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Chebyshev differentiation matrix

We conclude with a discussion on Chebyshev differencing. Starting with grid values at Chebyshev points , one constructs the Chebyshev interpolant
One can compute , efficiently via the cos-FFT with operations. Next, we differentiate in Chebyshev space
In this case, however, is not an eigenfunction of ; instead - being a polynomial of degree , can be expressed as a linear combination of (in fact is even/odd for even/odd k's): with we obtain
and hence
Rearranging we get (here, tex2html_wrap_inline11155 indicates halving the last term)
and similarly for the second derivative
The amount of work to carry out the differentiation in this form is operations which destroys the efficiency. Instead, we can employ the recursion relation which follows directly from (app_cheb.44)
To see this in a different way we note that

which leads to

and hence

as asserted. In general we have
With this, can be evaluated using operations, and the differentiated polynomial at the grid points is computed using another cos-FFT employing operations
with spectral/exponential error
The matrix representation of Chebyshev differentiation, , takes the almost antisymmetric form (here except for )

Eitan Tadmor
Thu Jan 22 19:07:34 PST 1998