从数据点计算多项式平方和
Computation of sum of squares polynomials from data points
摘要：
本文提出了一种在给定插值点上逼近给定数据的多项式平方和的数值计算的迭代算法。该方法是基于定义的凸泛函$G$所产生的二元化的二次回归的Cholesky因素的平方和分解。为了证明该构造的合理性，对$G$的定义域、边界以及无穷远处的行为进行了详细的分析。当数据插值一个正的一元多项式，我们表明，在Lukacs的平方和表示的上下文中，$G$是强制性的和严格凸产生一个唯一的临界点和相应的分解平方和。对于多元多项式，其中承认在平方和分解，直到一个小扰动的大小$\varepsilon$，$G^\varepsilon$总是强制性的，所以它最小产生一个近似的平方和分解。提出了各种无约束下降算法来最小化$G$。数值例子提供，单变量和双变量多项式。
英文摘要：
We propose an iterative algorithm for the numerical computation of sums of squares of polynomials approximating given data at prescribed interpolation points. The method is based on the definition of a convex functional $G$ arising from the dualization of a quadratic regression over the Cholesky factors of the sum of squares decomposition. In order to justify the construction, the domain of $G$, the boundary of the domain and the behavior at infinity are analyzed in details. When the data interpolate a positive univariate polynomial, we show that in the context of the Lukacs sum of squares representation, $G$ is coercive and strictly convex which yields a unique critical point and a corresponding decomposition in sum of squares. For multivariate polynomials which admit a decomposition in sum of squares and up to a small perturbation of size $\varepsilon$, $G^\varepsilon$ is always coercive and so it minimum yields an approximate decomposition in sum of squares. Various unconstrained descent algorithms are proposed to minimize $G$. Numerical examples are provided, for univariate and bivariate polynomials.
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