Minimizing the Condition Number of a Gram Matrix

The condition number of a Gram matrix defined by a polynomial basis and a set of points is often used to measure the sensitivity of the polynomial approximation. Given a polynomial basis, we consider the problem of finding a set of points and/or weights which minimizes the condition number of the Gram matrix. We use smoothing methods to solve such minimization problems and show the global convergence of smoothing methods. To illustrate applications of minimizing the condition number, we report numerical results for the Gram matrix defined by the weighted Vandermonde-like matrix for least squares polynomial approximation on an interval, and the Gram matrix defined by an orthonormal set of real spherical harmonics for interpolation on the sphere.