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annotate toolbox/ps.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 y = ps(A, m, tol, rl, marksize)
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2 %PS Dot plot of a pseudospectrum.
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3 % PS(A, M, TOL, RL) plots an approximation to a pseudospectrum
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4 % of the square matrix A, using M random perturbations of size TOL.
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5 % M defaults to a SIZE(A)-dependent value and TOL to 1E-3.
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6 % RL defines the type of perturbation:
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7 % RL = 0 (default): absolute complex perturbations of 2-norm TOL.
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8 % RL = 1: absolute real perturbations of 2-norm TOL.
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9 % RL = -1: componentwise real perturbations of size TOL.
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10 % The eigenvalues of A are plotted as crosses `x'.
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11 % PS(A, M, TOL, RL, MARKSIZE) uses the specified marker size instead
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12 % of a size that depends on the figure size, the matrix order, and M.
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13 % If MARKSIZE < 0, the plot is suppressed and the plot data is returned
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14 % as an output argument.
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15 % PS(A, 0) plots just the eigenvalues of A.
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16
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17 % For a given TOL, the pseudospectrum of A is the set of
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18 % pseudo-eigenvalues of A, that is, the set
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19 % { e : e is an eigenvalue of A+E, for some E with NORM(E) <= TOL }.
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20 %
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21 % Reference:
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22 % L.N. Trefethen, Pseudospectra of matrices, in D.F. Griffiths and
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23 % G.A. Watson, eds, Numerical Analysis 1991, Proceedings of the 14th
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24 % Dundee Conference, vol. 260, Pitman Research Notes in Mathematics,
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25 % Longman Scientific and Technical, Essex, UK, 1992, pp. 234-266.
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26
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27 if diff(size(A)), error('Matrix must be square.'), end
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28 n = max(size(A));
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29
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30 if nargin < 5, marksize = 0; end
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31 if nargin < 4, rl = 0; end
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32 if nargin < 3, tol = 1e-3; end
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33 if nargin < 2, m = max(1, round( 25*exp(-0.047*n) )); end
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34
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35 if m == 0
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36 e = eig(A);
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37 ax = cpltaxes(e);
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38 plot(real(e), imag(e), 'xg')
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39 axis(ax);
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40 axis('square');
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41 return
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42 end
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43
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44 x = zeros(m*n,1);
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45 i = sqrt(-1);
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46
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47 for j = 1:m
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48 if rl == -1 % Componentwise.
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49 dA = -ones(n) + 2*rand(n); % Uniform random numbers on [-1,1].
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50 dA = tol * A .* dA;
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51 else
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52 if rl == 0 % Complex absolute.
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53 dA = randn(n) + i*randn(n);
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54 else % Real absolute.
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55 dA = randn(n);
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56 end
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57 dA = tol/norm(dA)*dA;
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58 end
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59 e = eig(A + dA);
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60 x((j-1)*n+1:j*n) = e;
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61 end
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62
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63 if marksize >= 0
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64
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65 ax = cpltaxes(x);
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66 h = plot(real(x),imag(x),'.');
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67 axis(ax);
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68 axis('square');
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69
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70 % Next block adapted from SPY.M.
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71 if marksize == 0
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72 units = get(gca,'units');
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73 set(gca,'units','points');
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74 pos = get(gca,'position');
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75 nps = 2.4*sqrt(n*m); % Factor based on number of pseudo-ei'vals plotted.
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76 myguess = round(3*min(pos(3:4))/nps);
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77 % [nps myguess]
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78 marksize = max(1,myguess);
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79 set(gca,'units',units);
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80 end
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81 set(h,'markersize',marksize);
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82 % set(h,'linemarkersize',marksize);
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83
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84 hold on
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85 e = eig(A);
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86 plot(real(e),imag(e),'xw');
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87 set(h,'markersize',marksize);
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88 hold off
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89
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90 else
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91
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92 y = x;
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93
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94 end