Add DIIS.mli

Utils/DIIS.mli

@@ -0,0 +1,58 @@
+(** Direct Inversion of the Iterative Subspace algorithm.
+
+At each iteration, one has:
+
+- {% $\mathbf{p}$ %}, a vector of parameters
+- {% $\mathbf{e}$ %}, an approximate error vector
+
+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) \\
+\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 {% $\lambda$ %}.
+
+{%
+\begin{align*}
+\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 {% $B_{ij} = \langle \mathbf{e}_i | \mathbf{e}_j \rangle$ %}.
+
+Equating zero to the derivatives of {% $\mathcal{L}$ %} with respect to {% $c_i$ %} and {% $\lambda$ %} leads to 
+
+{% \begin{equation*}
+\begin{bmatrix}
+B_{11} & B_{12} & B_{13} & ... & B_{1m} & -1 \\
+B_{21} & B_{22} & B_{23} & ... & B_{2m} & -1 \\
+B_{31} & B_{32} & B_{33} & ... & B_{3m} & -1 \\
+\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
+B_{m1} & B_{m2} & B_{m3} & ... & B_{mm} & -1 \\
+1      & 1      & 1      & ... & 1      & 0
+\end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \\ c_3 \\ \vdots \\ c_m \\ \lambda \end{bmatrix}=
+\begin{bmatrix} 0 \\ 0 \\ 0 \\ \vdots \\ 0 \\ 1 \end{bmatrix}
+\end{equation*}
+
+%}
+
+The coefficients are then used to update {% $\mathbf{p}$ %} as
+{% $$
+\mathbf{p}_{m+1}=\sum_{i=1}^m c_i\mathbf{p}_i.
+$$ %}
+
+*)
+
+type t
+
+val make : unit -> t
+(** Initialize DIIS *)
+
+val append : p:Lacaml.D.Vec.t -> e:Lacaml.D.Vec.t -> t
+(** Append a parameter vector [p] and the corresponding error vector [e]. *)
+
+val next : t -> Lacaml.D.Vec.t
+(** Returns a new parameter vector. *)
+
+