Geodesic
The problem of projecting the origin on a quadric
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 1 (2013), pp. 11-17

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The problem of finding a point on a quadric with the least Euclidean norm is considered. This is a classical optimization problem for whose solving there exist a lot of methods, e.g., the method of Lagrange multipliers. In this paper a method for solving the stated problem is proposed. Depending on the sign of the constant term of the quadratic function defining the quadric, the original problem is reduced to one of two types of problems, each of them constructs a polynomial of degree $2n$ and finds its positive roots. Such roots always exist. For the positive numbers thus constructed, the points lying on the quadric and having the smallest Euclidean norm are determined. If the given set is an ellipsoid defined by a quadratic function with a negative constant term, the method allows to determine not only the points with the minimal norm, but also the points which are the most remoted from the origin (having the maximal distance from the origin). Bibliogr. 7. Il. 2.
Mots-clés : quadric, Euclidean norm
Keywords: the Lagrange function, positive roots of a polynomial. 
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D. M. Lebedev; L. N. Polyakova. The problem of projecting the origin on a quadric. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 1 (2013), pp. 11-17. http://geodesic.mathdoc.fr/item/VSPUI_2013_1_a1/