function x_trrqr = trrqr(Q,R,Pi,p,b)%  trrqr --> Solves a least squares problem using the RRQR decomposition.%%  <Synopsis>%    x_trrqr = trrqr(Q,R,Pi,p,b)%%  <Description>%    Solves the near-rank deficient least squares problem%%           min_x || b-A*x ||_2%%    using the RRQR decomposition. Here, A*Pi = Q*R is the RRQR%    decomposition of A, p is the numerical rank of A, and the%    TRRQR solution is defined by%%           x_trrqr = Pi(:,1:p)*inv(R(1:p,1:p))*Q(:,1:p)'*b.%%  <See Also>%    tulv --> Solves a least squares problem using the ULV decomposition.%    turv --> Solves a least squares problem using the URV decomposition.%  <Revision>%    Ricardo D. Fierro, California State University San Marcos%    Per Christian Hansen, IMM, Technical University of Denmark%    Peter S.K. Hansen, IMM, Technical University of Denmark%%    Last revised: June 22, 1999%-----------------------------------------------------------------------% Need the pseudoinverse of R(1:p,:); get it via QR factorization.[P,RR] = qr(R(1:p,:)',0);% Transform right hand side.beta = Q(:,1:p)'*b;% Back-substitution.y = (RR')\beta;% Compute TRRQR solution.x_trrqr = Pi*(P*y);%-----------------------------------------------------------------------% End of function trrqr%-----------------------------------------------------------------------