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63 lines
1.9 KiB
OCaml
63 lines
1.9 KiB
OCaml
(** Direct Inversion of the Iterative Subspace algorithm.
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At iteration {% $m$ %}, one has:
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- {% $\mathbf{p}_m$ %}, a vector of parameters
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- {% $\mathbf{e}_m$ %}, an approximate error vector
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The DIIS approximate solution for iteration {% $m+1$ %} is given by
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{% \begin{align*}
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\mathbf{p}_{m+1} & = \sum_{i=1}^m c_i (\mathbf{p}^f + \mathbf{e}_i) \\
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& = \sum_{i=1}^m c_i \mathbf{p}^f + \sum_i c_i \mathbf{e}_i
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\end{align*} %}
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where {% $\mathbf{p}^f$ %} is the exact solution. One wants to minimize the
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norm of the error vector imposing the constraint that {% $\sum_{i=1}^m c_i = 1$ %}
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with a Langrange multiplier {% $\lambda$ %}.
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{%
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\begin{align*}
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\mathcal{L} & = ||\sum_i c_i \mathbf{e}_i||^2 - \lambda \left(\sum_i c_i - 1\right) \\
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& = \sum_{ij} c_i c_j B_{ij} - \lambda \left(\sum_{i=1}^m c_i - 1\right)
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\end{align*}
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%}
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with
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{% $B_{ij} = \langle \mathbf{e}_i | \mathbf{e}_j \rangle$ %}.
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Equating zero to the derivatives of {% $\mathcal{L}$ %} with respect to {% $c_i$ %} and {% $\lambda$ %} leads to
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{% \begin{equation*}
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\begin{bmatrix}
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B_{11} & B_{12} & B_{13} & ... & B_{1m} & 1 \\
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B_{21} & B_{22} & B_{23} & ... & B_{2m} & 1 \\
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B_{31} & B_{32} & B_{33} & ... & B_{3m} & 1 \\
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\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
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B_{m1} & B_{m2} & B_{m3} & ... & B_{mm} & 1 \\
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1 & 1 & 1 & ... & 1 & 0
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\end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \\ c_3 \\ \vdots \\ c_m \\ -\lambda \end{bmatrix}=
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\begin{bmatrix} 0 \\ 0 \\ 0 \\ \vdots \\ 0 \\ 1 \end{bmatrix}
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\end{equation*}
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%}
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The coefficients are then used to update {% $\mathbf{p}$ %} as
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{% $$
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\mathbf{p}_{m+1}=\sum_{i=1}^m c_i\mathbf{p}_i.
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$$ %}
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*)
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type t
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val make : ?mmax:int -> unit -> t
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(** Initialize DIIS with a maximum size.*)
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val append : p:Lacaml.D.Vec.t -> e:Lacaml.D.Vec.t -> t -> t
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(** Append a parameter vector [p] and the corresponding error vector [e]. *)
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val next : t -> Lacaml.D.Vec.t
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(** Returns a new parameter vector. *)
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