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% L_DETAIL Step by step explainations of a Matlab implementation of the Lambda 
%                   method.  We follow the notation introduced in Strang and Borre,  
%                   pages 495--499 
 
% Written by Kai Borre 
% March 23, 2002 
 
fprintf('\n') 
echo on 
% Test example 
I_hat = [5.45;3.1;2.97]; 
%I_hat =[0; 1]; 
n = size(I_hat,1); 
Sigma = [6.29 5.978 .544; 5.978 6.292 2.34; .544 2.34 6.288]; 
%Sigma = [53.40 38.40;38.40 28.00]; 
echo off 
 
% Given float ambiguities 
fprintf('\nFloat ambiguities I_hat') 
for i = 1:n 
    fprintf('\n%12.3f', I_hat(i,1)) 
end 
 
% Original covariance matrix for I_hat 
fprintf('\n\nCovariance matrix for I_hat Sigma')  
for i = 1:n 
    fprintf('\n') 
    for j = 1:n 
        fprintf('%12.3f', Sigma(i,j)) 
    end 
end 
 
fprintf('\n\n') 
echo on 
% Shift of I_hat in order to secure that -1 < I_hat <= +1 
echo off 
 
shifts = I_hat-rem(I_hat,1); 
fprintf('\n\nInteger shifts of I_hat') 
for i = 1:n 
    fprintf('\n%12.0f', shifts(i,1)) 
end 
 
% Remainders of I_hat 
I_hat = rem(I_hat,1); 
fprintf('\n\nRemainders of I_hat') 
for i = 1:n 
    fprintf('\n%12.3f', I_hat(i,1)) 
end 
 
% L^T*D*L factorization of Sigma 
[L,D] = ldldecom(Sigma); 
fprintf('\n\nSigma is factorized into L^T*D*L') 
fprintf('\n\nThe lower triangular L') 
for i = 1:n 
    fprintf('\n') 
    for j = 1:n 
        fprintf('%12.3f',L(i,j)) 
    end 
end 
fprintf('\n\nThe diagonal matrix D\n') 
for i = 1:n 
    fprintf('%12.3f',D(i)) 
end 
 
% Computing the size of the search volume 
chi2 = chistart(D,L,I_hat,1); 
fprintf('\n\nchi^2 = %5.3f\n',chi2) 
 
% Doing the decorrelation 
[Sigma_t,Z,L_t,D_t] = decorrel(Sigma,I_hat); 
fprintf('\nInteger transformation matrix Z') 
for i = 1:n 
    fprintf('\n') 
    for j = 1:n 
        fprintf('%12.0f',Z(i,j)) 
    end 
end 
 
% I_hat transformed: z = Z'*I_hat 
fprintf('\n\nTransformed, shifted ambiguities z = Z^T*I_hat') 
z = Z'*I_hat; 
for i = 1:n 
    fprintf('\n%12.3f',z(i)) 
end 
 
% The transformed, decorrelated covariance matrix : Sigma_t = Z^T*Sigma*Z 
fprintf('\n\nThe transformed, decorrelated covariance matrix Sigma_t = Z^T*Sigma*Z.') 
fprintf('\n\nSigma_t = Z^T*Sigma*Z') 
for i = 1:n 
    fprintf('\n') 
    for j = 1:n 
        fprintf('%12.3f',Sigma_t(i,j)) 
    end 
end 
fprintf('\nNote the deminished off-diagonal terms!') 
fprintf('\n\nThe lower triangular L_t used in the search') 
for i = 1:n 
    fprintf('\n') 
    for j = 1:n 
        fprintf('%12.3f',L_t(i,j)) 
    end 
end 
fprintf('\n\nThe diagonal matrix D_t used in the seach\n') 
for i = 1:n 
    fprintf('%12.3f',D_t(i)) 
end 
 
fprintf('\n\n') 
echo on 
% Determining the size of the search volume for the transformed L_t and D_t 
echo off 
 
chi2 = chistart(D_t,L_t,z,1);  
fprintf('\n\nchi^2 for the transformed problem  =  %5.3f\n',chi2) 
 
fprintf('\nThe search domain is defined as') 
fprintf('\n          (I-I_hat)^T*(Z^T*Sigma*Z)*(I-I_hat) < chi^2,  for I integer') 
 
[I_bar,sqnorm,ierr] = lsearch(z,L_t,D_t,chi2,1); 
I_bar = (I_bar' * inv(Z))'+shifts; 
fprintf('\n\nFixed ambiguities I_bar') 
for i = 1:n 
    fprintf('\n%12.0f', I_bar(i,1)) 
end 
%%%%%%%%%%%%%%%%%%%%%%%%%%% end l_detail.m  %%%%%%%%%%%%