The Pseudo-Inverse of the Derivative Operator in Polynomial Spectral Methods
Josip Lončarić · 1997
The matrix D kI in polynomial approximations of order N is similar to a large Jordan block which isinvertible for nonzero k but extremely sensitive to perturbation. Solving the problem (D kI)f = g involves similarity transforms whose condition number grows as N!, which exceeds typical machine precision for N>17. By using orthogonal projections, we reformulate the problem in terms of Q, the pseudo-inverse of D, and therefore its optimal preconditioner. The matrix Q in commonly used Chebyshev or Legendre representations is a simple tridiagonal matrix and its eigenvalues are small and imaginary. The particular solution of (I kQ)f = Qg can be found for all real k at high resolutions and low computational cost (O(N) times faster than the commonly used Lanczos tau method). Boundary conditions are applied later by adding a multiple of the known homogeneous solution. In Chebyshev representation, machine precision results are achieved at modest resolution requirements. Multidimensional and higher order di erential operators can also take advantage of the simple form of Q by factoring or preconditioning.