Added Householder algorithm and introduced Householder QR (got R, gotta find Q)

Direct Methods for Solving Linear Systems/Alg11.m

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+function [Q R] = Alg11(A)
+% Algorithm 11: QR factorization via Housholder algorithm.
+
+[m, n] = size(A);
+
+if ~ (m>=n)
+   error("m has to be greater or equal n!") 
+end
+
+for j = 1:n
+    [v, beta] = householder(A(j:m, j))
+    A(j:m, j:n) = (eye(m-j+1)-beta*(v*v.'))*A(j:m, j:n)
+    if j < m
+        A(j+1:m, j) = v(2:m-j+1)
+    end
+end
+
+R = triu(A)
+
+end
+

Direct Methods for Solving Linear Systems/Alg13.m

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+function [Q R] = Alg13(A)
+
+
+
+end
+

Direct Methods for Solving Linear Systems/householder.m

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+function [v,beta] = householder(x)
+% householder takes vertical vector x
+
+if ~iscolumn(x)
+    x = x.';
+end
+
+n = length(x);
+sig = x(2:n).' * x(2:n);
+v = [1; 
+     x(2:n)];
+
+ if isempty(sig) || (sig == 0)
+     beta = 0;
+ else
+     mu = sqrt(x(1)^2+sig);
+     if x(1) <= 0
+         v(1) = x(1) - mu;
+     else
+         v(1) = -sig/(x(1) + mu);
+     end
+     beta = 2*v(1)^2/(sig + v(1)^2);
+     v = v/v(1);
+ end
+
+end
+