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Update DIIS.mli
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(** Direct Inversion of the Iterative Subspace algorithm.
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At each iteration, one has:
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At iteration {% $m$ %}, one has:
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- {% $\mathbf{p}$ %}, a vector of parameters
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- {% $\mathbf{e}$ %}, an approximate error vector
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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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The DIIS approximate solution is given by
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{% \begin{align*}
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\mathbf{p}^{\text{DIIS}} & = \sum_i c_i (\mathbf{p}^f + \mathbf{e}_i) \\
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& = \sum_i c_i \mathbf{p}^f + \sum_i c_i \mathbf{e}_i) \\
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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, so one wants to minimize
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the norm of the error vector imposing the constraint that $\sum_i c_i = 1$ with
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a Langrange multiplier {% $\lambda$ %}.
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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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\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 c_i - 1\right)
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\end{align*}
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with {% $B_{ij} = \langle \mathbf{e}_i | \mathbf{e}_j \rangle$ %}.
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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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