The Tau method and a new preconditioner

In relations to the order of linear ordinary differential equations, using a modified form of the Chebyshev or Legendre and Gegenbauer polynomials, some particular integral operators are introduced. These are used to give a factorization of the operators arising from the application of the Chebyshev or Legendre Tau method. The New-Tau method presented in this article is then compared with the standard Tau method and preconditioned method of Cabos. The New-Tau method shows a superior performance. An analysis of error and a bound for condition number is given. Numerical examples applying iterative solvers show dramatic reduction in condition number and improved convergence for the Tau method with the new preconditioner.