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annotate toolbox/chebspec.m @ 2:c124219d7bfa draft

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Re-add the 1995 toolbox after noticing the statement in the ~higham/mctoolbox/ webpage.
author Antonio Pino Robles <data.script93@gmail.com>
date Thu, 07 May 2015 18:36:24 +0200
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1 function C = chebspec(n, k)
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2 %CHEBSPEC Chebyshev spectral differentiation matrix.
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3 % C = CHEBSPEC(N, K) is a Chebyshev spectral differentiation
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4 % matrix of order N. K = 0 (the default) or 1.
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5 % For K = 0 (`no boundary conditions'), C is nilpotent, with
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6 % C^N = 0 and it has the null vector ONES(N,1).
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7 % C is similar to a Jordan block of size N with eigenvalue zero.
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8 % For K = 1, C is nonsingular and well-conditioned, and its eigenvalues
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9 % have negative real parts.
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10 % For both K, the computed eigenvector matrix X from EIG is
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11 % ill-conditioned (MESH(REAL(X)) is interesting).
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12
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13 % References:
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14 % C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral
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15 % Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988; p. 69.
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16 % L.N. Trefethen and M.R. Trummer, An instability phenomenon in
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17 % spectral methods, SIAM J. Numer. Anal., 24 (1987), pp. 1008-1023.
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18 % D. Funaro, Computing the inverse of the Chebyshev collocation
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19 % derivative, SIAM J. Sci. Stat. Comput., 9 (1988), pp. 1050-1057.
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21 if nargin == 1, k = 0; end
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22
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23 % k = 1 case obtained from k = 0 case with one bigger n.
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24 if k == 1, n = n + 1; end
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25
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26 n = n-1;
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27 C = zeros(n+1);
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28
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29 one = ones(n+1,1);
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30 x = cos( (0:n)' * (pi/n) );
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31 d = ones(n+1,1); d(1) = 2; d(n+1) = 2;
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32
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33 % eye(size(C)) on next line avoids div by zero.
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34 C = (d * (one./d)') ./ (x*one'-one*x' + eye(size(C)));
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36 % Now fix diagonal and signs.
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37
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38 C(1,1) = (2*n^2+1)/6;
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39 for i=2:n+1
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40 if rem(i,2) == 0
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41 C(:,i) = -C(:,i);
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42 C(i,:) = -C(i,:);
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43 end
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44 if i < n+1
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45 C(i,i) = -x(i)/(2*(1-x(i)^2));
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46 else
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47 C(n+1,n+1) = -C(1,1);
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48 end
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49 end
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51 if k == 1
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52 C = C(2:n+1,2:n+1);
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53 end