#+TITLE: Sherman-Morrison-Woodbury
#+SETUPFILE: ../tools/theme.setup
#+INCLUDE: ../tools/lib.org

Low- and high-level functions that use the Sherman-Morrison and Woodbury matrix inversion formulas to update the
 inverse of a non-singualr matrix

* Headers
  #+begin_src elisp :noexport :results none
(org-babel-lob-ingest "../tools/lib.org")
#+end_src

  #+begin_src c :comments link :tangle (eval c_test) :noweb yes
#include "qmckl.h"
#include "assert.h"
#ifdef HAVE_CONFIG_H
#include "config.h"
#endif
#include <cmath>
#ifndef THRESHOLD
#define THRESHOLD 1e-3
#endif

int main() {
  qmckl_context context;
  context = qmckl_context_create();

  #+end_src
  
* Naïve   Sherman-Morrison

** ~qmckl_sherman_morrison~
   :PROPERTIES:
   :Name:     qmckl_sherman_morrison
   :CRetType: qmckl_exit_code
   :FRetType: qmckl_exit_code
   :END:

   The Sherman-Morrison formula
   \begin{align}
	   S_k^{-1} &= (S_l + U_k)^-1 \\
			      &= S_l^{-1} - \frac{S_l^{-1}U_kS_l}{1+\underline{v}_k^tS_l^{-1}\underline{u}_k}
   \end{align}


   #+NAME: qmckl_sherman_morrison_args
   | qmckl_context | context | in | Global state |
   | double        | Slater_inv[Dim*Dim]    | inout | Array containing the inverse of a Slater-matrix      |
   | uint          | Dim                    | in    | Leading dimension of Slater_inv                      |
   | uint          | N_updates              | in    | Number of rank-1 updates to be applied to Slater_inv |
   | double        | Updates[N_updates*Dim] | in    | Array containing the updates                         |
   | double        | Updates_index          | in    | Array containing the rank-1 updates                  |

*** Requirements

      Add description of the input variables. (see for e.g. qmckl_distance.org)

*** C header

    #+CALL: generate_c_header(table=qmckl_sherman_morrison_args,rettyp=get_value("CRetType"),fname=get_value("Name"))

    #+RESULTS:
    #+begin_src c :tangle (eval h_func) :comments org
    qmckl_exit_code qmckl_sherman_morrison (
          const qmckl_context context,
          double *Slater_inv,
          unsigned int Dim,
          unsigned int N_updates,
          double *Updates,
          unsigned int *Updates_index );
    #+end_src

*** Source

    #+begin_src c :tangle (eval c_func) :comments org
bool qmckl_sherman_morrison(double *Slater_inv, unsigned int Dim, unsigned int N_updates,
         double *Updates, unsigned int *Updates_index) {
#ifdef DEBUG
  std::cerr << "Called qmckl_sherman_morrison with " << N_updates << " updates" << std::endl;
#endif

  double C[Dim];
  double D[Dim];

  unsigned int l = 0;
  // For each update
  while (l < N_updates) {
    // C = A^{-1} x U_l
    for (unsigned int i = 0; i < Dim; i++) {
      C[i] = 0;
      for (unsigned int j = 0; j < Dim; j++) {
        C[i] += Slater_inv[i * Dim + j] * Updates[l * Dim + j];
      }
    }

    // Denominator
    double den = 1 + C[Updates_index[l] - 1];
    if (std::fabs(den) < threshold()) {
      return false;
    }
    double iden = 1 / den;

    // D = v^T x A^{-1}
    for (unsigned int j = 0; j < Dim; j++) {
      D[j] = Slater_inv[(Updates_index[l] - 1) * Dim + j];
    }

    // A^{-1} = A^{-1} - C x D / den
    for (unsigned int i = 0; i < Dim; i++) {
      for (unsigned int j = 0; j < Dim; j++) {
        double update = C[i] * D[j] * iden;
        Slater_inv[i * Dim + j] -= update;
      }
    }

    l += 1;
  }
  return true;
}

    #+end_src



*** Performance

** C interface                                                     :noexport:

   #+CALL: generate_c_interface(table=qmckl_sherman_morrison_args,rettyp=get_value("FRetType"),fname=get_value("Name"))

   #+CALL: generate_f_interface(table=qmckl_sherman_morrison_args,rettyp=get_value("FRetType"),fname=get_value("Name"))

*** Test                                                           :noexport:

[TODO: FMJC] Write tests for the Sherman-Morrison part.

* End of files

   #+begin_src c :comments link :tangle (eval c_test)
  assert (qmckl_context_destroy(context) == QMCKL_SUCCESS);
  return 0;
}

   #+end_src


# -*- mode: org -*-
# vim: syntax=c