  This paper presents a new distance geometry algorithm for calculating atomic
coordinates from estimates of the interatomic distances, which maintains the
positions of the atoms in a known rigid substructure. Given an $M \times 3$
matrix of coordinates for the rigid substructure $\mathbf X$, this problem
consists of finding the $N \times 3$ matrix $\mathbf Y$ that yields of global
minimum of the so-called STRAIN, i.e. \[ \min_{\mathbf Y} \left\|
\begin{bmatrix} \mathbf{XX}^\top & \mathbf{XY}^\top \\ \mathbf{YX}^\top &
\mathbf{YY}^\top \end{bmatrix} \,-\, \begin{bmatrix} \mathbf A & \mathbf B \\
\mathbf B^\top & \mathbf C \end{bmatrix} \right\|_{\mathsf F}^2 ~, \] where
$\mathbf A = \mathbf{XX}^\top$ , and $\mathbf B, \mathbf C$ are matrices of
inner products calculated from the estimated distances.
  The vanishing of the gradient of the STRAIN is shown to be equivalent to a
system of only six nonlinear equations in six unknowns for the inertial tensor
associated with the solution Y . The entire solution space is characterized in
terms of the geometry of the intersection curves between the unit sphere and
certain variable ellipsoids. Upon deriving tight bilateral bounds on the
moments of inertia of any possible solution, we construct a search procedure
that reliably locates the global minimum. The effectiveness of this method is
demonstrated on realistic simulated and chemical test problems.
