qmckl/org/qmckl_sherman_morrison_woodbury.org

#+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 :exports 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 <math.h>

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

  qmckl_exit_code rc;
  #+end_src


* Helper Functions
  
Helper functions that are used by the Sherman-Morrison-Woodbury kernels. 
These functions can only be used internally by higher level functions.

** ~qmckl_slagel_splitting~
   :PROPERTIES:
   :Name:     qmckl_slagel_splitting
   :CRetType: double
   :FRetType: double precision
   :END:

 ~qmckl_slagel_splitting~ is used internally to perform a J.~Slagel style split of rank-1 updates.
  
   For a given $u_j$ we define a threshold $\epsilon_j$, which is the minimum value of 
   $1+v_j^TS^{-1}u_j$ for the matrix to be considered nonsingular. If $1+v_j^TS^{-1}u_j \geq \epsilon_j$, 
   the update is applied as usual. Otherwise, $u_j$ will be redefined as $\frac{u_j}{2}$, and the other half 
   (to be applied at the end) will be defined using vectors $\frac{u_{j'}}{2}$ and $v_{j'}^T=v_{j'}^T$.
   
   #+NAME: qmckl_slagel_splitting_args
   | uint64_t | Dim                            | in    | Leading dimension of Slater_inv                               |
   | uint64_t | N_updates                      | in    | Number of rank-1 updates to be applied to Slater_inv          |
   | double   | Updates[N_updates*Dim]         | in    | Array containing the rank-1 updates                           |
   | uint64_t | Updates_index[N_updates]       | in    | Array containing positions of the rank-1 updates              |
   | double   | breakdown                      | in    | Break-down parameter on which to fail or not                  |
   | double   | Slater_inv[Dim*Dim]            | inout | Array containing the inverse Slater-matrix                    |
   | double   | later_updates[Dim * N_updates] | inout | Array containing the split updates for later                  |
   | uint64_t | later_index[N_updates]         | inout | Array containing the positions of the split updates for later |
   | uint64_t | later                          | inout | Number of split updates for later                             |
  

*** Requirements

    - ~Dim >= 2~
    - ~N_updates >= 1~
    - ~Updates~ is allocated with at least $1 \times 2 \times 8$ bytes
    - ~Updates_index~ is allocated with at least $1 \times 8$ bytes
    - ~breakdown~ is a small number such that $0 < breakdown << 1$
    - ~Slater_inv~ is allocated with at least $Dim \times Dim \times 8$ bytes
    - ~later_updates~ is allocated with at least $1 \times 2 \times 8$ bytes
    - ~later_index~ is allocated with at least $1 \times 8$ bytes
    - ~later >= 0~ 

*** C header

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

    #+RESULTS:
    #+begin_src c :tangle (eval h_func) :comments org
    qmckl_exit_code qmckl_slagel_splitting_c (
	  const uint64_t Dim,
	  const uint64_t N_updates,
	  const double* Updates,
	  const uint64_t* Updates_index,
	  const double breakdown,
	  double* Slater_inv,
	  double* later_updates,
	  uint64_t* later_index,
	  uint64_t* later ); 
    #+end_src

*** Source Fortran

*** Source C

    #+begin_src c :tangle (eval c) :comments org
#include <stdbool.h>
#include <math.h>
#include "qmckl.h"

qmckl_exit_code qmckl_slagel_splitting_c(uint64_t Dim,
                                              uint64_t N_updates,
                                              const double *Updates, 
                                              const uint64_t *Updates_index,
                                              const double breakdown,
                                              double *Slater_inv,
                                              double *later_updates, 
                                              uint64_t *later_index,
                                              uint64_t *later) {
// #ifdef DEBUG //  Leave commented out since debugging information is not yet implemented in QMCkl.
//   std::cerr << "Called slagel_splitting with " << N_updates << " updates" << std::endl;
// #endif

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

  uint64_t l = 0;
  // For each update
  while (l < N_updates) {
    // C = S^{-1} x U_l
    for (uint64_t i = 0; i < Dim; i++) {
      C[i] = 0;
      for (uint64_t 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 (fabs(den) < breakdown) {

      // U_l = U_l / 2 (do the split)
      for (uint64_t i = 0; i < Dim; i++) {
        later_updates[*later * Dim + i] = Updates[l * Dim + i] / 2.0;
        C[i] /= 2.0;
      }
      later_index[*later] = Updates_index[l];
      (*later)++;

      den = 1 + C[Updates_index[l] - 1];
    }
    double iden = 1 / den;

    // D = v^T x S^{-1}
    for (uint64_t j = 0; j < Dim; j++) {
      D[j] = Slater_inv[(Updates_index[l] - 1) * Dim + j];
    }

    // S^{-1} = S^{-1} - C x D / den
    for (uint64_t i = 0; i < Dim; i++) {
      for (uint64_t j = 0; j < Dim; j++) {
        double update = C[i] * D[j] * iden;
        Slater_inv[i * Dim + j] -= update;
      }
    }
    l += 1;
  }

  return QMCKL_SUCCESS;
}

    #+end_src
    
*** Performance

This function performce better for cycles with 1 rank-1 update and with a high fail-rate.


** C interface                                                     :noexport:
   
   #+CALL: generate_c_interface(table=qmckl_slagel_splitting_args,rettyp=get_value("FRetType"),fname=get_value("Name"))

   #+RESULTS:
   #+begin_src f90 :tangle (eval f) :comments org :exports none
   integer(c_int32_t) function qmckl_slagel_splitting &
       (Dim, N_updates, Updates, Updates_index, breakdown, Slater_inv, later_updates, later_index, later) &
       bind(C) result(info)

     use, intrinsic :: iso_c_binding
     implicit none

     integer (c_int64_t) , intent(in)  , value :: Dim
     integer (c_int64_t) , intent(in)  , value :: N_updates
     real    (c_double ) , intent(in)          :: Updates(N_updates*Dim)
     integer (c_int64_t) , intent(in)          :: Updates_index(N_updates)
     real    (c_double ) , intent(in)  , value :: breakdown
     real    (c_double ) , intent(inout)        :: Slater_inv(Dim*Dim)
     real    (c_double ) , intent(inout)        :: later_updates(Dim * N_updates)
     integer (c_int64_t) , intent(inout)        :: later_index(N_updates)
     integer (c_int64_t) , intent(inout)        :: later

     integer(c_int32_t), external :: qmckl_slagel_splitting_c
     info = qmckl_slagel_splitting_c &
	    (Dim, N_updates, Updates, Updates_index, breakdown, Slater_inv, later_updates, later_index, later)

   end function qmckl_slagel_splitting
   #+end_src

*** Test                                                           :noexport:

  This function does not have an explicit test as it is only used internally by higher level Sherman-Morrison-Woodbury functions.


* Naïve Sherman-Morrison
  
** ~qmckl_sherman_morrison~
   :PROPERTIES:
   :Name:     qmckl_sherman_morrison
   :CRetType: qmckl_exit_code
   :FRetType: qmckl_exit_code
   :END:

   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 zero when an update is evaluated.
   It will exit with an error code of the denominator is too close to zero.

   The formula that is applied is
   \[
    (S + uv^T)^{-1} = S^{-1} - \frac{S^{-1} uv^T S^{-1}}{1 + v^T S^{-1} u}
   \]

   where
   $S$ is the Slater-matrix,
   $u$ and $v^T$ are the column and row vectors containing the updates,
   $S^{-1}$ is the inverse of the Slater-matrix.

   #+NAME: qmckl_sherman_morrison_args
   | qmckl_context | context                  | in    | Global state                                         |
   | uint64_t      | Dim                      | in    | Leading dimension of Slater_inv                      |
   | uint64_t      | N_updates                | in    | Number of rank-1 updates to be applied to Slater_inv |
   | double        | Updates[N_updates*Dim]   | in    | Array containing the updates                         |
   | uint64_t      | Updates_index[N_updates] | in    | Array containing the rank-1 updates                  |
   | double        | breakdown                | in    | Break-down parameter on which to fail or not         |
   | double        | Slater_inv[Dim*Dim]      | inout | Array containing the inverse of a Slater-matrix      |

*** Requirements
    
    - ~context~ is not ~QMCKL_NULL_CONTEXT~
    - ~Dim >= 2~
    - ~N_updates >= 1~
    - ~Updates~ is allocated with at least $1 \times 2 \times 8$ bytes
    - ~Updates_index~ is allocated with at least $1 \times 8$ bytes
    - ~breakdown~ is a small number such that $0 < breakdown << 1$
    - ~Slater_inv~ is allocated with at least $Dim \times Dim \times 8$ bytes
 
*** 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_c (
	  const qmckl_context context,
	  const uint64_t Dim,
	  const uint64_t N_updates,
	  const double* Updates,
	  const uint64_t* Updates_index,
    const double breakdown,
	  double* Slater_inv ); 
    #+end_src

*** Source Fortran

    #+begin_src f90 :tangle (eval f)
integer function qmckl_sherman_morrison_f(context, Dim, N_updates,    &
         Updates, Updates_index, breakdown, Slater_inv) result(info) 
  use qmckl
  implicit none
  integer(qmckl_context)  , intent(in)  :: context
  integer*8  , intent(in), value  :: Dim, N_updates
  integer*8  , intent(in)  :: Updates_index(N_updates)
  real*8     , intent(in) :: Updates(N_updates*Dim)
  real*8     , intent(in) :: breakdown
  real*8     , intent(inout)  :: Slater_inv(Dim*Dim)
  !logical, external :: qmckl_sherman_morrison_f
  info = qmckl_sherman_morrison(context, Dim, N_updates, Updates, Updates_index, breakdown, Slater_inv)
end function qmckl_sherman_morrison_f
    #+end_src

*** Source C

    #+begin_src c :tangle (eval c) :comments org
#include <stdbool.h>
#include "qmckl.h"

qmckl_exit_code qmckl_sherman_morrison_c(const qmckl_context context, 
                                const uint64_t Dim,
                                const uint64_t N_updates,
                                const double* Updates,
                                const uint64_t* Updates_index,
                                const double breakdown,
                                double * Slater_inv) {
// #ifdef DEBUG
//   std::cerr << "Called qmckl_sherman_morrison with " << N_updates << " updates" << std::endl;
// #endif

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

  uint64_t l = 0;
  // For each update
  while (l < N_updates) {
    // C = A^{-1} x U_l
    for (uint64_t i = 0; i < Dim; i++) {
      C[i] = 0;
      for (uint64_t 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 (fabs(den) < breakdown) {
      return QMCKL_FAILURE;
    }
    double iden = 1 / den;

    // D = v^T x A^{-1}
    for (uint64_t j = 0; j < Dim; j++) {
      D[j] = Slater_inv[(Updates_index[l] - 1) * Dim + j];
    }

    // A^{-1} = A^{-1} - C x D / den
    for (uint64_t i = 0; i < Dim; i++) {
      for (uint64_t j = 0; j < Dim; j++) {
        double update = C[i] * D[j] * iden;
        Slater_inv[i * Dim + j] -= update;
      }
    }

    l += 1;
  }

  return QMCKL_SUCCESS;
}

    #+end_src



*** Performance

    This function performs better when there is only 1 rank-1 update in the update cycle and the fail-rate of rank-1 updates is high.


** C interface                                                     :noexport:

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

   #+RESULTS:
   #+begin_src f90 :tangle (eval f) :comments org :exports none
   integer(c_int32_t) function qmckl_sherman_morrison &
       (context, Dim, N_updates, Updates, Updates_index, breakdown, Slater_inv) &
       bind(C) result(info)

     use, intrinsic :: iso_c_binding
     implicit none

     integer (c_int64_t) , intent(in)  , value :: context
     integer (c_int64_t) , intent(in)  , value :: Dim
     integer (c_int64_t) , intent(in)  , value :: N_updates
     real    (c_double ) , intent(in)          :: Updates(N_updates*Dim)
     integer (c_int64_t) , intent(in)          :: Updates_index(N_updates)
     real    (c_double ) , intent(in)          :: breakdown
     real    (c_double ) , intent(inout)       :: Slater_inv(Dim*Dim)

     integer(c_int32_t), external :: qmckl_sherman_morrison_c
     info = qmckl_sherman_morrison_c &
	    (context, Dim, N_updates, Updates, Updates_index, breakdown, Slater_inv)

   end function qmckl_sherman_morrison
   #+end_src

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

   #+RESULTS:
   #+begin_src f90 :tangle (eval fh_func) :comments org :exports none
   interface
     integer(c_int32_t) function qmckl_sherman_morrison &
	 (context, Dim, N_updates, Updates, Updates_index, breakdown, Slater_inv) &
	 bind(C)
       use, intrinsic :: iso_c_binding
       import
       implicit none

       integer (c_int64_t) , intent(in)  , value :: context
       integer (c_int64_t) , intent(in)  , value :: Dim
       integer (c_int64_t) , intent(in)  , value :: N_updates
       real    (c_double ) , intent(in)          :: Updates(N_updates*Dim)
       integer (c_int64_t) , intent(in)          :: Updates_index(N_updates)
       real    (c_double ) , intent(in)          :: breakdown
       real    (c_double ) , intent(inout)        :: Slater_inv(Dim*Dim)

     end function qmckl_sherman_morrison
   end interface
   #+end_src

*** Test                                                           :noexport:

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


     #+begin_src c :tangle (eval c_test)
const uint64_t Dim = 2;
const uint64_t N_updates = 2;
const uint64_t Updates_index[2] = {0, 0};
const double Updates[4] = {0.0, 0.0, 0.0, 0.0};
const double breakdown = 1e-3;
double Slater_inv[4] = {0.0, 0.0, 0.0, 0.0};

// [TODO : FMJC ] add realistic tests

rc = qmckl_sherman_morrison_c(context, Dim, N_updates, Updates, Updates_index, breakdown, Slater_inv);
assert(rc == QMCKL_SUCCESS);
     #+end_src

     
* Woodbury 2x2

** ~qmckl_woodbury_2~
   :PROPERTIES:
   :Name:     qmckl_woodbury_2
   :CRetType: qmckl_exit_code
   :FRetType: qmckl_exit_code
   :END:

   The simplest version of the generalised Sherman-Morrison-Woodbury kernels. It is used to apply two
   rank-1 updates at once. The formula used in this algorithm is called the Woodbury Matrix Identity
   \[
     (S + U V)^{-1}  =  S^{-1} - C B^{-1} D
   \]
   where
   $S$ is the Slater-matrix
   $U$ and $V$ are the matrices containing the updates and the canonical basis matrix
   $S^{-1}$ is the inverse of the Slater-matrix
   $C:= S^{-1}U$, a Dim $\times 2$ matrix
   $B := 1 + VC$, the $2 \times 2$ matrix that is going to be inverted
   $D := VS^{-1}$, a $2 \times Dim$ matrix
   

   #+NAME: qmckl_woodbury_2_args
   | qmckl_context | context             | in    | Global state                                    |
   | uint64_t      | Dim                 | in    | Leading dimension of Slater_inv                 |
   | double        | Updates[2*Dim]      | in    | Array containing the updates                    |
   | uint64_t      | Updates_index[2]    | in    | Array containing the rank-1 updates             |
   | double        | breakdown           | in    | Break-down parameter on which to fail or not    |
   | double        | Slater_inv[Dim*Dim] | inout | Array containing the inverse of a Slater-matrix |

*** Requirements

    - ~context~ is not ~qmckl_null_context~
    - ~dim >= 2~
    - ~updates~ is allocated with at least $2 \times 2 \times 8$ bytes
    - ~updates_index~ is allocated with $2 \times 8$ bytes
    - ~breakdown~ is a small number such that $0 < breakdown << 1$
    - ~slater_inv~ is allocated with at least $dim \times dim \times 8$ bytes

*** C header

    #+CALL: generate_c_header(table=qmckl_woodbury_2_args,rettyp=get_value("CRetType"),fname=get_value("Name"))

    #+RESULTS:
    #+begin_src c :tangle (eval h_func) :comments org
    qmckl_exit_code qmckl_woodbury_2_c (
	  const qmckl_context context,
	  const uint64_t Dim,
	  const double* Updates,
	  const uint64_t* Updates_index,
	  const double breakdown,
	  double* Slater_inv ); 
    #+end_src

*** Source Fortran

    #+begin_src f90 :tangle (eval f)
integer function qmckl_woodbury_2_f(context, Dim,    &
         Updates, Updates_index, breakdown, Slater_inv) result(info) 
  use qmckl
  implicit none
  integer(qmckl_context)  , intent(in)  :: context
  integer*8  , intent(in), value  :: Dim
  integer*8  , intent(in)  :: Updates_index(2)
  real*8     , intent(in) :: Updates(2*Dim)
  real*8     , intent(in) :: breakdown
  real*8     , intent(inout)  :: Slater_inv(Dim*Dim)
  !logical, external :: qmckl_woodbury_2_f
  info = qmckl_woodbury_2(context, Dim, Updates, Updates_index, breakdown, Slater_inv)
end function qmckl_woodbury_2_f
    #+end_src

*** Source C

    #+begin_src c :tangle (eval c) :comments org
#include <stdbool.h>
#include "qmckl.h"

qmckl_exit_code qmckl_woodbury_2_c(const qmckl_context context, 
                                const uint64_t Dim,
                                const double* Updates,
                                const uint64_t* Updates_index,
                                const double breakdown,
                                double * Slater_inv) {
/*
    C := S^{-1} * U,    dim x 2
    B := 1 + V * C,     2 x 2
    D := V * S^{-1},    2 x dim
*/
// #ifdef DEBUG //  Leave commented out since debugging information is not yet implemented in QMCkl.
//   std::cerr << "Called Woodbury 2x2 kernel" << std::endl;
// #endif

  const uint64_t row1 = (Updates_index[0] - 1);
  const uint64_t row2 = (Updates_index[1] - 1);

  // Compute C = S_inv * U  !! NON-STANDARD MATRIX MULTIPLICATION BECAUSE
  // OF LAYOUT OF 'Updates' !!
  double C[2 * Dim];
  for (uint64_t i = 0; i < Dim; i++) {
    for (uint64_t j = 0; j < 2; j++) {
      C[i * 2 + j] = 0;
      for (uint64_t k = 0; k < Dim; k++) {
        C[i * 2 + j] += Slater_inv[i * Dim + k] * Updates[Dim * j + k];
      }
    }
  }

  // Compute B = 1 + V * C
  const double B0 = C[row1 * 2] + 1;
  const double B1 = C[row1 * 2 + 1];
  const double B2 = C[row2 * 2];
  const double B3 = C[row2 * 2 + 1] + 1;

  // Check if determinant of inverted matrix is not zero
  double det = B0 * B3 - B1 * B2;
  if (fabs(det) < breakdown) {
    return QMCKL_FAILURE;
  }

  // Compute B^{-1} with explicit formula for 2x2 inversion
  double Binv[4], idet = 1.0 / det;
  Binv[0] = idet * B3;
  Binv[1] = -1.0 * idet * B1;
  Binv[2] = -1.0 * idet * B2;
  Binv[3] = idet * B0;

  // Compute tmp = B^{-1} x (V.S^{-1})
  double tmp[2 * Dim];
  for (uint64_t i = 0; i < 2; i++) {
    for (uint64_t j = 0; j < Dim; j++) {
      tmp[i * Dim + j] = Binv[i * 2] * Slater_inv[row1 * Dim + j];
      tmp[i * Dim + j] += Binv[i * 2 + 1] * Slater_inv[row2 * Dim + j];
    }
  }

  // Compute (S + U V)^{-1} = S^{-1} - C x tmp
  for (uint64_t i = 0; i < Dim; i++) {
    for (uint64_t j = 0; j < Dim; j++) {
      Slater_inv[i * Dim + j] -= C[i * 2] * tmp[j];
      Slater_inv[i * Dim + j] -= C[i * 2 + 1] * tmp[Dim + j];
    }
  }

  return QMCKL_SUCCESS;
}

    #+end_src


    
*** Performance

    This function is most efficient when used in cases where there are only 2 rank-1 updates
    
** C interface                                                     :noexport:

   #+CALL: generate_c_interface(table=qmckl_woodbury_2_args,rettyp=get_value("FRetType"),fname=get_value("Name"))

   #+RESULTS:
   #+begin_src f90 :tangle (eval f) :comments org :exports none
   integer(c_int32_t) function qmckl_woodbury_2 &
       (context, Dim, Updates, Updates_index, breakdown, Slater_inv) &
       bind(C) result(info)

     use, intrinsic :: iso_c_binding
     implicit none

     integer (c_int64_t) , intent(in)  , value :: context
     integer (c_int64_t) , intent(in)  , value :: Dim
     real    (c_double ) , intent(in)          :: Updates(2*Dim)
     integer (c_int64_t) , intent(in)          :: Updates_index(2)
     real    (c_double ) , intent(in)          :: breakdown
     real    (c_double ) , intent(inout)        :: Slater_inv(Dim*Dim)

     integer(c_int32_t), external :: qmckl_woodbury_2_c
     info = qmckl_woodbury_2_c &
	    (context, Dim, Updates, Updates_index, breakdown, Slater_inv)

   end function qmckl_woodbury_2
   #+end_src

   #+CALL: generate_f_interface(table=qmckl_woodbury_2_args,rettyp=get_value("FRetType"),fname=get_value("Name"))

   #+RESULTS:
   #+begin_src f90 :tangle (eval fh_func) :comments org :exports none
   interface
     integer(c_int32_t) function qmckl_woodbury_2 &
	 (context, Dim, Updates, Updates_index, breakdown, Slater_inv) &
	 bind(C)
       use, intrinsic :: iso_c_binding
       import
       implicit none

       integer (c_int64_t) , intent(in)  , value :: context
       integer (c_int64_t) , intent(in)  , value :: Dim
       real    (c_double ) , intent(in)          :: Updates(2*Dim)
       integer (c_int64_t) , intent(in)          :: Updates_index(2)
       real    (c_double ) , intent(in)          :: breakdown
       real    (c_double ) , intent(inout)        :: Slater_inv(Dim*Dim)

     end function qmckl_woodbury_2
   end interface
   #+end_src

*** Test                                                           :noexport:

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


     #+begin_src c :tangle (eval c_test)
const uint64_t woodbury_Dim = 2;
const uint64_t woodbury_Updates_index[2] = {1, 1};
const double woodbury_Updates[4] = {1.0, 1.0, 1.0, 1.0};
const double woodbury_breakdown = 1e-3;
double woodbury_Slater_inv[4] = {1.0, 1.0, 1.0, 1.0};

// [TODO : FMJC ] add realistic tests

rc = qmckl_woodbury_2_c(context, woodbury_Dim, woodbury_Updates, woodbury_Updates_index, woodbury_breakdown, woodbury_Slater_inv);
assert(rc == QMCKL_SUCCESS);
     #+end_src


* Woodbury 3x3
  
** ~qmckl_woodbury_3~
   :PROPERTIES:
   :Name:     qmckl_woodbury_3
   :CRetType: qmckl_exit_code
   :FRetType: qmckl_exit_code
   :END:

   The 3x3 version of the Woodbury 2x2 kernel. It is used to apply three
   rank-1 updates at once. The formula used in this kernel is the same as for Woodbury 2x2, with the exception of

   $C:= S^{-1}U$, a Dim $\times 3$ matrix
   $B := 1 + VC$, the $3 \times 3$ matrix that is going to be inverted
   $D := VS^{-1}$, a $3 \times Dim$ matrix

   #+NAME: qmckl_woodbury_3_args
   | qmckl_context | context             | in    | Global state                                    |
   | uint64_t      | Dim                 | in    | Leading dimension of Slater_inv                 |
   | double        | Updates[3*Dim]      | in    | Array containing the updates                    |
   | uint64_t      | Updates_index[3]    | in    | Array containing the rank-1 updates             |
   | double        | breakdown           | in    | Break-down parameter on which to fail or not    |
   | double        | Slater_inv[Dim*Dim] | inout | Array containing the inverse of a Slater-matrix |

*** Requirements


*** C header

    #+CALL: generate_c_header(table=qmckl_woodbury_3_args,rettyp=get_value("CRetType"),fname=get_value("Name"))

    #+RESULTS:
    #+begin_src c :tangle (eval h_func) :comments org
    qmckl_exit_code qmckl_woodbury_3_c (
	  const qmckl_context context,
	  const uint64_t Dim,
	  const double* Updates,
	  const uint64_t* Updates_index,
	  const double breakdown,
	  double* Slater_inv ); 
    #+end_src

*** Source Fortran

    #+begin_src f90 :tangle (eval f)
integer function qmckl_woodbury_3_f(context, Dim,    &
         Updates, Updates_index, breakdown, Slater_inv) result(info) 
  use qmckl
  implicit none
  integer(qmckl_context)  , intent(in)  :: context
  integer*8  , intent(in), value  :: Dim
  integer*8  , intent(in)  :: Updates_index(3)
  real*8     , intent(in) :: Updates(3*Dim)
  real*8     , intent(in) :: breakdown
  real*8     , intent(inout)  :: Slater_inv(Dim*Dim)
  !logical, external :: qmckl_woodbury_3_f
  info = qmckl_woodbury_3(context, Dim, Updates, Updates_index, breakdown, Slater_inv)
end function qmckl_woodbury_3_f
    #+end_src
    
*** Source C

    #+begin_src c :tangle (eval c) :comments org
#include <stdbool.h>
#include "qmckl.h"

qmckl_exit_code qmckl_woodbury_3_c(const qmckl_context context, 
                                const uint64_t Dim,
                                const double* Updates,
                                const uint64_t* Updates_index,
                                const double breakdown,
                                double * Slater_inv) {
/*
    C := S^{-1} * U,    dim x 3
    B := 1 + V * C,     3 x 3
    D := V * S^{-1},    3 x dim
,*/
// #ifdef DEBUG //  Leave commented out since debugging information is not yet implemented in QMCkl.
//   std::cerr << "Called Woodbury 3x3 kernel" << std::endl;
// #endif

  const uint64_t row1 = (Updates_index[0] - 1);
  const uint64_t row2 = (Updates_index[1] - 1);
  const uint64_t row3 = (Updates_index[2] - 1);

  // Compute C = S_inv * U  !! NON-STANDARD MATRIX MULTIPLICATION BECAUSE
  // OF LAYOUT OF 'Updates' !!
  double C[3 * Dim];
  for (uint64_t i = 0; i < Dim; i++) {
    for (uint64_t j = 0; j < 3; j++) {
      C[i * 3 + j] = 0;
      for (uint64_t k = 0; k < Dim; k++) {
        C[i * 3 + j] += Slater_inv[i * Dim + k] * Updates[Dim * j + k];
      }
    }
  }

  // Compute B = 1 + V.C
  const double B0 = C[row1 * 3] + 1;
  const double B1 = C[row1 * 3 + 1];
  const double B2 = C[row1 * 3 + 2];
  const double B3 = C[row2 * 3];
  const double B4 = C[row2 * 3 + 1] + 1;
  const double B5 = C[row2 * 3 + 2];
  const double B6 = C[row3 * 3];
  const double B7 = C[row3 * 3 + 1];
  const double B8 = C[row3 * 3 + 2] + 1;

  // Check if determinant of B is not too close to zero
  double det;
  det = B0 * (B4 * B8 - B5 * B7) - B1 * (B3 * B8 - B5 * B6) +
        B2 * (B3 * B7 - B4 * B6);
  if (fabs(det) < breakdown) {
    return QMCKL_FAILURE;
  }

  // Compute B^{-1} with explicit formula for 3x3 inversion
  double Binv[9], idet = 1.0 / det;
  Binv[0] = (B4 * B8 - B7 * B5) * idet;
  Binv[1] = -(B1 * B8 - B7 * B2) * idet;
  Binv[2] = (B1 * B5 - B4 * B2) * idet;
  Binv[3] = -(B3 * B8 - B6 * B5) * idet;
  Binv[4] = (B0 * B8 - B6 * B2) * idet;
  Binv[5] = -(B0 * B5 - B3 * B2) * idet;
  Binv[6] = (B3 * B7 - B6 * B4) * idet;
  Binv[7] = -(B0 * B7 - B6 * B1) * idet;
  Binv[8] = (B0 * B4 - B3 * B1) * idet;

  // Compute tmp = B^{-1} x (V.S^{-1})
  double tmp[3 * Dim];
  for (uint64_t i = 0; i < 3; i++) {
    for (uint64_t j = 0; j < Dim; j++) {
      tmp[i * Dim + j] = Binv[i * 3] * Slater_inv[row1 * Dim + j];
      tmp[i * Dim + j] += Binv[i * 3 + 1] * Slater_inv[row2 * Dim + j];
      tmp[i * Dim + j] += Binv[i * 3 + 2] * Slater_inv[row3 * Dim + j];
    }
  }

  // Compute (S + U V)^{-1} = S^{-1} - C x tmp
  for (uint64_t i = 0; i < Dim; i++) {
    for (uint64_t j = 0; j < Dim; j++) {
      Slater_inv[i * Dim + j] -= C[i * 3] * tmp[j];
      Slater_inv[i * Dim + j] -= C[i * 3 + 1] * tmp[Dim + j];
      Slater_inv[i * Dim + j] -= C[i * 3 + 2] * tmp[2 * Dim + j];
    }
  }

  return QMCKL_SUCCESS;
}

    #+end_src

*** Performance...

    - ~context~ is not ~qmckl_null_context~
    - ~dim >= 2~
    - ~updates~ is allocated with at least $3 \times 2 \times 8$ bytes
    - ~updates_index~ is allocated with $3 \times 8$ bytes
    - ~breakdown~ is a small number such that $0 < breakdown << 1$
    - ~slater_inv~ is allocated with at least $dim \times dim \times 8$ bytes


** C interface                                                     :noexport:

   #+CALL: generate_c_interface(table=qmckl_woodbury_3_args,rettyp=get_value("FRetType"),fname=get_value("Name"))

   #+RESULTS:
   #+begin_src f90 :tangle (eval f) :comments org :exports none
   integer(c_int32_t) function qmckl_woodbury_3 &
       (context, Dim, Updates, Updates_index, breakdown, Slater_inv) &
       bind(C) result(info)

     use, intrinsic :: iso_c_binding
     implicit none

     integer (c_int64_t) , intent(in)  , value :: context
     integer (c_int64_t) , intent(in)  , value :: Dim
     real    (c_double ) , intent(in)          :: Updates(3*Dim)
     integer (c_int64_t) , intent(in)          :: Updates_index(3)
     real    (c_double ) , intent(in)  , value :: breakdown
     real    (c_double ) , intent(inout)        :: Slater_inv(Dim*Dim)

     integer(c_int32_t), external :: qmckl_woodbury_3_c
     info = qmckl_woodbury_3_c &
	    (context, Dim, Updates, Updates_index, breakdown, Slater_inv)

   end function qmckl_woodbury_3
   #+end_src

   #+CALL: generate_f_interface(table=qmckl_woodbury_3_args,rettyp=get_value("FRetType"),fname=get_value("Name"))

   #+RESULTS:
   #+begin_src f90 :tangle (eval fh_func) :comments org :exports none
   interface
     integer(c_int32_t) function qmckl_woodbury_3 &
	 (context, Dim, Updates, Updates_index, breakdown, Slater_inv) &
	 bind(C)
       use, intrinsic :: iso_c_binding
       import
       implicit none

       integer (c_int64_t) , intent(in)  , value :: context
       integer (c_int64_t) , intent(in)  , value :: Dim
       real    (c_double ) , intent(in)          :: Updates(3*Dim)
       integer (c_int64_t) , intent(in)          :: Updates_index(3)
       real    (c_double ) , intent(in)  , value :: breakdown
       real    (c_double ) , intent(inout)        :: Slater_inv(Dim*Dim)

     end function qmckl_woodbury_3
   end interface
   #+end_src

*** Test                                                           :noexport:

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


     #+begin_src c :tangle (eval c_test)
const uint64_t woodbury3_Dim = 3;
const uint64_t woodbury3_Updates_index[3] = {1, 1, 1};
const double woodbury3_Updates[9] = {1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0};
const double woodbury3_breakdown = 1e-3;
double woodbury3_Slater_inv[9] = {1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0};

// [TODO : FMJC ] add realistic tests

rc = qmckl_woodbury_3_c(context, woodbury3_Dim, woodbury3_Updates, woodbury3_Updates_index, woodbury3_breakdown, woodbury3_Slater_inv);
assert(rc == QMCKL_SUCCESS);
     #+end_src


* Sherman-Morrison with update splitting
  
This is like naïve Sherman-Morrising, but whenever a denominator is
found that is too close to zero the update is split in half. Then one
half is applied immediately and the other have is ket for later. When
all the updates have been processed, the list of split updates that
have been kept for later are processed. If again applying an update
results in a denominator that is too close to zero, it is split in
half again. One half is applied immediately and one half is kept for
later. The algorithm is done when no more updates have been kept for
later. This recursion will always end in a finite number of steps,
unless the last original update causes a singular Slater-matrix.
  
** ~qmckl_sherman_morrison_splitting~
   :PROPERTIES:
   :Name:     qmckl_sherman_morrison_splitting
   :CRetType: qmckl_exit_code
   :FRetType: qmckl_exit_code
   :END:

   This is the simplest of the available Sherman-Morrison-Woodbury
   kernels in QMCkl. 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 zero (and exit with an
   error if it does) during the application of an update.

   #+NAME: qmckl_sherman_morrison_splitting_args
   | qmckl_context | context                  | in    | Global state                                         |
   | uint64_t      | Dim                      | in    | Leading dimension of Slater_inv                      |
   | uint64_t      | N_updates                | in    | Number of rank-1 updates to be applied to Slater_inv |
   | double        | Updates[N_updates*Dim]   | in    | Array containing the updates                         |
   | uint64_t      | Updates_index[N_updates] | in    | Array containing the rank-1 updates                  |
   | double        | breakdown                | in    | Break-down parameter on which to fail or not         |
   | double        | Slater_inv[Dim*Dim]      | inout | Array containing the inverse of a Slater-matrix      |

*** Requirements

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

*** C header

    #+CALL: generate_c_header(table=qmckl_sherman_morrison_splitting_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_splitting_c (
	  const qmckl_context context,
	  const uint64_t Dim,
	  const uint64_t N_updates,
	  const double* Updates,
	  const uint64_t* Updates_index,
	  const double breakdown,
	  double* Slater_inv ); 
    #+end_src

*** Source Fortran

    #+begin_src f90 :tangle (eval f)
integer function qmckl_sherman_morrison_splitting_f(context, Dim, N_updates,    &
         Updates, Updates_index, breakdown, Slater_inv) result(info) 
  use qmckl
  implicit none
  integer(qmckl_context)  , intent(in)  :: context
  integer*8  , intent(in), value  :: Dim, N_updates
  integer*8  , intent(in)  :: Updates_index(N_updates)
  real*8     , intent(in) :: Updates(N_updates*Dim)
  real*8     , intent(in) :: breakdown
  real*8     , intent(inout)  :: Slater_inv(Dim*Dim)
  info = qmckl_sherman_morrison_splitting(context, Dim, N_updates, Updates, Updates_index, breakdown, Slater_inv)
end function qmckl_sherman_morrison_splitting_f
    #+end_src


*** Source C

    #+begin_src c :tangle (eval c) :comments org
#include <stdbool.h>
#include "qmckl.h"

qmckl_exit_code qmckl_sherman_morrison_splitting_c(const qmckl_context context, 
                                const uint64_t Dim,
                                const uint64_t N_updates,
                                const double* Updates,
                                const uint64_t* Updates_index,
                                const double breakdown,
                                double * Slater_inv) {
// #ifdef DEBUG //  Leave commented out since debugging information is not yet implemented in QMCkl.
//   std::cerr << "Called qmckl_sherman_morrison_splitting with " << N_updates << " updates" << std::endl;
// #endif

  qmckl_exit_code rc;

  double later_updates[Dim * N_updates];
  uint64_t later_index[N_updates];
  uint64_t later = 0;

  rc = qmckl_slagel_splitting_c(Dim, N_updates, Updates, Updates_index,
                            breakdown, Slater_inv, later_updates, later_index, &later);
  
  if (later > 0) {
    rc = qmckl_sherman_morrison_splitting_c(context, Dim, later, 
                            later_updates, later_index, breakdown, Slater_inv);
  }

  return QMCKL_SUCCESS;
}

    #+end_src

*** Performance...

** C interface                                                     :noexport:

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

   #+RESULTS:
   #+begin_src f90 :tangle (eval f) :comments org :exports none
   integer(c_int32_t) function qmckl_sherman_morrison_splitting &
       (context, Dim, N_updates, Updates, Updates_index, breakdown, Slater_inv) &
       bind(C) result(info)

     use, intrinsic :: iso_c_binding
     implicit none

     integer (c_int64_t) , intent(in)  , value :: context
     integer (c_int64_t) , intent(in)  , value :: Dim
     integer (c_int64_t) , intent(in)  , value :: N_updates
     real    (c_double ) , intent(in)          :: Updates(N_updates*Dim)
     integer (c_int64_t) , intent(in)          :: Updates_index(N_updates)
     real    (c_double ) , intent(in)  , value :: breakdown
     real    (c_double ) , intent(inout)        :: Slater_inv(Dim*Dim)

     integer(c_int32_t), external :: qmckl_sherman_morrison_splitting_c
     info = qmckl_sherman_morrison_splitting_c &
	    (context, Dim, N_updates, Updates, Updates_index, breakdown, Slater_inv)

   end function qmckl_sherman_morrison_splitting
   #+end_src

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

   #+RESULTS:
   #+begin_src f90 :tangle (eval fh_func) :comments org :exports none
   interface
     integer(c_int32_t) function qmckl_sherman_morrison_splitting &
	 (context, Dim, N_updates, Updates, Updates_index, breakdown, Slater_inv) &
	 bind(C)
       use, intrinsic :: iso_c_binding
       import
       implicit none

       integer (c_int64_t) , intent(in)  , value :: context
       integer (c_int64_t) , intent(in)  , value :: Dim
       integer (c_int64_t) , intent(in)  , value :: N_updates
       real    (c_double ) , intent(in)          :: Updates(N_updates*Dim)
       integer (c_int64_t) , intent(in)          :: Updates_index(N_updates)
       real    (c_double ) , intent(in)  , value :: breakdown
       real    (c_double ) , intent(inout)        :: Slater_inv(Dim*Dim)

     end function qmckl_sherman_morrison_splitting
   end interface
   #+end_src


*** Test                                                           :noexport:

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


     #+begin_src c :tangle (eval c_test)
const uint64_t splitting_Dim = 3;
const uint64_t splitting_N_updates = 3;
const uint64_t splitting_Updates_index[3] = {1, 1, 1};
const double splitting_Updates[9] = {1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0};
const double splitting_breakdown = 1e-3;
double splitting_Slater_inv[9] = {1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0};

// [TODO : FMJC ] add realistic tests

rc = qmckl_sherman_morrison_splitting_c(context, splitting_Dim, splitting_N_updates, splitting_Updates, splitting_Updates_index, splitting_breakdown, splitting_Slater_inv);
assert(rc == QMCKL_SUCCESS);
     #+end_src


* Woodbury 2x2 with Sherman-Morrison and update splitting

This is like naïve Sherman-Morrising, but whenever a denominator is
found that is too close to zero the update is split in half. Then one
half is applied immediately and the other have is ket for later. When
all the updates have been processed, the list of split updates that
have been kept for later are processed. If again applying an update
results in a denominator that is too close to zero, it is split in
half again. One half is applied immediately and one half is kept for
later. The algorithm is done when no more updates have been kept for
later. This recursion will always end in a finite number of steps,
unless the last original update causes a singular Slater-matrix.
  
** ~qmckl_sherman_morrison_smw2s~
   :PROPERTIES:
   :Name:     qmckl_sherman_morrison_smw2s
   :CRetType: qmckl_exit_code
   :FRetType: qmckl_exit_code
   :END:

   This is the simplest of the available Sherman-Morrison-Woodbury
   kernels in QMCkl. 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 zero (and exit with an
   error if it does) during the application of an update.

   #+NAME: qmckl_sherman_morrison_smw2s_args
   | qmckl_context | context                  | in    | Global state                                         |
   | uint64_t      | Dim                      | in    | Leading dimension of Slater_inv                      |
   | uint64_t      | N_updates                | in    | Number of rank-1 updates to be applied to Slater_inv |
   | double        | Updates[N_updates*Dim]   | in    | Array containing the updates                         |
   | uint64_t      | Updates_index[N_updates] | in    | Array containing the rank-1 updates                  |
   | double        | breakdown                | in    | Break-down parameter on which to fail or not         |
   | double        | Slater_inv[Dim*Dim]      | inout | Array containing the inverse of a Slater-matrix      |

*** Requirements

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

*** C header

    #+CALL: generate_c_header(table=qmckl_sherman_morrison_smw2s_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_smw2s_c (
	  const qmckl_context context,
	  const uint64_t Dim,
	  const uint64_t N_updates,
	  const double* Updates,
	  const uint64_t* Updates_index,
	  const double breakdown,
	  double* Slater_inv ); 
    #+end_src


*** Source Fortran

    #+begin_src f90 :tangle (eval f)
integer function qmckl_sherman_morrison_smw2s_f(context, Slater_inv, Dim, N_updates,    &
         Updates, Updates_index) result(info) 
  use qmckl
  implicit none
  integer(qmckl_context)  , intent(in)  :: context
  integer*8  , intent(in), value  :: Dim, N_updates
  integer*8  , intent(in)  :: Updates_index(N_updates)
  real*8     , intent(in) :: Updates(N_updates*Dim)
  real*8     , intent(inout)  :: Slater_inv(Dim*Dim)
  info = qmckl_sherman_morrison_smw2s (context, Dim, N_updates, Updates, Updates_index, Slater_inv)
end function qmckl_sherman_morrison_smw2s_f
    #+end_src


*** Source C

    #+begin_src c :tangle (eval c) :comments org
#include <stdbool.h>
#include "qmckl.h"

qmckl_exit_code qmckl_sherman_morrison_smw2s_c(const qmckl_context context, 
                                const uint64_t Dim,
                                const uint64_t N_updates,
                                const double* Updates,
                                const uint64_t* Updates_index,
                                const double breakdown,
                                double * Slater_inv) {
// #ifdef DEBUG //  Leave commented out since debugging information is not yet implemented in QMCkl.
//   std::cerr << "Called qmckl_sherman_morrison_woodbury_2 with " << N_updates
//             << " updates" << std::endl;
// #endif

  qmckl_exit_code rc;

  uint64_t n_of_2blocks = N_updates / 2;
  uint64_t remainder = N_updates % 2;
  uint64_t length_2block = 2 * Dim;

  // Apply first 2*n_of_2blocks updates in n_of_2blocks blocks of 2 updates with
  // Woodbury 2x2 kernel
  double later_updates[Dim * N_updates];
  uint64_t later_index[N_updates];
  uint64_t later = 0;
  if (n_of_2blocks > 0) {
    for (uint64_t i = 0; i < n_of_2blocks; i++) {
      const double *Updates_2block = &Updates[i * length_2block];
      const uint64_t *Updates_index_2block = &Updates_index[i * 2];
      rc = qmckl_woodbury_2_c(context, Dim, Updates_2block, Updates_index_2block, breakdown, Slater_inv);
      if (rc != 0) { // Send the entire block to slagel_splitting
        uint64_t l = 0;
        rc = qmckl_slagel_splitting_c(Dim, 2, Updates_2block, Updates_index_2block,
                breakdown, Slater_inv, later_updates + (Dim * later), later_index + later, &l);
        later = later + l;
      }
    }
  }

  if (remainder == 1) { // Apply last remaining update with slagel_splitting
    const double *Updates_1block = &Updates[n_of_2blocks * length_2block];
    const uint64_t *Updates_index_1block = &Updates_index[2 * n_of_2blocks];
    uint64_t l = 0;
    rc = qmckl_slagel_splitting_c(Dim, 1, Updates_1block, Updates_index_1block,
            breakdown, Slater_inv, later_updates + (Dim * later), later_index + later, &l);
    later = later + l;
  }

  if (later > 0) {
    rc = qmckl_sherman_morrison_splitting_c(context, Dim, later, later_updates, later_index, breakdown, Slater_inv);
  }
  return QMCKL_SUCCESS;
}

    #+end_src

*** Performance...

** C interface                                                     :noexport:

   #+CALL: generate_c_interface(table=qmckl_sherman_morrison_smw2s_args,rettyp=get_value("FRetType"),fname=get_value("Name"))

   #+RESULTS:
   #+begin_src f90 :tangle (eval f) :comments org :exports none
   integer(c_int32_t) function qmckl_sherman_morrison_smw2s &
       (context, Dim, N_updates, Updates, Updates_index, Slater_inv) &
       bind(C) result(info)

     use, intrinsic :: iso_c_binding
     implicit none

     integer (c_int64_t) , intent(in)  , value :: context
     integer (c_int64_t) , intent(in)  , value :: Dim
     integer (c_int64_t) , intent(in)  , value :: N_updates
     real    (c_double ) , intent(in)          :: Updates(N_updates*Dim)
     integer (c_int64_t) , intent(in)          :: Updates_index(N_updates)
     real    (c_double ) , intent(inout)        :: Slater_inv(Dim*Dim)

     integer(c_int32_t), external :: qmckl_sherman_morrison_smw2s_c
     info = qmckl_sherman_morrison_smw2s_c &
	    (context, Dim, N_updates, Updates, Updates_index, Slater_inv)

   end function qmckl_sherman_morrison_smw2s
   #+end_src

   #+CALL: generate_f_interface(table=qmckl_sherman_morrison_smw2s_args,rettyp=get_value("FRetType"),fname=get_value("Name"))

   #+RESULTS:
   #+begin_src f90 :tangle (eval fh_func) :comments org :exports none
   interface
     integer(c_int32_t) function qmckl_sherman_morrison_smw2s &
	 (context, Dim, N_updates, Updates, Updates_index, Slater_inv) &
	 bind(C)
       use, intrinsic :: iso_c_binding
       import
       implicit none

       integer (c_int64_t) , intent(in)  , value :: context
       integer (c_int64_t) , intent(in)  , value :: Dim
       integer (c_int64_t) , intent(in)  , value :: N_updates
       real    (c_double ) , intent(in)          :: Updates(N_updates*Dim)
       integer (c_int64_t) , intent(in)          :: Updates_index(N_updates)
       real    (c_double ) , intent(inout)        :: Slater_inv(Dim*Dim)

     end function qmckl_sherman_morrison_smw2s
   end interface
   #+end_src

*** Test                                                           :noexport:

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


     #+begin_src c :tangle (eval c_test)
const uint64_t smw2s_Dim = 3;
const uint64_t smw2s_N_updates = 3;
const uint64_t smw2s_Updates_index[3] = {1, 1, 1};
const double smw2s_Updates[9] = {1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0};
const double smw2s_breakdown = 1e-3;
double smw2s_Slater_inv[9] = {1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0};

// [TODO : FMJC ] add realistic tests

rc = qmckl_sherman_morrison_smw2s_c(context, smw2s_Dim, smw2s_N_updates,
                  smw2s_Updates, smw2s_Updates_index, smw2s_breakdown, smw2s_Slater_inv);
assert(rc == QMCKL_SUCCESS);
     #+end_src


* Woodbury 3x3 with Sherman-Morrison and update splitting

This is like naïve Sherman-Morrising, but whenever a denominator is
found that is too close to zero the update is split in half. Then one
half is applied immediately and the other have is ket for later. When
all the updates have been processed, the list of split updates that
have been kept for later are processed. If again applying an update
results in a denominator that is too close to zero, it is split in
half again. One half is applied immediately and one half is kept for
later. The algorithm is done when no more updates have been kept for
later. This recursion will always end in a finite number of steps,
unless the last original update causes a singular Slater-matrix.
  
** ~qmckl_sherman_morrison_smw3s~
   :PROPERTIES:
   :Name:     qmckl_sherman_morrison_smw3s
   :CRetType: qmckl_exit_code
   :FRetType: qmckl_exit_code
   :END:

   This is the simplest of the available Sherman-Morrison-Woodbury
   kernels in QMCkl. 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 zero (and exit with an
   error if it does) during the application of an update.

   #+NAME: qmckl_sherman_morrison_smw3s_args
   | qmckl_context | context                  | in    | Global state                                         |
   | uint64_t      | Dim                      | in    | Leading dimension of Slater_inv                      |
   | uint64_t      | N_updates                | in    | Number of rank-1 updates to be applied to Slater_inv |
   | double        | Updates[N_updates*Dim]   | in    | Array containing the updates                         |
   | uint64_t      | Updates_index[N_updates] | in    | Array containing the rank-1 updates                  |
   | double        | breakdown                | in    | Break-down parameter on which to fail or not         |
   | double        | Slater_inv[Dim*Dim]      | inout | Array containing the inverse of a Slater-matrix      |

*** Requirements

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

*** C header

    #+CALL: generate_c_header(table=qmckl_sherman_morrison_smw3s_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_smw3s_c (
	  const qmckl_context context,
	  const uint64_t Dim,
	  const uint64_t N_updates,
	  const double* Updates,
	  const uint64_t* Updates_index,
	  const double breakdown,
	  double* Slater_inv ); 
    #+end_src


*** Source Fortran

    #+begin_src f90 :tangle (eval f)
integer function qmckl_sherman_morrison_smw3s_f(context, Slater_inv, Dim, N_updates,    &
         Updates, Updates_index) result(info) 
  use qmckl
  implicit none
  integer(qmckl_context)  , intent(in)  :: context
  integer*8  , intent(in), value  :: Dim, N_updates
  integer*8  , intent(in)  :: Updates_index(N_updates)
  real*8     , intent(in) :: Updates(N_updates*Dim)
  real*8     , intent(inout)  :: Slater_inv(Dim*Dim)
  !logical, external :: qmckl_sherman_morrison_f
  info = qmckl_sherman_morrison_smw3s(context, Dim, N_updates, Updates, Updates_index, Slater_inv)
end function qmckl_sherman_morrison_smw3s_f
    #+end_src


*** Source C

    #+begin_src c :tangle (eval c) :comments org
#include <stdbool.h>
#include "qmckl.h"

qmckl_exit_code qmckl_sherman_morrison_smw3s_c(const qmckl_context context, 
                                const uint64_t Dim,
                                const uint64_t N_updates,
                                const double* Updates,
                                const uint64_t* Updates_index,
                                const double breakdown,
                                double * Slater_inv) {
// #ifdef DEBUG //  Leave commented out since debugging information is not yet implemented in QMCkl.
//   std::cerr << "Called qmckl_sherman_morrison_woodbury_3 with " << N_updates
//             << " updates" << std::endl;
// #endif

  qmckl_exit_code rc;

  uint64_t n_of_3blocks = N_updates / 3;
  uint64_t remainder = N_updates % 3;
  uint64_t length_3block = 3 * Dim;

  // Apply first 3*n_of_3blocks updates in n_of_3blocks blocks of 3 updates with
  // Woodbury 3x3 kernel
  double later_updates[Dim * N_updates];
  uint64_t later_index[N_updates];
  uint64_t later = 0;
  if (n_of_3blocks > 0) {
    for (uint64_t i = 0; i < n_of_3blocks; i++) {
      const double *Updates_3block = &Updates[i * length_3block];
      const uint64_t *Updates_index_3block = &Updates_index[i * 3];
      rc = qmckl_woodbury_3_c(context, Dim, Updates_3block, Updates_index_3block, breakdown, Slater_inv);
      if (rc != 0) { // Send the entire block to slagel_splitting
        uint64_t l = 0;
        rc = qmckl_slagel_splitting_c(Dim, 3, Updates_3block, Updates_index_3block,
                breakdown, Slater_inv, later_updates + (Dim * later), later_index + later, &l);
        later = later + l;
      }
    }
  }

  if (remainder != 0) { // Apply last remaining block of 2 updates with Woodbury 2x2 kernel
    const double *Updates_remainder_block = &Updates[n_of_3blocks * length_3block];
    const uint64_t *Updates_index_remainder_block = &Updates_index[3 * n_of_3blocks];
    uint64_t l = 0;
    rc = qmckl_slagel_splitting_c(Dim, remainder, Updates_remainder_block, Updates_index_remainder_block,
            breakdown, Slater_inv, later_updates + (Dim * later), later_index + later, &l);
    later = later + l;
  }

  if (later > 0) {
    rc = qmckl_sherman_morrison_splitting_c(context, Dim, later, later_updates, later_index, breakdown, Slater_inv);
  }
  return QMCKL_SUCCESS;
}

    #+end_src

*** Performance...

** C interface                                                     :noexport:

   #+CALL: generate_c_interface(table=qmckl_sherman_morrison_smw3s_args,rettyp=get_value("FRetType"),fname=get_value("Name"))

   #+RESULTS:
   #+begin_src f90 :tangle (eval f) :comments org :exports none
   integer(c_int32_t) function qmckl_sherman_morrison_smw3s &
       (context, Dim, N_updates, Updates, Updates_index, Slater_inv) &
       bind(C) result(info)

     use, intrinsic :: iso_c_binding
     implicit none

     integer (c_int64_t) , intent(in)  , value :: context
     integer (c_int64_t) , intent(in)  , value :: Dim
     integer (c_int64_t) , intent(in)  , value :: N_updates
     real    (c_double ) , intent(in)          :: Updates(N_updates*Dim)
     integer (c_int64_t) , intent(in)          :: Updates_index(N_updates)
     real    (c_double ) , intent(inout)        :: Slater_inv(Dim*Dim)

     integer(c_int32_t), external :: qmckl_sherman_morrison_smw3s_c
     info = qmckl_sherman_morrison_smw3s_c &
	    (context, Dim, N_updates, Updates, Updates_index, Slater_inv)

   end function qmckl_sherman_morrison_smw3s
   #+end_src

   #+CALL: generate_f_interface(table=qmckl_sherman_morrison_smw3s_args,rettyp=get_value("FRetType"),fname=get_value("Name"))

   #+RESULTS:
   #+begin_src f90 :tangle (eval fh_func) :comments org :exports none
   interface
     integer(c_int32_t) function qmckl_sherman_morrison_smw3s &
	 (context, Dim, N_updates, Updates, Updates_index, Slater_inv) &
	 bind(C)
       use, intrinsic :: iso_c_binding
       import
       implicit none

       integer (c_int64_t) , intent(in)  , value :: context
       integer (c_int64_t) , intent(in)  , value :: Dim
       integer (c_int64_t) , intent(in)  , value :: N_updates
       real    (c_double ) , intent(in)          :: Updates(N_updates*Dim)
       integer (c_int64_t) , intent(in)          :: Updates_index(N_updates)
       real    (c_double ) , intent(inout)        :: Slater_inv(Dim*Dim)

     end function qmckl_sherman_morrison_smw3s
   end interface
   #+end_src

*** Test                                                           :noexport:

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


     #+begin_src c :tangle (eval c_test)
const uint64_t smw3s_Dim = 3;
const uint64_t smw3s_N_updates = 3;
const uint64_t smw3s_Updates_index[3] = {1, 1, 1};
const double smw3s_Updates[9] = {1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0};
const double smw3s_breakdown = 1e-3;
double smw3s_Slater_inv[9] = {1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0};

// [TODO : FMJC ] add realistic tests

rc = qmckl_sherman_morrison_smw3s_c(context, smw3s_Dim, smw3s_N_updates,
                  smw3s_Updates, smw3s_Updates_index, smw3s_breakdown, smw3s_Slater_inv);
assert(rc == QMCKL_SUCCESS);
     #+end_src


* Woodbury 3x3 and 2x2 with Sherman-Morrison and update splitting

This is like naïve Sherman-Morrising, but whenever a denominator is
found that is too close to zero the update is split in half. Then one
half is applied immediately and the other have is ket for later. When
all the updates have been processed, the list of split updates that
have been kept for later are processed. If again applying an update
results in a denominator that is too close to zero, it is split in
half again. One half is applied immediately and one half is kept for
later. The algorithm is done when no more updates have been kept for
later. This recursion will always end in a finite number of steps,
unless the last original update causes a singular Slater-matrix.
  
** ~qmckl_sherman_morrison_smw32s~
   :PROPERTIES:
   :Name:     qmckl_sherman_morrison_smw32s
   :CRetType: qmckl_exit_code
   :FRetType: qmckl_exit_code
   :END:

   This is the simplest of the available Sherman-Morrison-Woodbury
   kernels in QMCkl. 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 zero (and exit with an
   error if it does) during the application of an update.

   #+NAME: qmckl_sherman_morrison_smw32s_args
   | qmckl_context | context                  | in    | Global state                                         |
   | uint64_t      | Dim                      | in    | Leading dimension of Slater_inv                      |
   | uint64_t      | N_updates                | in    | Number of rank-1 updates to be applied to Slater_inv |
   | double        | Updates[N_updates*Dim]   | in    | Array containing the updates                         |
   | uint64_t      | Updates_index[N_updates] | in    | Array containing the rank-1 updates                  |
   | double        | breakdown                | in    | Break-down parameter on which to fail or not         |
   | double        | Slater_inv[Dim*Dim]      | inout | Array containing the inverse of a Slater-matrix      |

*** Requirements

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

*** C header

    #+CALL: generate_c_header(table=qmckl_sherman_morrison_smw32s_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_smw32s_c (
	  const qmckl_context context,
	  const uint64_t Dim,
	  const uint64_t N_updates,
	  const double* Updates,
	  const uint64_t* Updates_index,
	  const double breakdown,
	  double* Slater_inv ); 
    #+end_src


*** Source Fortran

    #+begin_src f90 :tangle (eval f)
integer function qmckl_sherman_morrison_smw32s_f(context, Slater_inv, Dim, N_updates,    &
         Updates, Updates_index) result(info) 
  use qmckl
  implicit none
  integer(qmckl_context)  , intent(in)  :: context
  integer*8  , intent(in), value  :: Dim, N_updates
  integer*8  , intent(in)  :: Updates_index(N_updates)
  real*8     , intent(in) :: Updates(N_updates*Dim)
  real*8     , intent(inout)  :: Slater_inv(Dim*Dim)
  !logical, external :: qmckl_sherman_morrison_f
  info = qmckl_sherman_morrison_smw32s(context, Dim, N_updates, Updates, Updates_index, Slater_inv)
end function qmckl_sherman_morrison_smw32s_f
    #+end_src


*** Source C

    #+begin_src c :tangle (eval c) :comments org
#include <stdbool.h>
#include "qmckl.h"

qmckl_exit_code qmckl_sherman_morrison_smw32s_c(const qmckl_context context, 
                                const uint64_t Dim,
                                const uint64_t N_updates,
                                const double* Updates,
                                const uint64_t* Updates_index,
                                const double breakdown,
                                double * Slater_inv) {
// #ifdef DEBUG //  Leave commented out since debugging information is not yet implemented in QMCkl.
//   std::cerr << "Called qmckl_sherman_morrison_woodbury_3 with " << N_updates
//             << " updates" << std::endl;
// #endif

  qmckl_exit_code rc;

  uint64_t n_of_3blocks = N_updates / 3;
  uint64_t remainder = N_updates % 3;
  uint64_t length_3block = 3 * Dim;

  // Apply first 3*n_of_3blocks updates in n_of_3blocks blocks of 3 updates with
  // Woodbury 3x3 kernel
  double later_updates[Dim * N_updates];
  uint64_t later_index[N_updates];
  uint64_t later = 0;
  if (n_of_3blocks > 0) {
    for (uint64_t i = 0; i < n_of_3blocks; i++) {
      const double *Updates_3block = &Updates[i * length_3block];
      const uint64_t *Updates_index_3block = &Updates_index[i * 3];
      rc = qmckl_woodbury_3_c(context, Dim, Updates_3block, Updates_index_3block, breakdown, Slater_inv);
      if (rc != 0) { // Send the entire block to slagel_splitting
        uint64_t l = 0;
        rc = qmckl_slagel_splitting_c(Dim, 3, Updates_3block, Updates_index_3block,
                breakdown, Slater_inv, later_updates + (Dim * later), later_index + later, &l);
        later = later + l;
      }
    }
  }

  if (remainder == 2) { // Apply last remaining block of 2 updates with Woodbury 2x2 kernel
    const double *Updates_2block = &Updates[n_of_3blocks * length_3block];
    const uint64_t *Updates_index_2block = &Updates_index[3 * n_of_3blocks];
    rc = qmckl_woodbury_2_c(context, Dim, Updates_2block, Updates_index_2block, breakdown, Slater_inv);
    if (rc != 0) { // Send the entire block to slagel_splitting
      uint64_t l = 0;
      rc = qmckl_slagel_splitting_c(Dim, 2, Updates_2block, Updates_index_2block,
              breakdown, Slater_inv, later_updates + (Dim * later), later_index + later, &l);
      later = later + l;
    }
  }
  else if (remainder == 1) { // Apply last remaining update with slagel_splitting
    const double *Updates_1block = &Updates[n_of_3blocks * length_3block];
    const uint64_t *Updates_index_1block = &Updates_index[3 * n_of_3blocks];
    uint64_t l = 0;
    rc = qmckl_slagel_splitting_c(Dim, 1, Updates_1block, Updates_index_1block,
            breakdown, Slater_inv, later_updates + (Dim * later), later_index + later, &l);
    later = later + l;
  }

  if (later > 0) {
    rc = qmckl_sherman_morrison_splitting_c(context, Dim, later, later_updates, later_index, breakdown, Slater_inv);
  }
  return QMCKL_SUCCESS;
}

    #+end_src

*** Performance...

** C interface                                                     :noexport:

   #+CALL: generate_c_interface(table=qmckl_sherman_morrison_smw32s_args,rettyp=get_value("FRetType"),fname=get_value("Name"))

   #+RESULTS:
   #+begin_src f90 :tangle (eval f) :comments org :exports none
   integer(c_int32_t) function qmckl_sherman_morrison_smw32s &
       (context, Dim, N_updates, Updates, Updates_index, Slater_inv) &
       bind(C) result(info)

     use, intrinsic :: iso_c_binding
     implicit none

     integer (c_int64_t) , intent(in)  , value :: context
     integer (c_int64_t) , intent(in)  , value :: Dim
     integer (c_int64_t) , intent(in)  , value :: N_updates
     real    (c_double ) , intent(in)          :: Updates(N_updates*Dim)
     integer (c_int64_t) , intent(in)          :: Updates_index(N_updates)
     real    (c_double ) , intent(inout)        :: Slater_inv(Dim*Dim)

     integer(c_int32_t), external :: qmckl_sherman_morrison_smw32s_c
     info = qmckl_sherman_morrison_smw32s_c &
	    (context, Dim, N_updates, Updates, Updates_index, Slater_inv)

   end function qmckl_sherman_morrison_smw32s
   #+end_src

   #+CALL: generate_f_interface(table=qmckl_sherman_morrison_smw32s_args,rettyp=get_value("FRetType"),fname=get_value("Name"))

   #+RESULTS:
   #+begin_src f90 :tangle (eval fh_func) :comments org :exports none
   interface
     integer(c_int32_t) function qmckl_sherman_morrison_smw32s &
	 (context, Dim, N_updates, Updates, Updates_index, Slater_inv) &
	 bind(C)
       use, intrinsic :: iso_c_binding
       import
       implicit none

       integer (c_int64_t) , intent(in)  , value :: context
       integer (c_int64_t) , intent(in)  , value :: Dim
       integer (c_int64_t) , intent(in)  , value :: N_updates
       real    (c_double ) , intent(in)          :: Updates(N_updates*Dim)
       integer (c_int64_t) , intent(in)          :: Updates_index(N_updates)
       real    (c_double ) , intent(inout)        :: Slater_inv(Dim*Dim)

     end function qmckl_sherman_morrison_smw32s
   end interface
   #+end_src

*** Test                                                           :noexport:

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


     #+begin_src c :tangle (eval c_test)
const uint64_t smw32s_Dim = 3;
const uint64_t smw32s_N_updates = 3;
const uint64_t smw32s_Updates_index[3] = {1, 1, 1};
const double smw32s_Updates[9] = {1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0};
const double smw32s_breakdown = 1e-3;
double smw32s_Slater_inv[9] = {1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0};

// [TODO : FMJC ] add realistic tests

rc = qmckl_sherman_morrison_smw32s_c(context, smw32s_Dim, smw32s_N_updates,
                  smw32s_Updates, smw32s_Updates_index, smw32s_breakdown, smw32s_Slater_inv);
assert(rc == QMCKL_SUCCESS);
     #+end_src




* 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