The Pseudo Inverse Of The Derivative Operator In Polynomial Spectral Methods

The Pseudo-Inverse of the Derivative Operator in Polynomial Spectral Methods

The matrix D - kI in polynomial approximations of order N is similar to a large Jordan block which is invertible for nonzero k but extremely sensitive to perturbation. Solving the problem (D - kI)f = g involves similarity transforms whose condition number grows as NI, 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 differential operators can also take advantage of the simple form of Q by factoring or preconditioning.

The Pseudo-inverse of the Derivative Operator in Polynomial Spectral Methods

Chebyshev and Fourier Spectral Methods

Completely revised text focuses on use of spectral methods to solve boundary value, eigenvalue, and time-dependent problems, but also covers Hermite, Laguerre, rational Chebyshev, sinc, and spherical harmonic functions, as well as cardinal functions, linear eigenvalue problems, matrix-solving methods, coordinate transformations, methods for unbounded intervals, spherical and cylindrical geometry, and much more. 7 Appendices. Glossary. Bibliography. Index. Over 160 text figures.