Existence of Solutions to Systems of Underdetermined Equations and Spherical Designs

• Published in: SIAM journal on numerical analysis Vol. 44; no. 6; pp. 2326 - 2341
• Main Authors: Chen, Xiaojun, Womersley, Robert S.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2006
• Subjects: Combinatorics; Combinatorics. Ordered structures; Degrees of polynomials; Designs and configurations; Determinants; Error bounds; Exact sciences and technology; Global analysis, analysis on manifolds; Jacobians; Mathematical intervals; Mathematics; Nonlinear equations; Numerical analysis; Numerical analysis. Scientific computation; Numerical integration; Polynomials; Sciences and techniques of general use; Sine function; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; numerical integration; underdetermined system; spherical designs; Error estimation; Existence of solution; Determinant; Equation system; Sphere; 65D30; interpolation; Numerical analysis; Fix point; Fixed point theorem; 65G20; 65D05; 65H10; extremal points; verification
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/050626636

This paper is concerned with proving the existence of solutions to an underdetermined system of equations and with the application to existence of spherical t-designs with (t + 1)² points on the unit sphere S² in R³ . We show that the construction of spherical designs is equivalent to solution of underdetermined equations. A new verification method for underdetermined equations is derived using Brouwer's fixed point theorem. Application of the method provides spherical t-designs which are close to extremal (maximum determinant) points and have the optimal order O(t²) for the number of points. An error bound for the computed spherical designs is provided.