A projection algorithm on the set of polynomials with two bounds

Abstract

The motivation of this work stems from the numerical approximation of bounded functions by polynomials satisfying the same bounds. The present contribution makes use of the recent algebraic characterization found in Després (Numer. Algorithms 76(3), 829–859, 2017) and Després and Herda (Numer. Algorithms 77(1), 309–311, 2018) where an interpretation of monovariate polynomials with two bounds is provided in terms of a quaternion algebra and the Euler four-squares formulas. Thanks to this structure, we generate a new nonlinear projection algorithm onto the set of polynomials with two bounds. The numerical analysis of the method provides theoretical error estimates showing stability and continuity of the projection. Some numerical tests illustrate this novel algorithm for constrained polynomial approximation.

Appendix: An algorithm for positive polynomial approximation

Here, we briefly describe the method used in the numerical tests to compute the positive (or nonnegative) polynomial approximations in Lukacs form. The problem is to find two polynomials a ∈ P_n and b ∈ P_n− 1 defining a positive polynomial p₀(x) = a(x)² + b(x)²w(x) such that given some data $(x_{r}, y_{r})_{r = 1,\dots ,R}$ (in general with $R = \dim (P_{2n}) = 2n+1$) the images (p₀(x_r))_r are a good approximation of (y_r)_r.

Our algorithm consists in a least-square minimization where a and b are “oscillating polynomials” parametrized by their roots. This parametrization is motivated by the method of [2] where a similar technique has been developed and analyzed for positive interpolation.

Mathematically, the method relies on the following optimization problem. Find

$$ (\alpha^{*}, {\upbeta}^{*})\ \in\ \text{argmin}_{\alpha\in{\mathbb{R}}^{n+1},\upbeta\in{\mathbb{R}}^{n}}J_{t}(\alpha,\upbeta) $$

where the objective function is

$$ J(\alpha,\upbeta)\ =\ \sum\limits_{r = 1}^{R}|a[\alpha](x_{r})^{2} + b[\upbeta](x_{r})^{2}w(x_{r}) - y_{r}|^{2} , $$

with a and b parametrized as follows,

$$ a[\alpha](x)\ =\ 2^{n-1} \alpha_{0} \prod\limits_{i=1}^{n} (x-\alpha_{i}) ,\qquad b[\upbeta](x)\ =\ 2^{n-1} {\upbeta}_{0} \prod\limits_{i=1}^{n-1} (x-{\upbeta}_{i}) . $$

The factor 2^n− 1 is taken so that α₀ and β₀ are of the same order as the other components of α and β. Then, the approximation polynomial p is defined by

$$ p_{0}(x) = a[\alpha^{*}](x)^{2} + b[{\upbeta}^{*}](x)^{2}w(x) $$

The optimization problem is nonlinear and non-convex. However, it can be solved efficiently in practice. Indeed, one can compute explicitly both the gradient and hessian of the functional J. In the numerical tests of Section 5, we used a Newton conjugate gradient trust-region algorithm. The initial couple (α,β ) is taken to be appropriate roots of Chebychev polynomials. In this way, the initial polynomials a[α] and b[β] are proportional to T_n and U_n, yielding a[α]²(x) + b[β]²(x)w(x) being some constant polynomial. In all the cases of Section 5, n = 5 and the algorithm converges after around 30 iterations.