% m15_backwardstab.m - backward stability of QR factorisation

  R = .001*eye(7) + 9*triu(ones(7),1 )  % ill-conditioned upper-triang R
  rng(4)                                % fix random number seed
  [Q,~] = qr(randn(7)); Q               % random orthogonal Q
  A = Q*R                               % product of these

  [Q2,R2] = qr(A);                      % computed QR factorisation
  R
  R2
  Q
  Q2
  A
  Q2*R2

  norm(R2-R)                            % error in factor R
  norm(Q2-Q)                            % error in factor Q
  norm(Q2*R2-A)                         % error in product Q*R

% Wilkinson's dictum: on a computer with rounding errors,
% a stable algorithm of numerical linear algebra gives a solution
% which is **the exact solution of a slightly perturbed problem**.