Update DIIS.mli

Utils/DIIS.mli

@@ -1,25 +1,29 @@
 (** Direct Inversion of the Iterative Subspace algorithm.
 
-At each iteration, one has:
+At iteration {% $m$ %}, one has:
 
-- {% $\mathbf{p}$ %}, a vector of parameters
-- {% $\mathbf{e}$ %}, an approximate error vector
+- {% $\mathbf{p}_m$ %}, a vector of parameters
+- {% $\mathbf{e}_m$ %}, an approximate error vector
+
+The DIIS approximate solution for iteration $m+1$ is given by
 
-The DIIS approximate solution is given by
 {% \begin{align*}
-\mathbf{p}^{\text{DIIS}} & = \sum_i c_i (\mathbf{p}^f + \mathbf{e}_i) \\
-                         & = \sum_i c_i \mathbf{p}^f + \sum_i c_i \mathbf{e}_i) \\
+\mathbf{p}_{m+1} & = \sum_{i=1}^m c_i (\mathbf{p}^f + \mathbf{e}_i) \\
+                         & = \sum_{i=1}^m c_i \mathbf{p}^f + \sum_i c_i \mathbf{e}_i 
 \end{align*} %}
-where {% $\mathbf{p}^f$ %} is the exact solution, so one wants to minimize
-the norm of the error vector imposing the constraint that $\sum_i c_i = 1$ with
-a Langrange multiplier {% $\lambda$ %}.
+
+where {% $\mathbf{p}^f$ %} is the exact solution. One wants to minimize the
+norm of the error vector imposing the constraint that {% $\sum_{i=1}^m c_i = 1$ %}
+with a Langrange multiplier {% $\lambda$ %}.
 
 {%
 \begin{align*}
-\mathcal{L} & = ||\sum_i c_i \mathbf{e}_i)||^2 - \lambda \left(\sum_i c_i - 1\right) \\
+\mathcal{L} & = ||\sum_i c_i \mathbf{e}_i||^2 - \lambda \left(\sum_i c_i - 1\right) \\
             & = \sum_{ij} c_i c_j B_{ij} - \lambda \left(\sum_i c_i - 1\right)
 \end{align*}
-with {% $B_{ij} = \langle \mathbf{e}_i | \mathbf{e}_j \rangle$ %}.
+%}
+with
+{% $B_{ij} = \langle \mathbf{e}_i | \mathbf{e}_j \rangle$ %}.
 
 Equating zero to the derivatives of {% $\mathcal{L}$ %} with respect to {% $c_i$ %} and {% $\lambda$ %} leads to