pinv_sigma

 PINV_SIGMA Computes the pseudoinverse of Sigma_H

 Computes the pseudoinverse of Sigma_H (Algorithm 6.4.4, pag. 399, CHQZ3) 

 Input : AGG = structure of local matrices A_{\Gamma_m,\Gamma_m}
         Amm = structure of local matrices (A_{m,m})^{-1}
         AGm = structure of local matrices A_{\Gamma_m,m}
         LGG = structure with maps from \Gamma_i to \Gamma
                 set in schur_local.m
         novg = restriction map from (\cup_i \Omega_i) to \Gamma
                 set in cosnovg.m
         Rgamma = matrix of size (ne, ngamma) (defined in CHQZ3, pag. 399)
                  (R_\Gamma)_ij: =  1/n_j     if  x_j \in \Gamma_i
                                 =  0         otherwise

 Output: PSH  = pseudoinverse of Sigma_H, i.e. Rgamma^T A_H^+ Rgamma


 References: CHQZ2 = C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang,
                    "Spectral Methods. Fundamentals in Single Domains"
                    Springer Verlag, Berlin Heidelberg New York, 2006.
             CHQZ3 = C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang,
                    "Spectral Methods. Evolution to Complex Geometries 
                     and Applications to Fluid DynamicsSpectral Methods"
                    Springer Verlag, Berlin Heidelberg New York, 2007.

0001 function [PSH]=pinv_sigma(AGG,Amm,AGm,LGG,novg,Rgamma);
0002 % PINV_SIGMA Computes the pseudoinverse of Sigma_H
0003 %
0004 % Computes the pseudoinverse of Sigma_H (Algorithm 6.4.4, pag. 399, CHQZ3)
0005 %
0006 % Input : AGG = structure of local matrices A_{\Gamma_m,\Gamma_m}
0007 %         Amm = structure of local matrices (A_{m,m})^{-1}
0008 %         AGm = structure of local matrices A_{\Gamma_m,m}
0009 %         LGG = structure with maps from \Gamma_i to \Gamma
0010 %                 set in schur_local.m
0011 %         novg = restriction map from (\cup_i \Omega_i) to \Gamma
0012 %                 set in cosnovg.m
0013 %         Rgamma = matrix of size (ne, ngamma) (defined in CHQZ3, pag. 399)
0014 %                  (R_\Gamma)_ij: =  1/n_j     if  x_j \in \Gamma_i
0015 %                                 =  0         otherwise
0016 %
0017 % Output: PSH  = pseudoinverse of Sigma_H, i.e. Rgamma^T A_H^+ Rgamma
0018 %
0019 %
0020 % References: CHQZ2 = C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang,
0021 %                    "Spectral Methods. Fundamentals in Single Domains"
0022 %                    Springer Verlag, Berlin Heidelberg New York, 2006.
0023 %             CHQZ3 = C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang,
0024 %                    "Spectral Methods. Evolution to Complex Geometries
0025 %                     and Applications to Fluid DynamicsSpectral Methods"
0026 %                    Springer Verlag, Berlin Heidelberg New York, 2007.
0027 
0028 %   Written by Paola Gervasio
0029 %   $Date: 2007/04/01$
0030 
0031 [ne,ngamma]=size(Rgamma);
0032 Sigma=zeros(ngamma,ngamma);
0033 for ie=1:ne
0034 lg=LGG{ie};
0035 ngammam=length(lg); nl=novg(lg,ie);
0036 agg=AGG{ie};agm=AGm{ie};amm=Amm{ie};
0037 Sigma_loc=agg-agm*amm*agm';
0038 Sigma(nl,nl)=Sigma(nl,nl)+Sigma_loc;
0039 end
0040 
0041 % A_H=Rgamma*Sigma*Rgamma'
0042 
0043 PSH=pinv(full(Rgamma*Sigma*Rgamma'));
0044 clear Sigma;
0045 % Sigma_H^+
0046 
0047 PSH=Rgamma'*PSH*Rgamma;