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Create local repository for matrix toolboxes. Step #0 done.
author Antonio Pino Robles <data.script93@gmail.com>
date Wed, 06 May 2015 14:56:53 +0200
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8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
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1 function A = randsvd(n, kappa, mode, kl, ku)
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2 %RANDSVD Random matrix with pre-assigned singular values.
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3 % RANDSVD(N, KAPPA, MODE, KL, KU) is a (banded) random matrix of order N
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4 % with COND(A) = KAPPA and singular values from the distribution MODE.
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5 % N may be a 2-vector, in which case the matrix is N(1)-by-N(2).
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6 % Available types:
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7 % MODE = 1: one large singular value,
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8 % MODE = 2: one small singular value,
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9 % MODE = 3: geometrically distributed singular values,
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10 % MODE = 4: arithmetically distributed singular values,
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11 % MODE = 5: random singular values with unif. dist. logarithm.
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12 % If omitted, MODE defaults to 3, and KAPPA defaults to SQRT(1/EPS).
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13 % If MODE < 0 then the effect is as for ABS(MODE) except that in the
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14 % original matrix of singular values the order of the diagonal entries
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15 % is reversed: small to large instead of large to small.
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16 % KL and KU are the lower and upper bandwidths respectively; if they
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17 % are omitted a full matrix is produced.
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18 % If only KL is present, KU defaults to KL.
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19 % Special case: if KAPPA < 0 then a random full symmetric positive
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20 % definite matrix is produced with COND(A) = -KAPPA and
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21 % eigenvalues distributed according to MODE.
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22 % KL and KU, if present, are ignored.
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23
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24 % Reference:
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25 % N.J. Higham, Accuracy and Stability of Numerical Algorithms,
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26 % Society for Industrial and Applied Mathematics, Philadelphia, PA,
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27 % USA, 1996; sec. 26.3.
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28
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29 % This routine is similar to the more comprehensive Fortran routine xLATMS
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30 % in the following reference:
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31 % J.W. Demmel and A. McKenney, A test matrix generation suite,
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32 % LAPACK Working Note #9, Courant Institute of Mathematical Sciences,
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33 % New York, 1989.
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34
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35 if nargin < 2, kappa = sqrt(1/eps); end
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36 if nargin < 3, mode = 3; end
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37 if nargin < 4, kl = n-1; end % Full matrix.
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38 if nargin < 5, ku = kl; end % Same upper and lower bandwidths.
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39
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40 if abs(kappa) < 1, error('Condition number must be at least 1!'), end
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41 posdef = 0; if kappa < 0, posdef = 1; kappa = -kappa; end % Special case.
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42
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43 p = min(n);
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44 m = n(1); % Parameter n specifies dimension: m-by-n.
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45 n = n(max(size(n)));
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46
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47 if p == 1 % Handle case where A is a vector.
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48 A = randn(m, n);
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49 A = A/norm(A);
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50 return
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51 end
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52
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53 j = abs(mode);
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54
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55 % Set up vector sigma of singular values.
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56 if j == 3
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57 factor = kappa^(-1/(p-1));
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58 sigma = factor.^[0:p-1];
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59
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60 elseif j == 4
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61 sigma = ones(p,1) - (0:p-1)'/(p-1)*(1-1/kappa);
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62
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63 elseif j == 5 % In this case cond(A) <= kappa.
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64 sigma = exp( -rand(p,1)*log(kappa) );
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65
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66 elseif j == 2
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67 sigma = ones(p,1);
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68 sigma(p) = 1/kappa;
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69
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70 elseif j == 1
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71 sigma = ones(p,1)./kappa;
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72 sigma(1) = 1;
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73 end
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74
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75 % Convert to diagonal matrix of singular values.
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76 if mode < 0
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77 sigma = sigma(p:-1:1);
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78 end
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79 sigma = diag(sigma);
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80
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81 if posdef % Handle special case.
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82 Q = qmult(p);
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83 A = Q'*sigma*Q;
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84 A = (A + A')/2; % Ensure matrix is symmetric.
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85 return
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86 end
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87
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88 if m ~= n
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89 sigma(m, n) = 0; % Expand to m-by-n diagonal matrix.
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90 end
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91
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92 if kl == 0 & ku == 0 % Diagonal matrix requested - nothing more to do.
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93 A = sigma;
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94 return
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95 end
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96
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97 % A = U*sigma*V, where U, V are random orthogonal matrices from the
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98 % Haar distribution.
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99 A = qmult(sigma');
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100 A = qmult(A');
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101
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102 if kl < n-1 | ku < n-1 % Bandwidth reduction.
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103 A = bandred(A, kl, ku);
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104 end