Figure 15.2:
The modelfree Hessian kite  a demonstration of the construction of the modelfree Hessian
∇^{2}χ^{2} for the global model
.
For each residue i a different matrix
∇^{2}χ^{2}_{i} is constructed.
The first element of the matrix represented by the two symbols
∂ (the red block) is the submatrix of chisquared second partial derivatives with respect to the diffusion tensor parameters
and
.
The orange blocks are the submatrices of chisquared second partial derivatives with respect to the diffusion parameter
and the modelfree parameter
.
The yellow blocks are the submatrices of chisquared second partial derivatives with respect to the modelfree parameters
and
.
For the residue dependent matrix
∇^{2}χ^{2}_{i} the second partial derivatives with respect to the modelfree parameters
and
where i≠l are zero.
In addition, the second partial derivatives with respect to the modelfree parameters
and
where i≠l are also zero.
These blocks of submatrices are left uncoloured.
The complete Hessian of
is the sum of the matrices
∇^{2}χ^{2}_{i}.

The construction of the Hessian for the models
,
,
, and
is very similar to the procedure used for the gradient.
The chisquared Hessian for the global models
and
is
∇^{2}χ^{2} = ∇^{2}χ^{2}_{i}. 
(15.12) 
Figure 15.2 demonstrates the construction of the full Hessian for the model
.
The Hessian for the model
is the sum of all the red blocks.
The Hessian for the model
is the combination of the single red block for residue i, the two orange blocks representing the submatrices of chisquared second partial derivatives with respect to the diffusion parameter
and the modelfree parameter
, and the single yellow block for that residue.
The Hessian for the modelfree model
is simply the submatrix for the residue i coloured yellow.
The relax user manual (PDF), created 20190614.