Existence of solutions to systems of underdetermined equations and spherical designs

Xiaojun Chen, Robert S. Womersley

Research output: Journal article publicationJournal articleAcademic researchpeer-review

39 Citations (Scopus)


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)2points on the unit sphere S2in 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.
Original languageEnglish
Pages (from-to)2326-2341
Number of pages16
JournalSIAM Journal on Numerical Analysis
Issue number6
Publication statusPublished - 1 Dec 2006
Externally publishedYes


  • Extremal points
  • Interpolation
  • Numerical integration
  • Spherical designs
  • Underdetermined system
  • Verification

ASJC Scopus subject areas

  • Numerical Analysis

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