Computing the Pseudoinverse of Specific Toeplitz Matrices Using Rank-One Updates

Algorithm 4

function alg42 = alg42(A,ll)

%***************************%

% General Information.    %

%***************************%

% Synopsis:

% alg42 = alg42(A)

% Input:

% A = the mxn initial matrix of interest.

% ll=n-m+1

% Output:

% alg42 = improvement of Algorithm 1 according to Theorem 6.

% This function follows Algorithm 2.

% The correct performance of alg42 requires the presence of yltXl function.

Astar = A’;

m,n = size(A);

f = A(1,1:ll);

E0 = zeros(1,m);

product = f’*f;

for i = 1:ll

  E0(1,i) = sum(diag(product,i-1));

end

E1=E0;

E = toeplitz(E1);

G0=zeros(ll-1);

Gnew = hankel(zeros(1,ll-2) f(end), flip(f(1,2:end)));

for i = 1:ll-1

  Gkoutnew = G0+Gnew(:,i)*Gnew(i,:);

  G0=Gkoutnew;

end

Gmnew = G0;

fullmatrix = zeros(m-ll+1) zeros(m-ll+1,ll-1);zeros(ll-1,m-ll+1) Gmnew;

Gkout = E-fullmatrix;

Y02 = Gkout∖A;

X02 = Y02’;

for l = m+1:n-1

yltout,Xlout = yltXl(Y02,X02,Astar,l,m,n);

Y02 = yltout;

X02 = Xlout;

end

alg42 = X02 - Astar*Y02(:,n)*Y02(:,n)’/(1+Astar(n,:)*Y02(:,n));