ASYMPTOTICS OF SOME ULTRA-SPHERICAL POLYNOMIALS AND THEIR EXTREMA MAGALI RIBOT Abstract. Motivated by questions on the preconditioning of spectral meth- ods, and independently of the extensive literature on the approximation of ze- roes of orthogonal polynomials, either by the Sturm method, or by the descent method, we develop a stationary phase-like technique for calculating asymp- totics of Legendre polynomials. The difference with the classical stationary phase method is that the phase is a nonlinear function of the large parameter and the integration variable, instead of being a product of the large parameter by a function of the integration variable. We then use an implicit functions theorem for approximating the zeroes of the derivatives of Legendre polyno- mials. This result is used for proving order and consistency of the residual smoothing scheme [1], [19]. 1. Introduction When we discretize implicitly in time a partial differential equation, we have to solve a linear system, where the matrix depends on the method used for the spatial discretization. Spectral methods are classical methods, but they produce matrices, which are not sparse and difficult to invert; therefore, their numerical efficiency depends on the introduction of appropriate preconditioners. A preconditioner P of a matrix M is a matrix, which can be more easily inverted than M and such that the condition number of P?1M , that is to say the product of the norm of the matrix P?1M by the norm of its inverse M?1P , is as close to 1 as possible.
- p1 finite
- let ?k
- stiffness matrix
- phase method
- parter's theorem
- legendre spectral
- precise estimates
- legendre polynomials