity of this algorithms takes only ON logn)+odn) 69) operations, where N = n 1 n 2...n d. Moreover, if we make natural assumption that n i 2 for i =1,2,...,d, then log 2 N d and the order of the algorithm can be reduced to ON logn)). It should be emphasized, that parts 53) and 66) of the algorithms presented in Theorems 1 and 2 are valid for arbitrary points x α,α Q n. However, we do not know if the remaining parts of these algorithms are true for the points defined in 11). References [1] Stoer J., Bulirsch R., Introduction to Numerical Analysis, Springer - Verlag, New York [2] Kapusta J., Smarzewski R., Fast algorithms for multivariate interpolation and evaluation at special points, Journal of Complexity ): 332. [3] Farouki R., Rajan V. T., Algorithms for polynomials in Bernstein form, Computer Aided Geometric Design ): 1. [4] Mainara E., Pe na J. M., Evaluation algorithms for multivariate polynomials in Bernstein-Bézier form, Journal of Approximation Theory 143 1) 2006): 44. [5] Peters J., Evaluation and approximate evaluation of the multivariate Bernstein-Bézier form on a regularly partitioned simplex, ACM Transactions on Mathematical Software 204) 1994): 460. [6] Phien H. N., Dejdumrong N., Efficient algorithms for Bé zier curves, Computer Aided Geometric Design ): 247. [7] Aho A., Hopcroft J., Ullman J., The design and analysis of computer algorithms, Addison-Wesley, London [8] Smarzewski R., Kapusta J., Fast Lagrange-Newton transformations, Journal of Complexity ): 336. [9] Bini D., Pan V. Y., Polynomial and matrix computations: fundamental algorithms, Birkhäuser Verlag, [10] Aho A., Steiglitz K., Ullman J., Evaluating polynomials at fixed sets of points, SIAM Journal Comput ): 533. [11] Borwein J. M., Borwein P. B., Pi and the AGM: A study in analytic number theory and computational complexity, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley and Sons, New York, Chichester, Brisbane, Toronto, Singapore, [12] Kapusta J., An efficient algorithm for multivariate Maclaurin-Newton transformation, Annales Informatica AI VIII 2) 2008): 5. [13] Andrews G. E., The theory of partitions Encyclopedia of mathematics and its applications), Addison-Wesley Publishing Company, Powered by TCPDF