[R,Lambda,Q] = PEVDToyProblem(Num,Disp);
[R,Lambda,Q] = PEVDToyProblem(Num,Disp) implements a number of defined toy
problems for a parahermitian matrix EVD. The parameter Num determines the
scenario; for Disp='on', the power spectra of the eigenvalues will be
displayed.
PEVDToyProblem(1) returns a 2x2 parahermitian matrix with spectrally
majorised eigenvalues; all factors are polynomial matrices.
PEVDToyProblem(2) is a 2x2 example from [Icart & Common, 2013] and [Weiss,
Pestana, & Proudler, TSP draft] with spectrally majorised eigenvalues. All
factors are transcendental matrices. The returned approximate eigenvectors
are NOT normalised, but can serve as a comparison w.r.t. subspace angles
of an alternative solution.
PEVDToyProblem(3) is another 2x2 exampled from [Icart & Common, 2013] and
[Weiss, Pestana, & Proudler, TSP draft] with spectrally unmajorised
eigenvalues. All factors are polynomial matrices.
PEVDToyProblem(4) is an `easy' 3x3 parahermitian matrix with spectrally
unmajorised eigenvalues. There are only pair-wise overlaps that are well-
separated. All factors are polynomial matrices.
PEVDToyProblem(5) is a `difficult' 3x3 parahermitian matrix, with
polynomial matrix factors but algebraic multiplicities up to 3, and some
closely-spaced intersections.
Input parameter:
Num toy problem number/index
Disp displays characteristics (optional for Disp='on')
Output parameter:
R parahermitian matrix
Lambda matrix of eigenvalues
Q parahermitian matrix
S. Weiss, UoS, 1/12/2017
0001 function [R,Lambda,Q] = PEVDToyProblem(ToyIndex,DisplayOn); 0002 %[R,Lambda,Q] = PEVDToyProblem(Num,Disp); 0003 % 0004 % [R,Lambda,Q] = PEVDToyProblem(Num,Disp) implements a number of defined toy 0005 % problems for a parahermitian matrix EVD. The parameter Num determines the 0006 % scenario; for Disp='on', the power spectra of the eigenvalues will be 0007 % displayed. 0008 % 0009 % PEVDToyProblem(1) returns a 2x2 parahermitian matrix with spectrally 0010 % majorised eigenvalues; all factors are polynomial matrices. 0011 % 0012 % PEVDToyProblem(2) is a 2x2 example from [Icart & Common, 2013] and [Weiss, 0013 % Pestana, & Proudler, TSP draft] with spectrally majorised eigenvalues. All 0014 % factors are transcendental matrices. The returned approximate eigenvectors 0015 % are NOT normalised, but can serve as a comparison w.r.t. subspace angles 0016 % of an alternative solution. 0017 % 0018 % PEVDToyProblem(3) is another 2x2 exampled from [Icart & Common, 2013] and 0019 % [Weiss, Pestana, & Proudler, TSP draft] with spectrally unmajorised 0020 % eigenvalues. All factors are polynomial matrices. 0021 % 0022 % PEVDToyProblem(4) is an `easy' 3x3 parahermitian matrix with spectrally 0023 % unmajorised eigenvalues. There are only pair-wise overlaps that are well- 0024 % separated. All factors are polynomial matrices. 0025 % 0026 % PEVDToyProblem(5) is a `difficult' 3x3 parahermitian matrix, with 0027 % polynomial matrix factors but algebraic multiplicities up to 3, and some 0028 % closely-spaced intersections. 0029 % 0030 % Input parameter: 0031 % Num toy problem number/index 0032 % Disp displays characteristics (optional for Disp='on') 0033 % 0034 % Output parameter: 0035 % R parahermitian matrix 0036 % Lambda matrix of eigenvalues 0037 % Q parahermitian matrix 0038 % 0039 % S. Weiss, UoS, 1/12/2017 0040 0041 switch ToyIndex 0042 case 1, 0043 %--------------------------------------------------------------------- 0044 % 2x2 spectrally majorised matrix with polynomial factors 0045 %--------------------------------------------------------------------- 0046 % one eigenvalue constant, the other highpass 0047 Lambda = zeros(2,2,3); 0048 Lambda(:,:,1) = [0 0; 0 -.2]; 0049 Lambda(:,:,2) = [1 0; 0 0.5]; 0050 Lambda(:,:,3) = Lambda(:,:,1)'; 0051 % eigenvectors via an elementary paraunitary operation 0052 v = [1 -1]'/sqrt(2); 0053 Q = zeros(2,2,2); 0054 Q(:,:,1) = eye(2) - v*v'; 0055 Q(:,:,2) = v*v'; 0056 % construction of the `space-time covariance' matrix 0057 R = PolyMatConv(Q,PolyMatConv(Lambda,ParaHerm(Q))); 0058 case 2, 0059 %--------------------------------------------------------------------- 0060 % 2x2 spectrally majorised matrix with transcendental factors 0061 %--------------------------------------------------------------------- 0062 % problem by Sylvie Icart, also in our TSP16 draft 0063 R = zeros(2,2,3); 0064 R(:,:,1) = [0 0; 0 -2]; 0065 R(:,:,2) = [1 1; 1 6]; 0066 R(:,:,3) = R(:,:,1)'; 0067 % determinant etc. 0068 R11 = [0 1 0]'; R12 = [0 1 0]'; R22 = [-2 6 -2]'; 0069 T = R11+R22; 0070 S2 = conv(R11-R22,R11-R22) + 4*conv(R12,flipud(conj(R12))); 0071 % transcendental eigenvalues ... truncated 0072 N = 20; 0073 a = ones(N+1,1); 0074 for n = 0:N, % MacLaurin series 0075 for i = 0:n, 0076 a(n+1) = a(n+1)*(0.5-i); 0077 end; 0078 a(n+1) = a(n+1)/factorial(n+1); 0079 end; 0080 a = [1; a]; 0081 xi = -0.355302 + j*0.198559; 0082 h1 = a.*(xi.^(0:N+1)'); 0083 h2 = a.*(conj(xi).^(0:N+1)'); 0084 h = conv(h1,h2); 0085 S = 2./abs(xi)*conv(h,flipud(conj(h))); 0086 L1 = S/2; 0087 L1(2*(N+1):2*(N+1)+2) = L1(2*(N+1):2*(N+1)+2) + T/2; 0088 L2 = -S/2; 0089 L2(2*(N+1):2*(N+1)+2) = L2(2*(N+1):2*(N+1)+2) + T/2; 0090 Lambda(1,1,:) = L1; 0091 Lambda(2,2,:) = L2; 0092 % transcendental eigenvectors ... truncated / approximated 0093 LL = (length(L1)-1)/2; 0094 Q = zeros(2,2,2*LL+1); 0095 % first EV 0096 Q(1,1,:) = L1; 0097 dummy = zeros(1,1,3); 0098 dummy(1,1,:) = R22; 0099 Q(1,1,LL:LL+2) = Q(1,1,LL:LL+2) - dummy; 0100 Q(2,1,LL:LL+2) = R12; 0101 % 2nd EV 0102 Q(1,2,LL:LL+2) = R12; 0103 Q(2,2,:) = L2; 0104 dummy(1,1,:) = R11; 0105 Q(2,2,LL:LL+2) = Q(2,2,LL:LL+2) - dummy; 0106 % these would now need normalisation -- divide by the square root :-( 0107 case 3, 0108 %--------------------------------------------------------------------- 0109 % 2x2 spectrally unmajorised matrix with polynomial factors 0110 %--------------------------------------------------------------------- 0111 Q = zeros(2,2,2); 0112 Q(:,:,1) = [1 1; 0 0]/sqrt(2); 0113 Q(:,:,2) = [0 0; 1 -1]/sqrt(2); 0114 Lambda = zeros(2,2,3); 0115 Lambda(:,:,1) = [1 0; 0 -j]; 0116 Lambda(:,:,2) = [3 0; 0 3]; 0117 Lambda(:,:,3) = Lambda(:,:,1)'; 0118 R = PolyMatConv(Q,PolyMatConv(Lambda,ParaHerm(Q))); 0119 case 4, 0120 %--------------------------------------------------------------------- 0121 % 'easy' 3x3 spectrally unmajorised matrix with polynomial factors 0122 %--------------------------------------------------------------------- 0123 Lambda = zeros(3,3,3); 0124 Lambda(1,1,:) = [j 4 -j]/4; 0125 Lambda(2,2,:) = [1/3 3 1/3]/4; 0126 Lambda(3,3,:) = [j 2 -j]/4; 0127 Q = zeros(3,3,2); 0128 v = [1 1 1]'/sqrt(3); 0129 Q(:,:,1) = eye(3) - v*v'; 0130 Q(:,:,2) = v*v'; 0131 R = PolyMatConv(Q,PolyMatConv(Lambda,ParaHerm(Q))); 0132 case 5, 0133 %--------------------------------------------------------------------- 0134 % 'hard' 3x3 spectrally unmajorised matrix with polynomial factors 0135 %--------------------------------------------------------------------- 0136 Lambda = zeros(3,3,5); 0137 Lambda(1,1,:) = [1 0 2 0 1]/4; 0138 Lambda(2,2,:) = [0 -1 2 -1 0]/4; 0139 Lambda(3,3,:) = [0 -j 4 j 0]/4; 0140 % eigenvectors via an elementary paraunitary operation 0141 V = [1 0 -1; 1 1 0; 1 0 1; -1 1 0]'/sqrt(2); 0142 Q = zeros(3,3,1); 0143 Q(:,:,1) = eye(3); 0144 Qi = zeros(3,3,2); 0145 for i = 1:size(V,2), 0146 Qi(:,:,1) = eye(3) - V(:,i)*V(:,i)'; 0147 Qi(:,:,2) = V(:,i)*V(:,i)'; 0148 Q = PolyMatConv(Q,Qi); 0149 end; 0150 % space-time covariance matrix 0151 R = PolyMatConv(Q,PolyMatConv(Lambda,ParaHerm(Q))); 0152 otherwise, 0153 disp('toy case not implemented'); 0154 R = 1; Lambda = 1; Q = 1; 0155 end; 0156 0157 %------------------------------------------------------------------------- 0158 % check for display option 0159 %------------------------------------------------------------------------- 0160 if exist('DisplayOn'), 0161 if strcmp(DisplayOn,'on')==1, 0162 P = PolyMatDiagSpec(Lambda,1024); 0163 plot((0:1023)/1024,abs(P)); 0164 xlabel('normalised frequency'); ylabel('magnitude'); 0165 end; 0166 end; 0167 0168