Structured Inversion of the Bernstein-Vandermonde Matrix

Home , Vandermonde matrix

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2 Inverse Formulas

2.1 General Formulas For integers m,n 0, let bn(x) n be a basis for the space of univariate polynomials of degree at ≥ { j }j=0 most n, and let x Rm+1 with x < x for each 0 imonomial basis is used, then the inverse of a Bézout matrix is a Hankel matrix. As a consequence of the arguments involved, one obtains a formula for the inverse of the monomial Vandermonde matrix in terms of its transpose, a diagonal matrix, and a particular Bézout matrix. Since none of the arguments require anything specific to the monomial basis, we can generalize the arguments to any basis, which we summarize here. For t R, let bn(t) denote the column vector (bn(t), . . . , bn(t))T . As a consequence of (3), if ∈ 0 n s = t, then 6 n n T n n n (b (s)) Bez(v, w)b (t)= Bezij(v, w)bi (s)bj (t) i,jX=0 v(s)w(t) v(t)w(s) = − . s t − This implies that (bn(t))T Bez(v, w)bn(t) = lim (bn(t))T Bez(v, w)bn(t + ε) ε→0 v(t + ε)w(t) v(t)w(t + ε) = lim − ε→0 ε = v′(t)w(t) v(t)w′(t). −

2 Therefore, if v has simple zeros at x and w(x ) = 0 for each 0 i n, then i i 6 ≤ ≤ n n T ′ n V (x) Bez(v, w) (V (x)) = diag(v (xj)w(xj ))j=0, (4) and so we have the following: Theorem 2.1. The inverse of V n(x) is given by 1 n (V n(x))−1 = Bez(v, w) (V n(x))T diag , (5) v′(x )w(x ) j j j=0 where v and w are any polynomials of degree n + 1 satisfying v(x ) = 0, v′(x ) = 0, and w(x ) = 0 i i 6 i 6 for each 0 i n. ≤ ≤ Corollary 2.2. The inverse of V n(x) is given by 1 n (V n(x))−1 = Bez(v, 1) (V n(x))T diag , (6) v′(x ) j j=0 where v is any polynomial of degree n + 1 that has simple zeros at x for each 0 i n. i ≤ ≤ 2.2 Bernstein–Vandermonde We now focus on the case where 0 x 1 for each 0 i n and the basis consists of the ≤ i ≤ ≤ ≤ polynomials Bn(x) n , where { j }j=0 n Bn(x)= xj(1 x)n−j (7) j j − are the Bernstein polynomials of degree n. In order to apply Theorem 2.1 or Corollary 2.2, we need a method for computing the entries of the Bernstein–Bézout matrix; in fact, they satisfy a recurrence relation involving the Bernstein coefficients of the polynomials. n+1 n+1 n+1 n+1 Theorem 2.3. If v(t) = i=0 viBi (t) and w(t) = i=0 wiBi (t) are polynomials of degree at most n + 1, then the entries b of the Bernstein–Bézout matrix Bez(v, w) generated by v and w P ij P satisfy 1 b = j(n i)b + (n + 1)2 (v w v w ) . (8) ij (i + 1)(n j + 1) − i+1,j−1 i+1 j − j i+1 − Proof. By (7), we have that j + 1 tBn(t)= Bn+1(t) (9) j n + 1 j+1 and n j + 1 j + 1 Bn(t)= − Bn+1(t)+ Bn+1(t). (10) j n + 1 j n + 1 j+1 Therefore, n n i + 1 s b Bn(s)Bn(t)= b Bn+1(s)Bn(t) ij i j n + 1 ij i+1 j i,jX=0 i,jX=0 n (i + 1)(n j + 1) = − b Bn+1(s)Bn+1(t) (n + 1)2 ij i+1 j i,jX=0 n (i + 1)(j + 1) + b Bn+1(s)Bn+1(t). (n + 1)2 ij i+1 j+1 i,jX=0

3 Similarly, we have that

n n j + 1 t b Bn(s)Bn(t)= b Bn(s)Bn+1(t) ij i j n + 1 ij i j+1 i,jX=0 i,jX=0 n (n i + 1)(j + 1) = − b Bn+1(s)Bn+1(t) (n + 1)2 ij i j+1 i,jX=0 n (i + 1)(j + 1) + b Bn+1(s)Bn+1(t). (n + 1)2 ij i+1 j+1 i,jX=0 This implies that

n n (i + 1)(n j + 1) (s t) b Bn(s)Bn(t)= − b Bn+1(s)Bn+1(t) − ij i j (n + 1)2 ij i+1 j i,jX=0 i,jX=0 n (n i + 1)(j + 1) − b Bn+1(s)Bn+1(t), − (n + 1)2 ij i j+1 i,jX=0 and so

n n+1 i(n j + 1) (s t) b Bn(s)Bn(t)= − b Bn+1(s)Bn+1(t) − ij i j (n + 1)2 i−1,j i j i,jX=0 i,jX=0 n+1 j(n i + 1) − b Bn+1(s)Bn+1(t). − (n + 1)2 i,j−1 i j i,jX=0 On the other hand,

n+1 v(s)w(t) v(t)w(s)= (v w v w )Bn+1(s)Bn+1(t), (11) − i j − j i i j i,jX=0 and so the result follows by comparing coeﬃcients in (3).

We can form the ﬁrst column and last row of Bez(v, w) by using (8) with j = 0 and i = n. The rest of the columns can then be built by applying the recurrence relation. Therefore,

Corollary 2.4. Bez(v, w) can be constructed in (n2) operations. O By repeatedly applying (8), we obtain a closed form for the entries of the Bernstein–B´ezout matrix.

n+1 n+1 n+1 n+1 Corollary 2.5. If v(t) = i=0 viBi (t) and w(t) = i=0 wiBi (t) are polynomials of degree at most n + 1, then the entries b of the Bernstein–B´ezout matrix Bez(v, w) generated by v and w P ij P are given by

mij 1 n + 1 n + 1 bij = n n (vi+k+1wj−k vj−kwi+k+1) , (12) i j i + k + 1 j k − Xk=0 − where m = min j, n i . ij { − }

4 Constructing Bez(v, w) by using (12) would require (n3) operations, and so Corollary 2.4 O n implies that (12) is not optimal for building Bez(v, w); however, since i = 0 whenever i>n, the upper limit of the sum in (12) can be replaced with k = n, and so we recognize that Bez(v, w) is given by a difference of matrix products. n+1 n+1 n+1 n+1 Corollary 2.6. Let v(t)= i=0 viBi (t) and w(t)= i=0 wiBi (t) be polynomials of degree at most n + 1. Define the Toeplitz matrices T v,n and T w,n by P P n + 1 n + 1 T v,n = v and T w,n = w , ij j i j−i ij j i j−i − − define the Hankel matrices Hv,n and Hw,n by n + 1 n + 1 Hv,n = v and Hw,n = w , ij i + j + 1 i+j+1 ij i + j + 1 i+j+1 n and let ∆n = diag n . Then the Bernstein–Bézout matrix Bez(v, w) generated by v and w is j j=0 given by Bez(v, w)=(∆n)−1 [Hv,nT w,n Hw,nT v,n] (∆n)−1 . (13) − Remark 2.7. In [3], it was shown that the inverse of the degree n Bernstein mass matrix M n has the decomposition (M n)−1 = (∆n)−1 T nHn T nHn (∆n)−1 , (14) − h i n where Hn and Hn are Hankel matrices, T n andeT n are Toeplitze matrices, and ∆n = diag n . j j=0 This decomposition was obtained from the monomial Bézout matrix corresponding to polynomials whosee coefficients are contained in thee last column of (M n)−1. It is interesting to note the similarities between (13) and (14) despite the different matrices and bases being considered. n+1 n+1 Given a polynomial v(t) = k=0 vkBk (t) of degree at most n + 1, we can express v in the monomial basis as v(t)= n+1 v tk. It was shown in [7] that k=0Pk k P 1 n ℓ + 1 ev = − v . (15) k n+1 k ℓ ℓ k Xℓ=0 − This implies that e

i+j+1 n + 1 n k + 1 Hv,n = v = − v ij i + j + 1 i+j+1 i + j k + 1 k Xk=0 − i+j+1 n k + 1 e = − v . n i j k Xk=0 − − Similarly, e j−i n k + 1 T v,n = − v . ij j i k k Xk=0 − − By Vieta’s formula (see, for example, [16]), e n n (x x )= ( 1)n−k+1σ (x)xk, (16) − i − n−k+1 Yi=0 Xk=0 5 where 1, if k = 0; σk(x)= (17) xi0 xi , otherwise; ( 0≤i0<···

6 where s(n, k) are the (signed) Stirling numbers of the ﬁrst kind [17]. In addition, we have that j!(n j)! (j/n i/n) = ( 1)n−j − . (21) − − nn i∈{0,...,nY}\{j} Corollary 2.9. Deﬁne the Hankel matrices Hn and Hn by

i+j+1 n + 1 e n k + 1 s(n + 1, k) Hn = and Hn = − , ij i + j + 1 ij n i j nn−k+1 Xk=0 − − e deﬁne the Toeplitz matrices T n and T n by

j−i n + 1 e n k + 1 s(n + 1, k) T n = and T n = − , ij j i ij j i k nn−k+1 − Xk=0 − − and deﬁne the diagonal matrices Dn and ∆n bye n n n Dn = diag ( 1)n−j [j!(n j)!] and ∆n = diag . − − j=0 j j=0 In addition, let V n be the scaled Bernstein–Vandermonde matrix

V n = ij(n i)n−j. e ij − n n Then the inverse of V = V (0, 1/n,...,e1) is given by T (V n)−1 = (∆n)−1 HnT n HnT n V n (Dn)−1 . (22) − h i 3 Applying the inverse e e e

Now, we describe several approaches to applying (V n(x))−1 to a vector.

LU factorization We can decompose V n(x) as

V n(x)= Ln(x)U n(x), (23) where Ln(x) and U n(x) are lower and upper triangular matrices, respectively. Widely available in libraries, computing Ln(x) and U n(x) requires (n3) operations, and each of the subsequent O triangular solves require (n2) operations to perform. O Exact inverse In light of Corollary 2.2, we can directly form (V n(x))−1. By Theorem 2.3, we can form the inverse in (n2) operations. The inverse can then be applied to any vector in (n2) O O operations using the standard algorithm.

DFT-based application By Theorem 2.8, we can express the inverse of V n(x) in terms of its (scaled) transpose and Hankel, Toeplitz, and diagonal matrices. The matrices Hn(x) and T n(x) can be formed in (n) operations, but the matrices Hn(x) and T n(x) require (n2) operations O O to form, since each entry involves a sum. Even though the Hankel and Toeplitz matrices can be applied to a vector in (n log n) operations via circulante embeddinge and a couple of FFT/iFFT O [18], the scaled transpose still requires (n2) operations to apply to a vector. O

7 Newton algorithm In [2], Ainsworth and Sanchez gave a fast algorithm for solving V n(x)c = b via a recursion relation for the B´ezier control points of the Newton form of the Lagrangian interpolant. The numerical results given suggest that the algorithm is stable even for high polynomial degree, and so we include it for comparison.

4 Conditioning and accuracy

In [3], we gave a partial explanation of the relatively high accuracy, compared to its large 2-norm condition number, of working with the Bernstein mass matrix. We will use similar reasoning to investigate the performance of the Bernstein–Vandermonde matrix. Recall that for an integer n 0, the Bernstein mass matrix M n is the (n + 1) (n + 1) matrix ≥ × given by 1 n n n Mij = Bi (x)Bj (x)dx. (24) Z0 The mass matrix (for Bernstein or any other family of polynomials) plays an important role in connecting the L2 topology on the finite-dimensional space to linear algebra. To see this, we first define mappings connecting polynomials of degree n to Rn+1. Given any c Rn+1, we let π(c) be ∈ the polynomial expressed in the Bernstein basis with coefficients contained in c:

n n π(c)(x)= ciBi (x). (25) Xi=0 We let Π be the inverse of this mapping, sending any polynomial of degree at most n to the vector of n + 1 coeﬃcients with respect to the Bernstein basis. Now, let p(x) and q(x) be polynomials of degree n with expansion coeﬃcients Π(p) = p and Π(q)= q. Then the L2 inner product of p and q is given by the M n-weighted inner product of p and q, for 1 n 1 n n T n p(x)q(x)dx = piqj Bi (x)Bj (x)dx = p M q. (26) 0 0 Z i,jX=0 Z Similarly, if p n = pT M np (27) k kM is the M n-weighted vector norm, then we have forpp = π(p),

p 2 = p n . (28) k kL k kM We can interpret the Bernstein–Vandermonde matrix V n(x) as an operator mapping polynomials of degree at most n to Rn+1 via p V n(x)Π(p). Therefore, it makes sense to measure p in 7→ the L2 norm and V n(x)Π(p) in the 2-norm. By (28), we can also consider Π(p) in the M n-norm, which leads us to deﬁne the operator norm

Ay 2 A M n→2 = max k k , (29) k k y6=0 y n k kM and going in the opposite direction,

Ay M n A 2→M n = max k k . (30) k k y6=0 y k k2

8 These two norms naturally combine to deﬁne a new condition number

−1 κ n (A)= A n A . (31) M →2 k kM →2 2→M n n Since M is symmetric and positive deﬁnite, it has a well-deﬁned positive square root via the spectral decomposition.

Lemma 4.0.1. n n n −1/2 κM n→2 (V (x)) = κ2 V (x) (M ) . (32) Proof. Observe that if y = 0, then 6 2 n −1 (V (x)) y T n −T n n −1 n y (V (x)) M (V (x)) y M = y 2 y 2 k k2 k k2 T (M n)1/2 (V n(x))−1 y (M n)1/2 (V n(x))−1 y = 2 y 2 k2 k (M n)1/2 (V n(x))−1 y = 2 , 2 y 2 k k and so (V n(x))−1 = (M n)1/2 (V n(x))−1 . (33) 2→M n 2

Let z = (M n)1/2 y. Then z = 0 and 6 n 2 T n T n V (x)y 2 y (V (x)) V (x)y k 2 k = T n y n y M y k kM T V n(x) (M n)−1/2 z V n(x) (M n)−1/2 z = zT z 2 V n(x) (M n)−1/2 z = 2 , 2 z 2 k k and so n n n −1/2 V (x) M n→2 = V (x) (M ) . (34) k k 2

It follows that

n n n −1 κM n→2 (V (x)) = V (x) M n→2 (V (x)) k k 2→M n

= V n(x) (M n)− 1/2 (M n) 1/2 (V n(x))−1 2 2 − 1 n n −1/2 n n −1/2 = V (x) (M ) V (x) (M ) 2 2 = κ V n(x) (M n)− 1/2 . 2

9 n n −1/2 To analyze κ2 V (x) (M ) , we will need reference to the Legendre polynomials, mapped from their typical home on [ 1, 1] to [0, 1]. For 0 j n, let Lj(x) denote the Legendre polynomial − ≤ ≤ of degree n over [0, 1], scaled so that Ln(1) = 1 and

j 2 1 L 2 = . (35) k kL 2j + 1

The Legendre–Vandermonde matrix is the (n + 1) (n + 1) matrix V n(x) given by × n j Vij (x)= L (xi). b (36)

Given any polynomial p of degree at mostb n, let Θ(p) denote the vector of n + 1 coeﬃcients with respect to the Legendre basis. Deﬁne T n to be the (n+1) (n+1) matrix satisfying T nΘ(p)= Π(p) × for all polynomials p of degree at most n. In particular, we have the relationship

V n(x)= V n(x)T n. (37)

Let λn n be the set of eigenvalues of M n. In [3], it was shown that M n admits the spectral { j }j=0 b decomposition M n = QnΛn (Qn)T , (38) where n n n n Q = T diag (2j + 1)λj (39) j=0 qn n n is an orthogonal matrix and Λ = diag λj contains the eigenvalues. This decomposition j=0 combined with Lemma 4.0.1 and the transformation given in (37) suggest a relationship between the M n 2 condition number of V n(x) and the 2-norm condition number of V n(x) scaled by a → diagonal matrix. b Lemma 4.0.2. n n n κM n→2 (V (x)) = κ2 V (x) diag 2j + 1 . (40) j=0 p Proof. We have that b

V n(x) (M n)−1/2 = V n(x)Qn (Λn)−1/2 (Qn)T n n n n n 1 n T = V (x)T diag (2j + 1)λj diag (Q ) j=0  n  λj q j=0 n q  = V n(x) diag 2j + 1 (Qn)T . j=0 p Since Qn is orthogonal, the resultb follows from Lemma 4.0.1.

In [9], Gautschi computed the condition number in the Frobenius norm for Vander- k · kF monde matrices associated with families of orthonormal polynomials. Since the diagonal matrix in Lemma 4.0.2 contains the reciprocal of the L2 norms of the Legendre polynomials, we can adapt −1 n x n his arguments to bound V ( ) diag √2j + 1 j=0 in the Frobenius norm. In addition, the j Legendre polynomials have the special property that L (x) 1 for all 0 x 1, and so we can b n x n | | ≤ ≤ ≤ bound the Frobenius norm of V ( ) diag √2j + 1 j=0. b 10 For each integer 0 j n, let ℓj,n be the jth Lagrange polynomial with respect to x; that is, ≤ ≤ x x ℓj,n(x)= − i . (41) xj xi i∈{0,...,nY}\{j} − Deﬁne wn Rn+1 by ∈ n j,n w = ℓ 2 . (42) j k kL Theorem 4.1. n 3/2 n κ n (V (x)) (n + 1) w . (43) M →2 ≤ k k2 Proof. Since Lj(x) 1 for all 0 x 1, we have that | |≤ ≤ ≤ 1/2 n n n j 2 V (x) diag 2j + 1 = (2j + 1) L (xi) j=0   F i,j=0 p X b  1/2  n (2j + 1) ≤   i,jX=0 = (n + 1)3/2. 

Since j,n 1, if i = j; ℓ (xi)= (44) (0, if i = j; 6

11 −1 we have that V n(x) = Θ ℓj,n . Therefore, by (35), we have that ij i b 1/2 n wn = wn 2 k k2  j  j=0 X   1/2 n 1 2 = ℓj,n(x) dx   j=0 Z0 X   1/2 n 1 n j,n j,n i k = Θ(ℓ )iΘ(ℓ )kL (x)L (x)dx  0  Xj=0 Z i,kX=0   1/2 n n 1 j,n j,n i k = Θ(ℓ )iΘ(ℓ )k L (x)L (x)dx  0  Xj=0 i,kX=0 Z  1/2  n 1 j,n 2 i 2 = Θ(ℓ )i (L (x)) dx  0  i,jX=0 Z  1/2  n Θ(ℓj,n)2 = i  2i + 1  i,jX=0  1  n −1 = diag V n(x) √2i + 1 i=0 F b n −1 = V n(x) diag 2j + 1 . j=0 F p b Since the condition number in the 2 norm is bounded above by the condition number in the Frobenius norm, the result follows from Lemma 4.0.2.

5 Higher dimensions

Now, in the case of equispaced nodes on the d-simplex, we can use block-recursive structure as discussed in [14] to give low-complexity algorithms for the inversion of simplicial Bernstein– Vandermonde matrices. This gives, at least for the equispaced lattice, an alternate algorithm for solving simplicial Bernstein interpolation problems to the one worked out in [1].

5.1 Notation and Preliminaries For an integer d 1, let S be a nondegenerate simplex in Rd. Let v d Rd be the vertices of ≥ d { i}i=0 ⊂ S , and let b d denote the barycentric coordinates of S . Each b is an aﬃne map from Rd to d { i}i=0 d i R such that 1, if i = j; bi(vj)= (45) (0, if i = j; 6

12 8 10 n κM n→2(V (x)) Estimate 6 n 10 κ2(V (x))

104

102 Condition Numbers

100 5 10 15 20 n

Figure 1: Comparsion of the condition number of V n in the M n 2 norm and the 2 norm for → 1 n 20, where V n is the matrix described in Subsection 2.3. We also include the estimate ≤ ≤ n given in Theorem 4.1. We use Lemma 4.0.2 to compute κM n→2(V (x)).

for each vertex vj. Each bi is nonnegative on Sd, and

d bi = 1. (46) Xi=0 A multiindex β of length d + 1 is a (d + 1)-tuple of nonnegative integers, written

β = (β0,...,βd). (47)

The order of β, denoted β , is given by | | d β = β . (48) | | i Xi=0 The factorial β! of a multiindex β is deﬁned by

d β!= βi. (49) Yi=0 For a multiindex β of length d + 1, denote by β′ the multiindex of length d given by

′ β = (β1,...,βd). (50)

′ Given a nonnegative integer b and a multiindex β = (β1,...,βd) of length d, deﬁne a new multiindex b β of length d +1 by ⊢ b β′ = (b, β ,...,β ). (51) ⊢ 1 d In particular, β = β β′. (52) 0 ⊢

13 Multiindices have a natural partial ordering given by

β β if and only if β β for all 0 i d. (53) ≤ i ≤ i ≤ ≤

The Bernstein polynomialse of degree n on the d-simplexe Sd are deﬁned by

d n! Bn = bβi . (54) β β! i Yi=0 The complete set of Bernstein polynomials Bn form a basis for polynomials in d variables { β }|β|=n of complete degree at most n. If n n, then any polynomial expressed in the basis Bn0 can also be expressed in the 0 ≤ { β }|β|=n0 n d,n0,n n+d n0+d basis Bβ |β|=n. We denote by E the d d matrix that maps the coefficients of the { } × d,n0,n degree n0 representation to the coefficients of the degree n representation. The matrix E is sparse and can be applied matrix-free [13], if desired. For a nonnegative integer m and a set of distinct nodes xm S , define the Bernstein– { α }|α|=m ⊂ d Vandermonde matrix V d,m,n to be the m+d n+d matrix given by d × d d,m,n n m Vαβ = Bβ (xα ) (55) Theorem 5.1. Let d,m,n , and n be nonnegative integers with d 1 and n n, and let 0 ≥ 0 ≤ xm be distinct nodes contained in the d-simplex S . Then { α }|α|=m d V d,m,nEd,n0,n = V d,m,n0 . (56)

Proof. Given a polynomial p Span Bn0 , the matrix V d,m,n0 evaluates p at each of the ∈ { β }|β|=n0 m d,m,n d,n0,n nodes xα . On the other hand, the matrix V E evaluates the degree n representation of p m at each of the nodes xα . Since we are evaluating at the same nodes and the polynomial has not been modiﬁed, both evaluations must give the same result.

The partial ordering (53) implies a natural way to order the entries of V d,m,n and also imposes d,m,n a block structure on V by dividing the matrix into sections where α0 and β0 are constant. For d,m,n m−α0+d n−β0+d integers 0 α0 m and 0 β0 n, let Vα0β0 denote the d d submatrix of d,m,n ≤ ≤ ≤ ≤ × V whose entries satisfy d,m,n d,m,n Vα0β0 = V ′ ′ . (57) α′β′ (α0⊢α )(β0⊢β ) 5.2 Degree Reduction We now consider the case of equispaced nodes on the d-simplex Sd. For this case, the entries of the Bernstein–Vandermonde matrix are given by

d n! 1 V d,m,n = αβi . (58) αβ β! mn i Yi=0 d,m,n This representation of the entries allows us to represent the submatrix Vα0β0 in terms of a lower- dimensional Bernstein–Vandermonde matrix.

14 Theorem 5.2. Let d> 1 and m,n 0 be integers, and let α , β be integers with 0 α m and ≥ 0 0 ≤ 0 ≤ 0 β0 n. Then ≤ ≤ d,m,n d−1,m−α0,n−β0 Vα0β0 = mα0β0 V , (59) where n m = (α /m)β0 (1 α /m)n−β0 . (60) α0β0 β 0 − 0 0 Proof. If α = m and β = n, then both sides equal zero and we are done. Otherwise, by (58), we 0 0 6 have that d d−1,m−α0,n−β0 (n β0)! 1 βi V ′ ′ = − α . (61) α β ′ n−β0 i β ! (m α0) − Yi=1 On the other hand,

d,m,n d,m,n Vα0β0 = Vαβ α′β′ d n! 1 = αβi β! mn i Yi=0 d n! (n β )! αβ0 (m α )n−β0 1 = − 0 0 − 0 αβi , ′ β0 n−β0 n−β0 i β0!(n β0)! β ! m m (m α0) − − Yi=1 where we have separated terms containing α and β and multiplied and divided by (n β )! and 0 0 − 0 (m α )n−β0 . Since − 0

β0 n−β0 n! α0 (m α0) n β0 n−β0 − = (α0/m) (1 α0/m) , (62) β !(n β )! mβ0 mn−β0 β − 0 − 0 0 we have the desired result.

The fact that the blocks are multiples of lower-dimensional matrices is analogous to what has been observed for other matrices related to Bernstein polynomials. For example, the Bernstein– Vandermonde matrix associated with the zeroes of Legendre polynomials has a similar structure [14], and so does the Bernstein mass matrix [13].

We recognize that mα0β0 describes a univariate Bernstein polynomial of degree n being evaluated at equispaced points. Therefore, if V n is the one-dimensional Bernstein–Vandermonde matrix associated with equispaced nodes as described in Subsection 2.3, then we have that V d,n,n admits the block structure V n V d−1,n,n V n V d−1,n,n−1 V n V d−1,n,0 00 01 · · · 0n V n V d−1,n−1,n V n V d−1,n−1,n−1 V n V d−1,n−1,0 d,n,n 10 11 1n V =  . . ·. · · .  . (63) . . .. .    V n V d−1,0,n V n V d−1,0,n−1 V n V d−1,0,0   n0 n1 · · · nn    We can use this block structure and Theorem 5.1 to perform block Gaussian elimination on V d,n,n. This method was used in [13] to obtain the block LU decomposition of the Bernstein mass matrix. In the same way, we have the following:

15 Theorem 5.3. Let V n be the one-dimensional Bernstein–Vandermonde matrix associated with equispaced nodes as described in Subsection 2.3. Suppose V n = LnU n is the LU decomposition of V n. Then V d,n,n = Ld,nU d,n, (64) where Ld,n is the block lower triangular matrix with blocks given by

d,n n d−1,n−α0,n−β0 Lα0β0 = Lα0β0 V (65) and U d,n is the block upper triangular matrix with blocks given by

d,n n d−1,n−β0,n−α0 Uα0β0 = Uα0β0 E . (66) Proof. By Theorem 5.1, if 0 α , β n and γ β , then ≤ 0 0 ≤ 0 ≤ 0 d,n d,n n n d−1,n−α0,n−γ0 d−1,n−β0,n−γ0 Lα0γ0 Uγ0β0 = Lα0γ0 Uγ0β0 V E n n d−1,n−α0,n−β0 = Lα0γ0 Uγ0β0 V .

Therefore,

β0 d,n d,n d,n d,n L U = Lα0γ0 Uγ0β0 α0β0 γ X0=0 β0 = Ln U n V d−1,n−α0,n−β0  α0γ0 γ0β0  γ0=0 X n d−1,n−α0,n−β0 = Vα0β0 V d,n,n = Vα0β0 .

6 Numerical results

Now, we consider the accuracy of the methods described above on several problems. For all of the problems, we chose random solution vectors, computed the right-hand side by matrix multiplication, and then attempted to recover the solution. The numerical results are run in double precision arithmetic on a 2014 Macbook Air running macOS 10.13 and using Python 2.7.13. Cholesky factorization and FFTs are performed using the numpy (v1.12.0) function calls. Also, because our code is a mix of pure Python and low-level compiled libraries, timings are not terribly informative. Consequently, we focus on assessing the stability and accuracy of our methods. If future work leads to more stable fast algorithms, greater care will be aﬀorded to tuning our implementations for performance. In Figure 2, we consider the case of equispaced nodes. When considering the Euclidean norm of the error (Figure 2a), the LU factorization and the Newton algorithm have the best perfomance, followed by the DFT-based application and multiplying by the inverse. We observe that the LU and Newton methods have comparable performance, as do the DFT-based algorithm and multiplying by the inverse. This is in contrast to what was observed in [3], where a similar DFT-based algorithm quickly became unstable. The same behavior can be observed when comparing the Euclidean norm of the residual (Figure 2c), although some separation does occur between the Newton method

16 10−3 10−3 B´ezout B´ezout DFT 10−6 DFT 10−6 LU n LU M 2 k k Newton

Newton c c

−9 k −9 k 10 10 / / n 2 k M c b k c −12 b −12 − 10 10 c − k c k 10−15 10−15

10−18 10−18 5 10 15 20 5 10 15 20 n n (a) Error in the 2-norm. (b) Error in the M n norm.

10−3 B´ezout

−6 DFT 10 LU

2 Newton k

b 10−9 − c b n −12 V 10 k

10−15

10−18 5 10 15 20 n (c) Residual in the 2-norm.

Figure 2: Error/residual in using the methods described in Section 3 to solve V nc = b for 1 ≤ n 20, where V n is the matrix described in Subsection 2.3 and b is a random vector in [ 1, 1]n+1. ≤ − B´ezout refers to Corollary 2.2, DFT refers to Corollary 2.9, LU refers to LU decomposition of V n, and Newton refers to the Ainsworth–Sanchez algorithm. We use c to denote the computed solution.

b and the LU factorization. Since the solution vector can be viewed as the Bernstein coefficients of the interpolation polynomial, we also measure the L2 difference between the exact and computed solutions (Figure 2b); equivalently, we measure the relative M n error, where M n is the Bernstein mass matrix given in (24). All four solution methods have very similar behavior in the M n norm as they do when considering the Euclidean norm of the residual. To ensure that the behavior of the solution methods does not depend on the choice of nodes, we also considered the case where the nodes xj are randomly selected from [j/(n+1), (j+1)/(n+1)) for each 0 j n (Figure 3); however, there was no significant difference in the quantities measured ≤ ≤ between this case and the equispaced case. We also used the block LU decomposition given in Theorem 5.3 combined with the one- dimensional LU algorithm to solve the interpolation problem for equispaced nodes on the d-simplex for d = 2 and d = 3 (Figure 4). Due to the recursive nature of the algorithm, the quantities measured are very similar to the ones observed for equispaced nodes.

17 10−3 10−3 B´ezout B´ezout DFT −6 DFT −6

10 n 10 LU LU M 2 k k Newton

Newton c c k

k −9 −9

10 / 10 / n 2 k M c b k c −12 b −12 − 10 10 c − k c k 10−15 10−15

10−18 10−18 5 10 15 20 5 10 15 20 n n (a) Error in the 2-norm. (b) Error in the M n norm.

10−3 B´ezout DFT −6 10 LU 2 k Newton b −9 − 10 c b ) x (

n 10−12 V k 10−15

10−18 5 10 15 20 n (c) Residual in the 2-norm.

Figure 3: Error/residual in using the methods described in Section 3 to solve V n(x)c = b for 1 n 20, where V n(x) is the Bernstein–Vandermonde matrix associated to x, the nodes x are ≤ ≤ j randomly selected from [j/(n + 1), (j + 1)/(n + 1)) for 0 j n, and b is a random vector in ≤ ≤ [ 1, 1]n+1. B´ezout refers to Corollary 2.2, DFT refers to Theorem 2.8, LU refers to LU decom- − position of V n, and Newton refers to the Ainsworth–Sanchez algorithm. We use c to denote the computed solution. b

18 10−12 10−9 d = 2 d = 2 d = 3 d = 3 10−13 d,n −11 M 2 10 k k −14 c c 10 k k / / 2 −13 k 10 d,n −15 c b 10 M k − c b c −16 k −15 10 − 10 c k 10−17 10−17 10−18 5 10 15 20 5 10 15 20 n n (a) Error in the 2-norm. (b) Error in the M d,n norm.

d = 2 10−15 d = 3 2 k b

− −16 c b 10 d,n,n V k 10−17

10−18 5 10 15 20 n (c) Residual in the 2-norm.

Figure 4: Error/residual in using the block LU decomposition given in Theorem 5.3 to solve V d,n,nc = b for 1 n 20 and d = 2, 3, where V d,n,n is the Bernstein–Vandermonde matrix given ≤ ≤ n+d in (58) and b is a random vector in [ 1, 1]( d ). We use c to denote the computed solution. − b

19 7 Conclusion

We have studied several algorithms for the inversion of the univariate Bernstein–Vandermonde matrix. These algorithms, while less stable than algorithms discovered previously, provide insight into the structure of the Bernstein–Vandermonde matrix and are remarkably similar to algorithms derived for the Bernstein mass matrix. In addition, we have used a block LU decomposition of the Bernstein–Vandermonde matrix corresponding to equispaced nodes on the d-simplex to give a recursive, block-structured algorithm with comparable accuracy to the one-dimensional algorithm. Moreover, we have given a new perspective on the conditioning of the Bernstein–Vandermonde matrix, indicating that the interpolation problem is better-conditioned with respect to the L2 norm than the Euclidean norm. In the future, we hope to expand this perspective to other polynomial problems and continue the development of fast and accurate methods for problems involving Bernstein polynomials.

20 References

[1] Mark Ainsworth, Gaelle Andriamaro, and Oleg Davydov. Bernstein–Bézier finite elements of arbitrary order and optimal assembly procedures. SIAM Journal on Scientific Computing, 33(6):3087–3109, 2011. [2] Mark Ainsworth and Manuel A. Sánchez. Computing the bézier control points of the lagrangain interpolant in arbitrary dimension. SIAM Journal on Scientific Computing, 38(3):A1682– A1700, 2016. [3] Larry Allen and Robert C. Kirby. Structured inversion of the bernstein mass matrix. SIAM Journal of Matrix Analysis and Applications, 41(2):413–431, April 2020. [4] Richard L. Burden and J. Douglas Faires. Numerical Methods. Cengage Learning, 4 edition, 2012. [5] Philip J. Davis. Interpolation and Approximation. Dover Publications, Inc., New York, 1975. [6] Michael G Duffy. Quadrature over a pyramid or cube of integrands with a singularity at a vertex. SIAM journal on Numerical Analysis, 19(6):1260–1262, 1982. [7] Rida T. Farouki. Legendre–Bernstein basis transformations. Journal of Computational and Applied Mathematics, 119(1-2):145–160, 2000. [8] Mariano Gasca and J. M. Peña. Total positivity and neville elimination. Linear Algebra and its Applications, 165:25–44, 1992. [9] Walter Gautschi. The condition of vandermonde-like matrices involving orthogonal polynomials. Linear Algebra and its Applications, 52/53:293–300, 1983. [10] Walter Gautschi, Gene H. Golub, and Gerhard Opfer. Applications and Computation of Or- thogonal Polynomials. Springer, 1999. [11] Georg Heinig and Karla Rost. Algebraic methods for Toeplitz-like matrices and operators, volume 13. Birkhäuser, 1984. [12] Shmuel Kaplan, Alexander Shapiro, and Mina Teicher. Several applications of bézout matrices, 2006. [13] Robert C. Kirby. Fast inversion of the simplicial Bernstein mass matrix. Numerische Mathe- matik, 135(1):73–95, 2017. [14] Robert C. Kirby and Kieu Tri Thinh. Fast simplicial quadrature-based finite element operators using Bernstein polynomials. Numerische Mathematik, 121(2):261–279, 2012. [15] Ana Marco and José-Javier Mart´ınez. A fast and accurate algorithm for solving bernstein– vandermonde linear systems. Linear Algebra and its Applications, 422(2):616–628, 2007. [16] Joseph J. Rotman. Advanced Modern Algebra. American Mathematical Society, 3 edition, 2015. [17] Irene A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathe- matical Tables. U.S. Government Printing Office, 1965. [18] Curtis R. Vogel. Computational Methods for Inverse Problems. SIAM, 2002.