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annotate toolbox/pscont.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 [x, y, z, m] = pscont(A, k, npts, ax, levels)
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2 %PSCONT Contours and colour pictures of pseudospectra.
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3 % PSCONT(A, K, NPTS, AX, LEVELS) plots LOG10(1/NORM(R(z))),
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4 % where R(z) = INV(z*I-A) is the resolvent of the square matrix A,
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5 % over an NPTS-by-NPTS grid.
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6 % NPTS defaults to a SIZE(A)-dependent value.
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7 % The limits are AX(1) and AX(2) on the x-axis and
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8 % AX(3) and AX(4) on the y-axis.
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9 % If AX is omitted, suitable limits are guessed based on the
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10 % eigenvalues of A.
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11 % The eigenvalues of A are plotted as crosses `x'.
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12 % K determines the type of plot:
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13 % K = 0 (default) PCOLOR and CONTOUR
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14 % K = 1 PCOLOR only
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15 % K = 2 SURFC (SURF and CONTOUR)
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16 % K = 3 SURF only
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17 % K = 4 CONTOUR only
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18 % The contours levels are specified by the vector LEVELS, which
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19 % defaults to -10:-1 (recall we are plotting log10 of the data).
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20 % Thus, by default, the contour lines trace out the boundaries of
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21 % the epsilon pseudospectra for epsilon = 1e-10, ..., 1e-1.
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22 % [X, Y, Z, NPTS] = PSCONT(A, ...) returns the plot data X, Y, Z
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23 % and the value of NPTS used.
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24 %
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25 % After calling this function you may want to change the
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26 % color map (e.g., type COLORMAP HOT - see HELP COLOR) and the
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27 % shading (e.g., type SHADING INTERP - see HELP INTERP).
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28 % For an explanation of the term `pseudospectra' see PS.M.
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29 % When A is real and the grid is symmetric about the x-axis, this
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30 % routine exploits symmetry to halve the computational work.
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31
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32 % Colour pseduospectral pictures of this type are referred to as
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33 % `spectral portraits' by Godunov, Kostin, and colleagues.
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34 % References:
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35 % V. I. Kostin, Linear algebra algorithms with guaranteed accuracy,
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36 % Technical Report TR/PA/93/05, CERFACS, Toulouse, France, 1993.
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37 % L.N. Trefethen, Pseudospectra of matrices, in D.F. Griffiths and
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38 % G.A. Watson, eds, Numerical Analysis 1991, Proceedings of the 14th
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39 % Dundee Conference, vol. 260, Pitman Research Notes in Mathematics,
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40 % Longman Scientific and Technical, Essex, UK, 1992, pp. 234-266.
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41
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42 Areal = ~norm(imag(A),1);
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43
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44 if nargin < 5, levels = -10:-1; end
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45 e = eig(A);
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46 if nargin < 4
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47 ax = cpltaxes(e);
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48 if Areal, ax(3) = -ax(4); end % Make sure region symmetric about x-axis.
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49 end
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50 n = max(size(A));
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51 if nargin < 3, npts = round( min(max(5, sqrt(20^2*10^3/n^3) ), 30)); end
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52 if nargin < 2, k = 0; end
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53
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54 nptsx = npts; nptsy = npts;
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55 Ysymmetry = (Areal & ax(3) == -ax(4));
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56
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57 x = seqa(ax(1), ax(2), npts);
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58 y = seqa(ax(3), ax(4), npts);
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59 if Ysymmetry % Exploit symmetry about x-axis.
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60 nptsy = ceil(npts/2);
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61 y1 = y;
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62 y = y(1:nptsy);
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63 end
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64
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65 [xx, yy] = meshgrid(x,y);
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66 z = xx + sqrt(-1)*yy;
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67 I = eye(n);
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68 Smin = zeros(nptsy, nptsx);
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69
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70 for j=1:nptsx
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71 for i=1:nptsy
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72 Smin(i,j) = min( svd( z(i,j)*I-A ) );
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73 end
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74 end
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75
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76 z = log10( Smin + eps );
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77 if Ysymmetry
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78 z = [z; z(nptsy-rem(npts,2):-1:1,:)];
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79 y = y1;
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80 end
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81
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82 if k == 0 | k == 1
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83 pcolor(x, y, z); hold on
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84 elseif k == 2
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85 surfc(x, y, z); hold on
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86 elseif k == 3
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87 surf(x, y, z); hold on
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88 end
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89
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90 if k == 0 | k == 4
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91 contour(x, y, z, levels); hold on
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92 end
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93
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94 if k ~= 2 & k ~= 3
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95 if k == 0 | k == 1
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96 s = 'k'; % Black.
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97 else
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98 s = 'w'; % White.
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99 end
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100 plot(real(e),imag(e),['x' s]);
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101 end
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102
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103 axis('square');
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104 hold off