One can examine the relationship between the parameters in the model to determine which pairs will have significant off-diagonal elements in the normal matrix. The pairs whose off-diagonal elements are predicted to be small can then be ignored. Such selective attention only pays off when the vast majority of elements are removed.
With some functions all the off-diagonal elements may be ignored where other functions do not allow any. One must treat functions on a case by case basis to determine which elements to use. An analysis of the residual function for x-ray diffraction shows that the size of the off-diagonal elements is related to the extent of electron density overlap of the two atoms. Since atoms are fairly compact all off-diagonal terms between parameters in atoms are negligible except for atoms bonded to one another, and the terms for those pairs are small. Since an atom has a large overlap with its own electrons the diagonal elements are very large compared to any off-diagonal ones.
The stereochemical restraints commonly used in protein refinement have a different pattern. Here the parameters of atoms connected by a bond distance or angle have strong correlation. Atoms not restrained to one another have no correlation at all. The off-diagonal terms which are nonzero are as significant as the diagonal ones.
This knowledge allows one to calculate the normal matrix as a sparse matrix, the vast majority of the off-diagonal elements are never calculated or even have computer memory allocated for their storage. The only elements calculated are the diagonal ones (including contributions from both the crystallographic and stereochemical restraints) and the off-diagonal elements for parameters from atoms directly connected by geometric restraints.
Even with the simplification of the normal matrix introduced by the sparse approximation the problem of inverting the matrix is difficult. There are a number of methods available for generating an approximation to the inverse of a sparse matrix. A discussion of these methods is beyond the scope of this paper. However it is important to note that each of them includes assumptions and approximations which should be understood when they are used.
The refinement program PROLSQ[2] uses the sparse matrix approximation to the normal matrix. PROLSQ inverts the matrix using a method called ``Conjugate Gradient'' which is unrelated to the Conjugate Gradient method used to minimize functions. It is a sign of confusion to state that X-PLOR[1] and PROLSQ both use the Conjugate Gradient method.
It is quite difficult to calculate the proper values for the elements of the normal matrix. To simplify these calculations Konnert and Hendrickson decided to implement all stereochemical restraints as distances. While this restructuring of the restraints does simplify the normal matrix it makes the restraints more difficult for the user to visualize and prevents the minimization method from seeing the true, underlying nature of the restraints.
While the minimization method used in PROLSQ is the most powerful of those used in large molecule refinement (and therefore the smallest radius of convergence) in practice it does not seem to work any better than simple the Conjugate Gradient method. Its limitations arise from the approximations made in the calculation of the normal matrix elements and the way the space matrix is inverted.