function X = basicqr_apply_qt (QRdata, X)
  [M, N] = size(QRdata.QR);
  if size(X,1) ~= M,
    error(sprintf("matrix must be conformal with QR, leading dimension %d",M));
  endif
  for j = 1:min(M,N),
    h=[1;QRdata.QR(j+1:M,j)];
    X(j:M,:) -= QRdata.tau(j)*h*h'*X(j:M,:);
  endfor
endfunction