riccati_validity_bound.m

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function [bound,valid] = riccati_validity_bound(model, reduced_data, rbsim)
%function [bound,valid] = riccati_estim_validity_bound(model, reduced_data,
%                            rbsim)
% 
%   This function performs a check, whether the validity bound for the 
%   application of the error estimator is valid.
%
%   This is done by checking the inequality
%
%        8 gamma^2 \|E\|^2 \|B\|^2 \| R \| <= 1
%
%   This function also takes care of performance improvements, when the 
%   matrices E,B are not parameter dependent.
%
%   Andreas Schmidt, 2015
%

if ~isfield(rbsim.estim, 'gamma')
    error('The field rbsim.gamma must be present!')
end
if ~isfield(rbsim, 'P_residual')
    error('The field rbsim.P_residual must be present')
end

model.decomp_mode = 2;
Ecoeff = model.E(model, reduced_data);
if length(Ecoeff) > 1
    error('Q_E > 1 currently not implemented.');
end
normE2 = (reduced_data.estim.normE*Ecoeff(1))^2;

Bcoeff = model.B(model, reduced_data);
if reduced_data.estim.normBBT_use2norm
    normBBT = (reduced_data.estim.normBBT*abs(Bcoeff(1)))^2;
else
    normBBT = 0;
    for i = 1:length(Bcoeff)
        for j = 1:length(Bcoeff)
            for k = 1:length(Bcoeff)
                for l = 1:length(Bcoeff)
                    normBBT = normBBT + Bcoeff(i)*Bcoeff(j)*Bcoeff(k)* ...
                        Bcoeff(l)*reduced_data.estim.normBBT{i,j,k,l};
                end
            end
        end
    end
end
 
bound = 8*rbsim.estim.gamma^2*rbsim.P_residual*normE2*normBBT;
valid = bound <= 1;
end