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Francois Coppens 2023-02-10 17:16:08 +01:00
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org/qmckl_sherman_morrison_woodbury.org
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@ -33,39 +33,39 @@ range(2, 22)
* Naïve Sherman-Morrison * Naïve Sherman-Morrison
:PROPERTIES: :PROPERTIES:
:Name: qmckl_sherman_morrison_naive :Name: qmckl_sherman_morrison_naive
:CRetType: qmckl_exit_code :CRetType: qmckl_exit_code
:FRetType: qmckl_exit_code :FRetType: qmckl_exit_code
:END: :END:
This is the simplest of the available Sherman-Morrison-Woodbury kernels. It applies rank-1 updates one by one in This is the simplest of the available Sherman-Morrison-Woodbury kernels. It applies rank-1 updates one by one in
the order that is given. It only checks if the denominator in the Sherman-Morrison formula is not too close to the order that is given. It only checks if the denominator in the Sherman-Morrison formula is not too close to
zero when an update is evaluated. It will exit with an error code of the denominator is too close to zero. zero when an update is evaluated. It will exit with an error code of the denominator is too close to zero.
The formula for any update $u_j$ (index $j$ is suppresed for clarity) that is applied is The formula for any update $u_j$ (index $j$ is suppresed for clarity) that is applied is
\[ \[
(S + uv^T)^{-1} = S^{-1} - \frac{S^{-1} uv^T S^{-1}}{1 + v^T S^{-1} u} (S + uv^T)^{-1} = S^{-1} - \frac{S^{-1} uv^T S^{-1}}{1 + v^T S^{-1} u}
\] \]
where where
$S$ is the Slater-matrix, $S$ is the Slater-matrix,
$u$ and $v^T$ are the column and row vectors containing the updates, $u$ and $v^T$ are the column and row vectors containing the updates,
$S^{-1}$ is the inverse of the Slater-matrix. $S^{-1}$ is the inverse of the Slater-matrix.
Even though the Slater-matrix $S$ with all updates applied at once is invertable, during the course of applying Even though the Slater-matrix $S$ with all updates applied at once is invertable, during the course of applying
updates to the inverse Slater-matrix $S^{-1}$ one-by-one it can happen that one of the intermediate inverse updates to the inverse Slater-matrix $S^{-1}$ one-by-one it can happen that one of the intermediate inverse
matrices $S^{-1}$ becomes singular. Therefore a global threshold value $\epsilon$ is defined that is used to matrices $S^{-1}$ becomes singular. Therefore a global threshold value $\epsilon$ is defined that is used to
evaluate each individual update $u_j$ when it is applied. evaluate each individual update $u_j$ when it is applied.
This value sets the lower bound for which the This value sets the lower bound for which the
denominator $1+v_j^TS^{-1}u_j$ is considered to be too small and will most probably result in a singular matrix denominator $1+v_j^TS^{-1}u_j$ is considered to be too small and will most probably result in a singular matrix
$S$, or at least in an inverse of $S$ of very poor numerical quality. Therefore, when $1+v_j^TS^{-1}u_j \geq \epsilon$, $S$, or at least in an inverse of $S$ of very poor numerical quality. Therefore, when $1+v_j^TS^{-1}u_j \geq \epsilon$,
the update is applied as usual and the kernel exits with return code \texttt{QMCKL_SUCCESS}. the update is applied as usual and the kernel exits with return code \texttt{QMCKL_SUCCESS}.
If $1+v_j^TS^{-1}u_j \leq \epsilon$ the update is rejected and the kernel exits with return code \texttt{QMCKL_FAILURE}. If $1+v_j^TS^{-1}u_j \leq \epsilon$ the update is rejected and the kernel exits with return code \texttt{QMCKL_FAILURE}.
If the determinant of the Slater-matrix is passed, it will be updated to the determinant resulting If the determinant of the Slater-matrix is passed, it will be updated to the determinant resulting
from applying the updates to the original matrix. from applying the updates to the original matrix.
#+NAME: qmckl_sherman_morrison_naive_args #+NAME: qmckl_sherman_morrison_naive_args
| Variable | Type | In/Out | Description | | Variable | Type | In/Out | Description |
@ -612,9 +612,9 @@ end interface
* End of files * End of files
#+begin_src c :comments link :tangle (eval c_test) #+begin_src c :comments link :tangle (eval c_test)
assert (qmckl_context_destroy(context) == QMCKL_SUCCESS); assert (qmckl_context_destroy(context) == QMCKL_SUCCESS);
return 0; return 0;
} }
#+end_src #+end_src