Extrap: added Householder reflections to MPE algo

Research/Extrapolation methods/Code/ALGO_extrap_MPE_v2_20211029.m

 function [x_out,r_out]=ALGO_extrap_MPE_v2_20211029(F,X)
-% MPE performs minimal polynomial extrapolation using Arnoldi iteration
+% MPE performs minimal polynomial extrapolation
 
 [d,k] = size(F);
 x_out=zeros(d,k); x_out(:,1)=X(:,1);

Research/Extrapolation methods/Code/ALGO_extrap_MPE_v3_20211119.m

+function [x_out,r_out]=ALGO_extrap_MPE_v3_20211119(F,X)
+% MPE performs minimal polynomial extrapolation using Arnoldi iteration
+% with Householder reflections
+
+[d,k] = size(F);
+I=eye(d); Q=I; H = F;
+x_out=zeros(d,k); r_out = zeros(k,1);
+for i=1:min(d,k)
+    w = SUB_extrap_Householder_v1_20211119(H(:,i),i);
+    Q = (I-2*w*w')*Q;
+    H = (I-2*w*w')*H;
+    u = zeros(i,1); u(end)=1; u=H(1:i,1:i)\u; u=u/sum(u);
+    x_out(:,i) = X(:,1:i)*u;
+    r_out(i) = norm(F(:,1:i)*u);
+end
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Research/Extrapolation methods/Code/EX_extrap_linearFPI_v1_20211022.m

@@ -10,7 +10,8 @@ end
 F = X(:,2:end) - X(:,1:n);
 
 [x_v1,r_v1] = ALGO_extrap_MPE_v1_20211022(F,X(:,1:n));
-[x_v2,r_v2] = ALGO_extrap_MPE_v2_20211029(F,X(:,1:n));
+% [x_v2,r_v2] = ALGO_extrap_MPE_v2_20211029(F,X(:,1:n));
+[x_v2,r_v2] = ALGO_extrap_MPE_v3_20211119(F,X(:,1:n));
 x = x_v1(:,end);
 Error_abs= norm(x - x_true)
 Error_rel= Error_abs/norm(x_true)

Research/Extrapolation methods/Code/SUB_extrap_Householder_v1_20211119.m

+function w = SUB_extrap_Householder_v1_20211119(v,i)
+% HOUSEHOLDER calculates component of Householder reflection
+
+w = v(i:end);
+w(1) = w(1) + sign(w(1))*norm(w);
+w = w/norm(w); w = [zeros(i-1,1); w];
+
+end
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