Part I: Background and Existing Work 8
3.2 Properties of the Bernstein Coefficients
From the point of view of bounding the ranges of polynomials, the Bernstein coefficients (3.9, 3.13) exhibit a number of important and useful properties, encapsulating the behaviour and properties of a polynomial over a box of interest. These coefficients may be collected in ann-dimensional array (i.e. a tensor); in the field of computer-aided geometric design such a construct is typically labelled as apatch. For simplicity, we will refer to the tensor of the Bernstein coefficients hereafter as an ‘array’.
Where the general case is considered, we shall assume that we have an n-dimensional polynomialp(3.6) of degree l= (l1, . . . , ln)over a boxX (3.11). In this case the number of coefficients appearing in this array is
#{bi} =
It is readily apparent that the2nBernstein coefficients occurring at a vertex of the array are identical to the values attained by p at the corresponding vertices of X. In the univariate case over the unit box:
3 Bernstein Expansion X (3.11). Then the Bernstein coefficients of p over the m-dimensional faces of X, where 0≤m≤l−1, are the same as the coefficients located at the corresponding m-dimensional faces of the array of Bernstein coefficients of p over X.
Proof (see also [GS01b]): We consider the case of the unit box I= [0,1]n without loss of generality. It suffices to prove the statement for m = l−1; the statement for smaller m then follows by repeated application. Here we indicate by a subscript (k) that the quantity under consideration is taken without the contribution of the kth component, e.g.
x(k)= (x1, . . . , xk−1, xk+1, . . . , xn). Theith Bernstein coefficient of p(x1, . . . , xk−1,0, xk+1, . . . , xn) = X
i≤l,ik=0
ai(xi)(k),
considered as a polynomial inn−1 variables, is given by X
which coincides withbi1...ik−10ik+1...in. Similarly, the ith Bernstein coefficient of
p(x1, . . . , xk−1,1, xk+1, . . . , xn) = X
An application of Lemma 3.1 for bounding the range of a polynomial over an edge of a box was given in [ZG98, Edge Lemma].
3 Bernstein Expansion
3.2.3 Linearity
Letp(x) =αp1(x) +βp2(x), wherep1 andp2 are polynomials andlis the degree ofp. Then bi = αb{pi 1}+βb{pi 2} ∀0≤i≤l,
whereb{pi 1} and b{pi 2} are theith coefficients of the degreel Bernstein expansions ofp1 and p2, respectively.
3.2.4 Range Enclosure
The essential property of the Bernstein expansion, given by Cargo and Shisha [CS66], is that the range ofp over X is contained within the interval spanned by the minimum and maximum Bernstein coefficients (which is called the Bernstein enclosure):
mini {bi} ≤ p(x) ≤ max
i {bi}, x∈X. (3.17)
The proof follows readily by firstly applying an affine transformation to the unit boxI and observing, as in [Riv70], from (3.7) that, for all x ∈ I, p(x) is a convex combination of b0, . . . , bl.
By computing all the Bernstein coefficients, one therefore obtains bounds on the range of poverX; this is central to the application of the Bernstein expansion in enclosure methods.
As discussed in Section 2.2, it is known [Sta95] that these bounds are in general tighter than those given by interval arithmetic and many centered forms. However it should be noted that the computational effort of generating all the Bernstein coefficients is often much greater than that of these alternatives.
3.2.5 Sharpness
The lower bound on the range of p overX provided by the minimum Bernstein coefficient is sharp, i.e. there is no underestimation, if and only if this coefficient occurs at a vertex of X. Likewise, the upper bound provided by the maximum Bernstein coefficient is sharp (in this case, there is no overestimation), if and only if the coefficient occurs at a vertex of X. If both bounds are sharp, then the Bernstein enclosure provides the exact range; this property allows one to easily test whether or not this is the case.
3.2.6 Convex Hull
This property is a generalisation of the range enclosing property. We firstly need to define thecontrol pointsassociated with the Bernstein coefficients, and then we shall consider their convex hull.
Definition 3.1(Control Points). Given the degreelBernstein expansion (i.e. the Bernstein coefficients) of ann-dimensional polynomialpover the unit boxI= [0,1]n, the control point
3 Bernstein Expansion
associated with the ith Bernstein coefficient bi is the point bi ∈Rn+1 given by bi :=
The abscissae of the control points thus form a uniform grid over the box and the ordinates are equal to the values of the Bernstein coefficients.
Theorem 3.4(Convex Hull Property). The graph ofpoverIis contained within the convex hull of the control points derived from the Bernstein coefficients, i.e.
{(x1, . . . , xn, p(x))|x∈I} ⊆ conv{bi|0≤i≤l}. (3.19) Proof: Consider the degree l Bernstein expansion of the identity function idj(x) :=xj. From (3.9) we have thatb{idi j} = ilj contained within their convex hull. 2
Figure 3.2 illustrates the convex hull property for a univariate polynomial of degree 5 over the unit interval. The property holds in the case of a general box X by simply adjusting the control points accordingly so that they form a uniform grid overX.
3.2.7 Inclusion Isotonicity
The Bernstein form is inclusion isotone (see Definition 2.12), i.e. if the intervalXis shrunk to a smaller interval then the Bernstein enclosure shrinks, too. This was proven for the univariate case in [HS95a]. Here we give a shorter proof for the multivariate case and an extension to show that the convex hull of the control points associated with the Bernstein coefficients is also inclusion isotone [GJS03a]. For simplicity we consider the case of the unit intervalI= [0,1].
Theorem 3.5 (Inclusion Isotonicity of the Convex Hull [GJS03a]). The convex hull of the control points associated with the Bernstein coefficients of a univariate polynomial is inclusion isotone.
Proof: Let p be a degree l univariate polynomial, with Bernstein coefficients b0, . . . , bl
over the interval [0,1]. It suffices to show that inclusion isotonicity holds if we shrink only one of the two endpoints. Letb∗0, . . . , b∗l andb†0, . . . , b†l be the Bernstein coefficients ofp over
3 Bernstein Expansion
bI
b2
b5
b1
b0 b4
b3
0.4 0.6 0.8 0.2
0 1 5
i
Figure 3.2: The graph of a degree5 polynomial and the convex hull (shaded) of its control points (marked by squares).
b2
b5 b0
bI
b1 b3
b4
0.4 0.6 0.8 0.2
0 1 5
i
0.5
Figure 3.3: The graph of the polynomial in Figure 3.2 with the convex hull (shaded light) of its control points and its Bernstein enclosure, together with the smaller convex hulls (shaded dark) and enclosures over sub-domains arising from a bisection.
3 Bernstein Expansion
in Subsection 3.3.2 and may use it to compute the Bernstein coefficients on the intervals [0,1−ε]and[ε,1]. In (3.27), only convex combinations are formed, therefore
b∗i ∈conv{b0, . . . ,bi} and b†i ∈conv{bi, . . . ,bl}, i= 0, . . . , l. 2 (3.21) In the multivariate case, the proof of the inclusion isotonicity of the convex hull of the con-trol points can be obtained similarly by shrinking the unit box in each dimension separately, thus forming convex combinations in only one direction.
As an immediate consequence, we also have:
Corollary 3.1 (Hong and Stahl [HS95a]). The enclosure for the range of a univariate polynomialp over an intervalX provided by the interval spanning its Bernstein coefficients b0, . . . , bl is inclusion isotone.
Figure 3.3 illustrates this property with the same polynomial as above, after performing a bisection about the midpoint. The inclusion isotonicity of convex-concave extensions and affine bounding functions based upon the Bernstein coefficients is considered in Chapter 10.
3.2.8 Partial Derivatives
As given by Farouki and Rajan [FR88], the partial derivative ofpwith respect toxr, where r= 1, . . . , n, is given by
∂p
∂xr
(x) = lr
X
i≤(l1,...,lr−1,...,ln)T
(bi1... ir+1...in−bi)Bil1... lr−1...ln(x). (3.22)