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

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

This is the range that determines the how many high performance kernel instantces will be generated, using the C-function templates defined in the sections below. If the name of the C-function template is called ~qmckl_kernel_{Dim}~, then ~range(K, L+1)~ will results in kernel instances from ~qmckl_kernel_K~ to ~qmckl_kernel_L~.
#+NAME:kernel_generator_range
#+begin_src python :noweb yes :exports none
range(2, 22)
#+end_src

#+RESULTS: kernel_generator_range
: None
  

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

*** Introduction
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.

#+TODO
Change the math notation so that the update vectors appear as row in the math
so that it is consistent with the representation in C (memory)

The formula for any update $u_j$ (index $j$ is suppresed for clarity) that is applied is
\[
(S + uv^T)^{-1} = S^{-1} - \frac{S^{-1} uv^T S^{-1}}{1 + v^T S^{-1} u}
\]

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.

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
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.

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
$S$, 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}.
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
from applying the updates to the original matrix.

*** API
#+NAME: qmckl_sm_naive_args
| Variable        | Type                    | In/Out | Description                                          |
|-----------------+-------------------------+--------+------------------------------------------------------|
| ~context~       | ~qmckl_context~         | in     | Global state                                         |
| ~LDS~           | ~uint64_t~              | in     | Leading dimension of Slater_inv                      |
| ~Dim~           | ~uint64_t~              | in     | Dimension of Slater_inv                              |
| ~N_updates~     | ~uint64_t~              | in     | Number of rank-1 updates to be applied to Slater_inv |
| ~Updates~       | ~double[N_updates*LDS]~ | in     | Array containing the updates                         |
| ~Updates_index~ | ~uint64_t[N_updates]~   | in     | Array containing the rank-1 updates                  |
| ~breakdown~     | ~double~                | in     | Break-down parameter on which to fail or not         |
| ~Slater_inv~    | ~double[Dim*LDS]~       | inout  | Array containing the inverse of a Slater-matrix      |
| ~determinant~   | ~double~                | inout  | Determinant of the Slater-matrix                     |

*** Requirements
- ~context~ is not ~QMCKL_NULL_CONTEXT~
- ~LDS >= 2~
- ~Dim >= 2~
- ~N_updates >= 1~
- ~Updates~ is allocated with $N_updates \times Dim$ elements
- ~Updates_index~ is allocated with $N_updates$ elements
- ~breakdown~ is a small number such that $0 < breakdown << 1$
- ~Slater_inv~ is allocated with $Dim \times Dim$ elements
- ~determinant > 0~

*** Pedagogical kernel source (in Fortran)
The following source code written in Fortran is inteded to illustrate how the kernel works. Even though the kernel is
able to do numerically correct computations, it does not do it in the most efficient way possible. It should therefore
not be used in real workloads.

#+begin_src f90 :tangle (eval f) :comment org :exports none
subroutine convert(upds, s_inv, Updates, Inverse, nupdates, lds, dim)
  implicit none
  integer*8 , intent(in)                             :: lds, dim, nupdates
  real*8    , intent(in)                             :: upds(nupdates * lds)
  real*8    , intent(in)                             :: s_inv(dim * lds)  
  real*8    , intent(out) , dimension(dim, nupdates) :: Updates
  real*8    , intent(out) , dimension(dim, dim)      :: Inverse

  integer*8                                          :: i, j

  ! Construct Updates: lds x nupdates
  do i = 1, nupdates
    do j = 1, dim
      Updates(j, i) = upds((i - 1) * lds + j)
    end do
  end do

  ! Construct Inverse: dim x lds
  do i = 1, dim
    do j = 1, dim
      Inverse(i, j) = s_inv((i - 1) * lds + j)
    end do
  end do
end subroutine convert
#+end_src

#+begin_src f90 :tangle (eval f) :comment org :exports none
subroutine copy_back_inv(Inverse, s_inv, lds, dim)
  implicit none
  integer*8 , intent(in)                       :: lds, dim
  real*8    , intent(in) , dimension(dim, dim) :: Inverse
  real*8    , intent(out)                      :: s_inv(dim * lds)  

  integer*8                                    :: i, j

  ! Copy updated inverse back to s_inv
  do i = 1, dim
    do j = 1, dim
      s_inv((i - 1) * lds + j) = Inverse(i, j)
    end do
  end do
end subroutine copy_back_inv
#+end_src

#+begin_src f90 :tangle (eval f) :comment org :exports none
subroutine copy_back_lu(Later_updates, later_upds, lds, dim, nupdates)
  implicit none
  integer*8 , intent(in)                            :: lds, dim, nupdates
  real*8    , intent(in) , dimension(dim, nupdates) :: Later_updates
  real*8    , intent(out)                           :: later_upds(nupdates * lds)  

  integer*8                                    :: i, j

  ! Copy updated inverse back to s_inv
  do i = 1, nupdates
    do j = 1, dim
      later_upds((i - 1) * lds + j) = Later_updates(j, i)
    end do
  end do
end subroutine copy_back_lu
#+end_src

#+begin_src f90 :tangle (eval f)
integer function qmckl_sm_naive_doc_f(context, &
    lds, dim, &
    nupdates, &
    upds, &
    updates_index, &
    breakdown, &
    s_inv, &
    determinant) result(info)

  use qmckl
  implicit none
  integer*8 , intent(in)    :: context
  integer*8 , intent(in)    :: lds, dim
  integer*8 , intent(in)    :: nupdates
  integer*8 , intent(in)    :: updates_index(nupdates)
  real*8    , intent(in)    :: upds(nupdates * lds)
  real*8    , intent(in)    :: breakdown
  real*8    , intent(inout) :: s_inv(dim * lds)
  real*8    , intent(inout) :: determinant

  real*8 , dimension(dim, nupdates) :: Updates
  real*8 , dimension(dim, dim)      :: Inverse
  real*8 , dimension(dim)           :: C
  real*8 , dimension(dim)           :: D
  real*8                            :: denominator, idenominator, update
  integer*8                         :: i, j, l, row

  info = QMCKL_FAILURE

  if (context == QMCKL_NULL_CONTEXT) then
    info = QMCKL_INVALID_CONTEXT
    return
  endif

  ! Convert 'upds' and 's_inv' into the more easily readable Fortran
  ! matrices 'Updates' and 'Inverse'.
  call convert(upds, s_inv, Updates, Inverse, nupdates, lds, dim)

  l = 1;
  ! For each update do...
  do while (l < nupdates + 1)

    ! Compute C = S^{-1}U(l)
    do i = 1, dim
      C(i) = 0
      do j = 1, dim
        C(i) = C(i) + Inverse(i, j) * Updates(j, l)
      end do
    end do

    ! Compute denominator = 1 + V(l)^TC
    row = updates_index(l)
    denominator = 1 + C(row)

    ! Return early if denominator is too small
    if (abs(denominator) < breakdown) return
    idenominator = 1 / denominator

    ! Update det(S)
    determinant = determinant * denominator

    ! selecting column: v_l^T * S_inv
    D = Inverse(row, :)

    ! A^{-1} = A^{-1} - C x D / denominator
    do i = 1, dim
      do j = 1, dim
        update = C(i) * D(j) * idenominator
        Inverse(i, j) = Inverse(i, j) - update
      end do
    end do

    l = l + 1
  end do

  ! Copy updated inverse back to s_inv
  call copy_back_inv(Inverse, s_inv, lds, dim)

  info = QMCKL_SUCCESS

end function qmckl_sm_naive_doc_f
#+end_src

**** C interface (not directly exposed)
The following Fortran function ~qmckl_sm_naive_doc~ makes sure
that the pedagogical kernel ~qmckl_sm_naive_doc_f~, written in
Fortran, can be called from C using the ~ISO_C_BINDING~. The Fortran function ~qmckl_sm_naive_doc~ will be exposed in the header file 'qmckl.h'
for C users and in the module file 'qmckl_f.F90' for Fortran users.

#+CALL: generate_c_interface(table=qmckl_sm_naive_args,rettyp=get_value("CRetType"),fname="qmckl_sm_naive_doc")

#+RESULTS:
#+begin_src f90 :tangle (eval f) :comments org :exports none
integer(c_int32_t) function qmckl_sm_naive_doc &
    (context, LDS, Dim, N_updates, Updates, Updates_index, breakdown, Slater_inv, determinant) &
    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 :: LDS
  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*LDS)
  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*LDS)
  real    (c_double ) , intent(inout)        :: determinant

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

end function qmckl_sm_naive_doc
#+end_src

*** C headers (exposed in qmckl.h)

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

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

#+CALL: generate_private_c_header(table=qmckl_sm_naive_args,rettyp=get_value("CRetType"),fname="qmckl_sm_naive_hpc")

#+RESULTS:
#+begin_src c :tangle (eval h_private_func) :comments org
qmckl_exit_code qmckl_sm_naive_hpc (
      const qmckl_context context,
      const uint64_t LDS,
      const uint64_t Dim,
      const uint64_t N_updates,
      const double* Updates,
      const uint64_t* Updates_index,
      const double breakdown,
      double* Slater_inv,
      double* determinant ); 
#+end_src

#+CALL: generate_c_header(table=qmckl_sm_naive_args,rettyp=get_value("CRetType"),fname="qmckl_sm_naive_doc")

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

*** C sources
Common includes and macros used by all the Sherman-Morrison-Woodbury kernels.
#+begin_src c :tangle (eval c) :comments org
#include <stdbool.h>
#include <math.h>
#include "qmckl.h"
#include "config.h"
#include "assert.h"
#include "stdio.h"

#+end_src

~qmckl_sm_naive_hpc~ is a high performance variation of
~qmckl_sm_naive~ written in C. It is used in cases when ~Dim~ is
smaller than the leading dimension ~LDS~, irrespective of whetether ~LDS~
includes zero padding to benefit from SIMD instructions or not. Cases like this
include situations where one wants to apply updates to a square submatrix of the
full matrix.
It takes advantage of memory aligned data and assumes no data dependencies
inside the loops. The loops are fully vectorised whenever ~Dim~ is an integer
multiple of ~SIMD_LENGTH~.
#+begin_src c :tangle (eval c) :comments org
qmckl_exit_code qmckl_sm_naive_hpc(
    const qmckl_context context,
    const uint64_t LDS,
    const uint64_t Dim,
    const uint64_t N_updates,
    const double* restrict Updates,
    const uint64_t* restrict Updates_index,
    const double breakdown,
    double* restrict Slater_inv,
    double* restrict determinant) {

  if (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
      return qmckl_failwith( context,
          QMCKL_NULL_CONTEXT,
          "qmckl_sm_naive_hpc",
          NULL);
  }

  double __attribute__((aligned(8))) C[Dim];
  double __attribute__((aligned(8))) D[LDS];

  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] = 0.0f;
      IVDEP
      ALIGNED
      for (uint64_t j = 0; j < Dim; j++) {
        C[i] += Slater_inv[i * LDS + j] * Updates[l * LDS + j];
      }
    }

    // Denominator: v_l^T * C
    const int cui = Updates_index[l] - 1;
    double den = 1.0f + C[cui];

    if (fabs(den) < breakdown)
      return QMCKL_FAILURE;

    double iden = 1.0f / den;

    // Update det(A)
    if (determinant)
      *determinant *= den;

    // selecting column: v_l^T * S_inv
    IVDEP
    ALIGNED
    for (uint64_t j = 0; j < Dim; j++) {
      D[j] = Slater_inv[cui * LDS + j];
    }

    // A^{-1} = A^{-1} - C x D / den
    for (uint64_t i = 0; i < Dim; i++) {
      IVDEP
      ALIGNED
      for (uint64_t j = 0; j < Dim; j++) {
        const double update = C[i] * D[j] * iden;
        Slater_inv[i * LDS + j] -= update;
      }
    }
    l += 1;
  }
  return QMCKL_SUCCESS;
}
#+end_src

~qmckl_exit_code qmckl_sm_naive_{Dim}~ is a C function-template that is used to genereate instances of C fucntions based on the range given above. The advantage of this method is that for each of these instances all the dimensions and loop-bounds are known at compile time, allowing the compiler to optimize more aggressively.
#+NAME:naive_template_code
#+begin_src c
  static inline qmckl_exit_code qmckl_sm_naive_{Dim}(
      const qmckl_context context,
      const uint64_t N_updates,
      const double* restrict Updates,
      const uint64_t* restrict Updates_index,
      const double breakdown,
      double* restrict Slater_inv,
      double* restrict determinant) {

    if (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
      return qmckl_failwith(context,
          QMCKL_NULL_CONTEXT,
          "qmckl_sm_naive_{Dim}",
          NULL);
    }

    #define D{Dim}_P ((1+({Dim}-1)/SIMD_LENGTH)*SIMD_LENGTH)

    double __attribute__((aligned(8))) C[{Dim}];
    double __attribute__((aligned(8))) D[D{Dim}_P];

    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;
        IVDEP
        ALIGNED
        for (uint64_t j = 0; j < D{Dim}_P; j++) {
          C[i] += Slater_inv[i * D{Dim}_P + j] * Updates[l * D{Dim}_P + j];
        }
      }

      // Denominator
      const int cui = Updates_index[l] - 1;
      double den = 1.0f + C[cui];

      if (fabs(den) < breakdown) {
        return QMCKL_FAILURE;
      }
      double iden = 1.0f / den;

      // Update det(A)
      if (determinant)
        *determinant *= den;

      // selecting column: D = v_l^T * S_inv
      IVDEP
      ALIGNED
      for (uint64_t j = 0; j < D{Dim}_P; j++) {
        D[j] = Slater_inv[cui * D{Dim}_P + j];
      }

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

      l += 1;
    }

    return QMCKL_SUCCESS;
  }
  #+end_src

This is the kernel generator written in Python. It uses the kernel generator range and templates defined above to generate the C kernel instances.
#+NAME:naive_kernel_generator
#+begin_src python :noweb yes
text="""
<<naive_template_code>>
"""
result = []
for Dim in <<kernel_generator_range>>:
    Dim=str(Dim)
    result.append(text.replace("{Dim}",Dim))

return ''.join(result)
#+end_src

Python script that generated C switch cases that call individual kernel instances.
#+NAME:naive_switch-case_generator
#+begin_src python :noweb yes
text="""
case {Dim}:  
  return qmckl_sm_naive_{Dim}(context,
    N_updates,
    Updates,
    Updates_index,
    breakdown,
    Slater_inv,
    determinant);"""
result = []
for Dim in <<kernel_generator_range>>:
    Dim=str(Dim)
    result.append(text.replace("{Dim}",Dim))

return ''.join(result)
#+end_src

#+begin_src c :tangle (eval c) :comments org :noweb yes
<<naive_kernel_generator()>>
#+end_src

~qmckl_sm_naive~ is a generic function that contains decision making logic that calls the proper kernel based on the used library configuration (~--enable-doc~ and ~--enable-hpc~) and the passed array dimensions ~LDS~ and ~Dim~.
#+begin_src c :tangle (eval c) :comments org :noweb yes
qmckl_exit_code qmckl_sm_naive(const qmckl_context context,
    const uint64_t LDS,
    const uint64_t Dim,
    const uint64_t N_updates,
    const double* Updates,
    const uint64_t* Updates_index,
    const double breakdown,
    double* Slater_inv,
    double* determinant) {

  if (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
    return qmckl_failwith(
        context,
        QMCKL_NULL_CONTEXT,
        "qmckl_sm_naive",
        NULL);
  }

  #ifdef HAVE_HPC__BROKEN_WITH_CRAY
  if (LDS == (1+(Dim-1)/SIMD_LENGTH)*SIMD_LENGTH) { // Most cases
    switch (Dim) {
      <<naive_switch-case_generator()>>
    }
  }
  else
  { // Updating smaller sub-matrix
    return qmckl_sm_naive_hpc(
        context,
        LDS,
        Dim,
        N_updates,
        Updates,
        Updates_index,
        breakdown,
        Slater_inv,
        determinant);
  }
  #else
  return qmckl_sm_naive_doc(
      context,
      LDS,
      Dim,
      N_updates,
      Updates,
      Updates_index,
      breakdown,
      Slater_inv,
      determinant);
  #endif
  
  return QMCKL_FAILURE;
}
#+end_src

*** Fortran interfaces (exposed in qmckl_f.F90)
   :PROPERTIES:
   :Name:     qmckl_sm_naive
   :CRetType: qmckl_exit_code
   :FRetType: qmckl_exit_code
   :END:

#+CALL: generate_f_interface(table=qmckl_sm_naive_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_sm_naive &
      (context, LDS, Dim, N_updates, Updates, Updates_index, breakdown, Slater_inv, determinant) &
      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 :: LDS
    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*LDS)
    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*LDS)
    real    (c_double ) , intent(inout)        :: determinant

  end function qmckl_sm_naive
end interface
#+end_src

#+CALL: generate_f_interface(table=qmckl_sm_naive_args,rettyp=get_value("FRetType"),fname="qmckl_sm_naive_hpc")

#+RESULTS:
#+begin_src f90 :tangle (eval fh_func) :comments org :exports none
interface
  integer(c_int32_t) function qmckl_sm_naive_hpc &
      (context, LDS, Dim, N_updates, Updates, Updates_index, breakdown, Slater_inv, determinant) &
      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 :: LDS
    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*LDS)
    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*LDS)
    real    (c_double ) , intent(inout)        :: determinant

  end function qmckl_sm_naive_hpc
end interface
#+end_src

#+CALL: generate_f_interface(table=qmckl_sm_naive_args,rettyp=get_value("FRetType"),fname="qmckl_sm_naive_doc")

#+RESULTS:
#+begin_src f90 :tangle (eval fh_func) :comments org :exports none
interface
  integer(c_int32_t) function qmckl_sm_naive_doc &
      (context, LDS, Dim, N_updates, Updates, Updates_index, breakdown, Slater_inv, determinant) &
      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 :: LDS
    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*LDS)
    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*LDS)
    real    (c_double ) , intent(inout)        :: determinant

  end function qmckl_sm_naive_doc
end interface
#+end_src

*** Performance
This function performs best when there is only 1 rank-1 update in the update cycle. It is 
not useful to use Sherman-Morrison with update splitting for these cycles since splitting 
can never resolve a situation where applying the update causes singular behaviour.
*** Tests
The tests for the kernels are executed on datasets that are extracted from a run of 
QMC=Chem on Benzene (21 spin-up/21 spin down electrons) using 329 unique alpha determinants. 
The tests are run such that the kernels reject the computed inverse whenever the computed 
intermediate determinants or denominators are smaller than 1e-3. This is the default value in 
QMC=Chem. The tests will return QMCKL_SUCCESS whenever all the elements of the final matrix 
$R=S.S^-1 - 1$ are smaller than the given tolerance value of 1e-3, and will return 
QMCKL_FAILURE if the values are larger than this tolerance value.

#+begin_src c :tangle (eval c_test)
const uint64_t Dim = 21;
const uint64_t LDS = (1+(Dim-1)/SIMD_LENGTH)*SIMD_LENGTH;
const double breakdown = 1e-3;
const double tolerance = 1e-3;
double res[441];

#include "sm_test.h"

assert(Updates1 != NULL);
assert(Updates_index1 != NULL);
assert(Slater_inv1 != NULL);

// original determinant of Slater1 (before applying updates)
double det = 3.407025646103221e-10;
rc = qmckl_sm_naive(context,
                            LDS,
                            Dim,
                            N_updates1,
                            Updates1,
                            Updates_index1,
                            breakdown,
                            Slater_inv1,
                            &det);

// Check that the determinant is updated properly
assert(fabs(det + 4.120398385068217e-10) < 1e-15);

for (unsigned int i = 0; i < Dim; i++) {
  for (unsigned int j = 0; j < Dim; j++) {
    res[i * Dim + j] = 0;
    for (unsigned int k = 0; k < Dim; k++) {
      res[i * Dim + j] += Slater1[i * Dim + k] * Slater_inv1[k * LDS + j];
    }
  }
}
rc = QMCKL_SUCCESS;
for (unsigned int i = 0; i < Dim; i++) {
  for (unsigned int j = 0; j < Dim; j++) {
    if (i == j && fabs(res[i * Dim + j] - 1) > tolerance) {
      rc = QMCKL_FAILURE;
    }
    if (i != j && fabs(res[i * Dim + j]) > tolerance) {
      rc = QMCKL_FAILURE;
    }
  }
}
assert(rc == QMCKL_SUCCESS);
#+end_src


* Sherman-Morrison with Slagel Splitting (core)
** ~qmckl_sm_splitting_core~
:PROPERTIES:
:Name:     qmckl_sm_splitting_core
:CRetType: qmckl_exit_code
:FRetType: qmckl_exit_code
:END:

*** Introduction
~qmckl_sm_splitting_core~ is the inner core part of 'Sherman-Morrison with update splitting' in the next section.
It is not normally used by itself but it is possible to use it nonetheless.

It has three extra parameters in its API:
- ~later_updates~ initially empty array that will contain the second halves of updates that were split during kernel execution
- ~later_index~ initially empty array that will contain the row/column numbers of the updates that were split during execution
- ~later~ initially zero integer that records the number of updates that were split during exection.

It is up to the user to decide what to do with these updates once the kernel returns. Normally ~qmckl_sm_splitting_core~ is
used as the core part of a recursive function, as is done in ~qmckl_sm_splitting~ or as part of a more complex
kernel like ~qmckl_sherman_morrison_smw32s~.  
  
If the determinant is passed it will only be partially updated if there were any update splits.

*** API
#+NAME: qmckl_sm_splitting_core_args
| Variable        | Type                    | In/Out | Description                                                   |
|-----------------+-------------------------+--------+---------------------------------------------------------------|
| ~context~       | ~qmckl_context~         | in     | Global state                                                  |
| ~LDS~           | ~uint64_t~              | in     | Leading dimension of Slater_inv                               |
| ~Dim~           | ~uint64_t~              | in     | Dimension of Slater_inv                                       |
| ~N_updates~     | ~uint64_t~              | in     | Number of rank-1 updates to be applied to Slater_inv          |
| ~Updates~       | ~double[LDS*N_updates]~ | in     | Array containing the rank-1 updates                           |
| ~Updates_index~ | ~uint64_t[N_updates]~   | in     | Array containing positions of the rank-1 updates              |
| ~breakdown~     | ~double~                | in     | Break-down parameter on which to fail or not                  |
| ~Slater_inv~    | ~double[Dim*LDS]~       | inout  | Array containing the inverse Slater-matrix                    |
| ~later_updates~ | ~double[LDS*N_updates]~ | inout  | Array containing the split updates for later                  |
| ~later_index~   | ~uint64_t[N_updates]~   | inout  | Array containing the positions of the split updates for later |
| ~later~         | ~uint64_t~              | inout  | Number of split updates for later                             |
| ~determinant~   | ~double~                | inout  | Determinant of the Slater-matrix                              |

*** Requirements
  * ~LDS >= 2~
  * ~Dim >= 2~
  * ~N_updates >= 1~
  * ~Updates~ is allocated with $N_updates \times Dim$ elements
  * ~Updates_index~ is allocated with $N_updates$ elements
  * ~breakdown~ is a small number such that $0 < breakdown << 1$
  * ~Slater_inv~ is allocated with $Dim \times Dim$ elements
  * ~later_updates~ is allocated with $later \times Dim$ elements
  * ~later_index~ is allocated with $N_updates$ elements
  * ~later >= 0~

*** Pedagogical kernel source (in Fortran)
The following source code written in Fortran is inteded to illustrate how the kernel works. Even though the kernel is
able to do numerically correct computations, it does not do it in the most efficient way possible. It should therefore
not be used in real workloads.

#+begin_src f90 :tangle (eval f)
integer function qmckl_sm_splitting_core_doc_f( &
    context, &
    lds, dim, &
    nupdates, &
    upds, &
    updates_index, &
    breakdown, &
    s_inv, &
    later_upds, &
    Later_index, &
    Later, &
    determinant) result(info)

  use qmckl
  implicit none
  integer*8 , intent(in)            :: context
  integer*8 , intent(in)            :: lds, dim
  integer*8 , intent(in)            :: nupdates
  integer*8 , intent(in)            :: updates_index(nupdates)
  real*8    , intent(in)            :: upds(lds * nupdates)
  real*8    , intent(in)            :: breakdown
  real*8    , intent(inout)         :: s_inv(dim * lds)
  real*8    , intent(inout)         :: determinant
  integer*8 , intent(inout)         :: Later
  integer*8 , intent(inout)         :: Later_index(nupdates)
  real*8    , intent(inout)         :: later_upds(lds * nupdates)

  real*8 , dimension(dim, nupdates) :: Updates
  real*8 , dimension(dim, nupdates) :: Later_updates
  real*8 , dimension(dim, dim)      :: Inverse
  real*8 , dimension(dim)           :: C
  real*8 , dimension(dim)           :: D
  real*8                            :: denominator, idenominator, update
  integer*8                         :: i, j, l, row

  info = QMCKL_FAILURE

  if (context == QMCKL_NULL_CONTEXT) then
    info = QMCKL_INVALID_CONTEXT
    return
  endif

  ! Convert 'upds' and 's_inv' into the more easily readable Fortran
  ! matrices 'Updates' and 'Inverse'.
  call convert(upds, s_inv, Updates, Inverse, nupdates, lds, dim)

  l = 1;
  ! For each update do...
  do while (l < nupdates + 1)

    ! Compute C = S^{-1}U(l)
    do i = 1, dim
      C(i) = 0
      do j = 1, dim
        C(i) = C(i) + Inverse(i, j) * Updates(j, l)
      end do
    end do

    ! Compute denominator = 1 + V(l)^TC
    row = updates_index(l)
    denominator = 1 + C(row)

    ! If denominator is too close to zero:
    ! - Split update in 2 before storing in Later_updates
    ! - Split previously computed vector C in 2
    ! - Recompute the denominator
    if (abs(denominator) < breakdown) then
      do i = 1, dim
        Later_updates(i, l) = Updates(i, l) / 2
        C(i) = C(i) / 2
      end do
      Later_index(Later + 1) = updates_index(l)
      Later = Later + 1
      denominator = 1 + C(row)
    end if

    idenominator = 1 / denominator

    ! Update det(S)
    determinant = determinant * denominator

    ! selecting column: v_l^T * S_inv
    D = Inverse(row, :)

    ! A^{-1} = A^{-1} - C x D / denominator
    do i = 1, dim
      do j = 1, dim
        update = C(i) * D(j) * idenominator
        Inverse(i, j) = Inverse(i, j) - update
      end do
    end do

    l = l + 1
  end do

  ! Copy updated inverse and later updates
  ! back to s_inv and later_upds
  call copy_back_inv(Inverse, s_inv, lds, dim)
  call copy_back_lu(Later_Updates, later_upds, lds, dim, nupdates)

  info = QMCKL_SUCCESS

end function qmckl_sm_splitting_core_doc_f
#+end_src

**** C interface to the pedagogical kernel (not directly exposed)
The function ~qmckl_sm_splitting_core_doc~ makes sure that
~qmckl_sm_splitting_core_doc_f~ can be called from C using the
~ISO_C_BINDING~. Function ~qmckl_sm_splitting_core_doc~ will be
exposed in ~qmckl.h~ and ~qmckl_f.F90~, but
~qmckl_sm_splitting_core_doc_f~ will not.

#+CALL: generate_c_interface(table=qmckl_sm_splitting_core_args,rettyp=get_value("CRetType"),fname="qmckl_sm_splitting_core_doc")

#+RESULTS:
#+begin_src f90 :tangle (eval f) :comments org :exports none
integer(c_int32_t) function qmckl_sm_splitting_core_doc &
    (context, &
     LDS, &
     Dim, &
     N_updates, &
     Updates, &
     Updates_index, &
     breakdown, &
     Slater_inv, &
     later_updates, &
     later_index, &
     later, &
     determinant) &
    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 :: LDS
  integer (c_int64_t) , intent(in)  , value :: Dim
  integer (c_int64_t) , intent(in)  , value :: N_updates
  real    (c_double ) , intent(in)          :: Updates(LDS*N_updates)
  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*LDS)
  real    (c_double ) , intent(inout)        :: later_updates(LDS*N_updates)
  integer (c_int64_t) , intent(inout)        :: later_index(N_updates)
  integer (c_int64_t) , intent(inout)        :: later
  real    (c_double ) , intent(inout)        :: determinant

  integer(c_int32_t), external :: qmckl_sm_splitting_core_doc_f
  info = qmckl_sm_splitting_core_doc_f &
         (context, &
     LDS, &
     Dim, &
     N_updates, &
     Updates, &
     Updates_index, &
     breakdown, &
     Slater_inv, &
     later_updates, &
     later_index, &
     later, &
     determinant)

end function qmckl_sm_splitting_core_doc
#+end_src

*** C headers (exposed in qmckl.h)
#+CALL: generate_c_header(table=qmckl_sm_splitting_core_args,rettyp=get_value("CRetType"),fname=get_value("Name"))

#+RESULTS:
#+begin_src c :tangle (eval h_func) :comments org
qmckl_exit_code qmckl_sm_splitting_core (
      const qmckl_context context,
      const uint64_t LDS,
      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,
      double* determinant ); 
#+end_src

#+CALL: generate_c_header(table=qmckl_sm_splitting_core_args,rettyp=get_value("CRetType"),fname="qmckl_sm_splitting_core_hpc")

#+RESULTS:
#+begin_src c :tangle (eval h_private_func) :comments org
qmckl_exit_code qmckl_sm_splitting_core_hpc (
      const qmckl_context context,
      const uint64_t LDS,
      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,
      double* determinant ); 
#+end_src

#+CALL: generate_c_header(table=qmckl_sm_splitting_core_args,rettyp=get_value("CRetType"),fname="qmckl_sm_splitting_core_doc")

#+RESULTS:
#+begin_src c :tangle (eval h_func) :no-expand comments org
qmckl_exit_code qmckl_sm_splitting_core_doc (
      const qmckl_context context,
      const uint64_t LDS,
      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,
      double* determinant ); 
#+end_src

*** C sources
#+begin_src c :tangle (eval c) :comments org
qmckl_exit_code qmckl_sm_splitting_core_hpc(
    const qmckl_context context,
    uint64_t LDS,
    uint64_t Dim,
    uint64_t N_updates,
    const double* restrict Updates,
    const uint64_t* restrict Updates_index,
    const double breakdown,
    double* restrict Slater_inv,
    double* restrict later_updates,
    uint64_t* restrict later_index,
    uint64_t* restrict later,
    double* restrict determinant) {

  if (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
    return qmckl_failwith(
        context,
        QMCKL_NULL_CONTEXT,
        "qmckl_sm_splitting_core_hpc",
        NULL);
  }

  double __attribute__((aligned(8))) C[LDS];
  double __attribute__((aligned(8))) D[LDS];

  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] = 0.0f;
      IVDEP
      ALIGNED
      for (uint64_t j = 0; j < LDS; j++) {
        C[i] += Slater_inv[i * LDS + j] * Updates[l * LDS + j];
      }
    }

    // Denominator
    const int cui = Updates_index[l] - 1;
    double den = 1.0f + C[cui];
    if (fabs(den) < breakdown) {
      // U_l = U_l / 2: split the update in 2 equal halves and save the
      // second halve in later_updates
      IVDEP
      ALIGNED
      for (uint64_t i = 0; i < LDS; i++) {
        later_updates[*later * LDS + i] = Updates[l * LDS + i] * 0.5f;
        C[i] *= 0.5f;
      }
      later_index[*later] = Updates_index[l];
      (*later)++;

      den = 1.0f + C[cui];
    } // From here onwards we continue with applying the first halve of the
      // update to Slater_inv
    double iden = 1.0f / den;

    if (determinant)
      *determinant *= den;

    // D = v^T x S^{-1} : 1 x LDS
    IVDEP
    ALIGNED
    for (uint64_t j = 0; j < LDS; j++) {
      D[j] = Slater_inv[cui * LDS + j];
    }

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

  return QMCKL_SUCCESS;
}
    #+end_src

#+NAME:slagel_splitting_template_code
#+begin_src c
static inline qmckl_exit_code qmckl_sm_splitting_core_{Dim}(
    const qmckl_context context,
    uint64_t N_updates,
    const double* restrict Updates,
    const uint64_t* restrict Updates_index,
    const double breakdown,
    double* restrict Slater_inv,
    double* restrict later_updates,
    uint64_t* restrict later_index,
    uint64_t* restrict later,
    double* restrict determinant) {

  if (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
    return qmckl_failwith(
        context,
        QMCKL_NULL_CONTEXT,
        "qmckl_sm_splitting_core_{Dim}",
        NULL);
  }

  double __attribute__((aligned(8))) C[D{Dim}_P];
  double __attribute__((aligned(8))) D[D{Dim}_P];

  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] = 0.0f;
      IVDEP
      ALIGNED
      for (uint64_t j = 0; j < D{Dim}_P; j++) {
        C[i] += Slater_inv[i * D{Dim}_P + j] * Updates[l * D{Dim}_P + j];
      }
    }

    // Denominator
    const int cui = Updates_index[l] - 1;
    double den = 1.0f + C[cui];
    if (fabs(den) < breakdown) {
      // U_l = U_l / 2: split the update in 2 equal halves and save the
      // second halve in later_updates
      IVDEP
      ALIGNED
      for (uint64_t i = 0; i < D{Dim}_P; i++) {
        later_updates[*later * D{Dim}_P + i] = Updates[l * D{Dim}_P + i] * 0.5f;
        C[i] *= 0.5f;
      }
      later_index[*later] = Updates_index[l];
      (*later)++;

      den = 1.0f + C[cui];
    } // From here onwards we continue with applying the first halve of the
      // update to Slater_inv
    double iden = 1.0f / den;

    if (determinant)
      *determinant *= den;

    // D = v^T x S^{-1} : 1 x D{Dim}_P
    IVDEP
    ALIGNED
    for (uint64_t j = 0; j < D{Dim}_P; j++) {
      D[j] = Slater_inv[cui * D{Dim}_P + j];
    }

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

  return QMCKL_SUCCESS;
}
#+end_src

#+NAME:slagel_splitting_kernel_generator
#+begin_src python :noweb yes
text="""
<<slagel_splitting_template_code>>
"""
result = []
for Dim in <<kernel_generator_range>>:
  Dim=str(Dim)
  result.append(text.replace("{Dim}",Dim) )

return ''.join(result)
#+end_src

#+NAME:slagel_splitting_switch-case_generator
#+begin_src python :noweb yes
text="""
case {Dim}: {
  return qmckl_sm_splitting_core_{Dim}(
      context,
      N_updates,
      Updates,
      Updates_index,
      breakdown,
      Slater_inv,
      later_updates,
      later_index,
      later,
      determinant);
}"""
result = []
for Dim in <<kernel_generator_range>>:
  Dim=str(Dim)
  result.append(text.replace("{Dim}",Dim) )

return ''.join(result)
#+end_src

#+begin_src c :tangle (eval c) :comments org :noweb yes
<<slagel_splitting_kernel_generator()>>
#+end_src

#+begin_src c :tangle (eval c) :comments org :noweb yes
qmckl_exit_code qmckl_sm_splitting_core(
    const qmckl_context context,
    const uint64_t LDS,
    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,
    double* determinant) {

  #ifdef HAVE_HPC__BROKEN_WITH_CRAY
  if (LDS == (1+(Dim-1)/SIMD_LENGTH)*SIMD_LENGTH) { // Most cases
    switch (Dim) {
      <<slagel_splitting_switch-case_generator()>>
      default: {
        assert(0 == 1 && "TEMPLATE NOT IMPLEMENTED!");
        break;
      }
    }
  }
  else { // Updating smaller sub-matrix
    return qmckl_sm_splitting_core_hpc(
        context,
        LDS,
        Dim,
        N_updates,
        Updates,
        Updates_index,
        breakdown,
        Slater_inv,
        later_updates,
        later_index,
        later,
        determinant);
  }
  #else
  return qmckl_sm_splitting_core_doc(
      context,
      LDS,
      Dim,
      N_updates,
      Updates,
      Updates_index,
      breakdown,
      Slater_inv,
      later_updates,
      later_index,
      later,
      determinant);
  #endif

  return QMCKL_FAILURE;
}
#+end_src

*** Fortran interfaces (exposed in qmckl_f.F90)
:PROPERTIES:
:Name:     qmckl_sm_splitting_core
:CRetType: qmckl_exit_code
:FRetType: qmckl_exit_code
:END:

#+CALL: generate_f_interface(table=qmckl_sm_splitting_core_args,rettyp=get_value("FRetType"),fname="qmckl_sm_splitting_core_hpc")

#+RESULTS:
#+begin_src f90 :tangle (eval fh_func) :comments org :exports none
interface
  integer(c_int32_t) function qmckl_sm_splitting_core_hpc &
      (context, LDS, Dim, N_updates, Updates, Updates_index, breakdown, &
      Slater_inv, later_updates, later_index, later, determinant) &
      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 :: LDS
    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*LDS)
    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*LDS)
    real    (c_double ) , intent(inout)        :: later_updates(N_updates*LDS)
    integer (c_int64_t) , intent(inout)        :: later_index(N_updates)
    integer (c_int64_t) , intent(inout)        :: later
    real    (c_double ) , intent(inout)        :: determinant

  end function qmckl_sm_splitting_core_hpc
end interface
#+end_src

#+CALL: generate_f_interface(table=qmckl_sm_splitting_core_args,rettyp=get_value("FRetType"),fname="qmckl_sm_splitting_core_doc")

#+RESULTS:
#+begin_src f90 :tangle (eval fh_func) :comments org :exports none
interface
  integer(c_int32_t) function qmckl_sm_splitting_core_doc &
      (context, LDS, Dim, N_updates, Updates, Updates_index, breakdown, &
      Slater_inv, later_updates, later_index, later, determinant) &
      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 :: LDS
    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*LDS)
    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*LDS)
    real    (c_double ) , intent(inout)        :: later_updates(N_updates*LDS)
    integer (c_int64_t) , intent(inout)        :: later_index(N_updates)
    integer (c_int64_t) , intent(inout)        :: later
    real    (c_double ) , intent(inout)        :: determinant

  end function qmckl_sm_splitting_core_doc
end interface
#+end_src

#+CALL: generate_f_interface(table=qmckl_sm_splitting_core_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_sm_splitting_core &
      (context, LDS, Dim, N_updates, Updates, Updates_index, breakdown, &
      Slater_inv, later_updates, later_index, later, determinant) &
      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 :: LDS
    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*LDS)
    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*LDS)
    real    (c_double ) , intent(inout)        :: later_updates(N_updates*LDS)
    integer (c_int64_t) , intent(inout)        :: later_index(N_updates)
    integer (c_int64_t) , intent(inout)        :: later
    real    (c_double ) , intent(inout)        :: determinant

  end function qmckl_sm_splitting_core
end interface
#+end_src

*** Performance
This function cannot be used by itself and is used in Sherman-Morrison with update splitting and Woodbury 3x3 and 2x2
with Sherman-Morrison and update splitting. Please look at the performance reccomendations for those two kernels.


* Woodbury 2x2
** ~qmckl_woodbury_2x2~
:PROPERTIES:
:Name:     qmckl_woodbury_2x2
:CRetType: qmckl_exit_code
:FRetType: qmckl_exit_code
:END:

*** Introduction
The Woodbury 2x2 kernel. It is used to apply two rank-1 updates at once. The formula used in
this algorithm is called the Woodbury Matrix Id
\[
(S + U V)^{-1}  =  S^{-1} - C B^{-1} D
\]
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

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.

*** API
#+NAME: qmckl_woodbury_2x2_args
| Variable        | Type                | In/Out | Description                                  |
|-----------------+-------------------+-------+-------------------------------------------------|
| ~context~       | ~qmckl_context~   | in    | Global state                                    |
| ~LDS~           | ~uint64_t~        | in    | Leading dimension of Slater_inv                 |
| ~Dim~           | ~uint64_t~        | in    | Dimension of Slater_inv                         |
| ~Updates~       | ~double[2*Dim]~   | in    | Array containing the updates                    |
| ~Updates_index~ | ~uint64_t[2]~     | in    | Array containing the rank-1 updates             |
| ~breakdown~     | ~double~          | in    | Break-down parameter on which to fail or not    |
| ~Slater_inv~    | ~double[LDS*Dim]~ | inout | Array containing the inverse of a Slater-matrix |
| ~determinant~   | ~double~          | inout | Determinant of Slater-matrix                    |

*** Requirements
- ~context~ is not ~qmckl_null_context~
- ~LDS >= 2~
- ~Dim >= 2~
- ~Updates~ is allocated with $2 \times Dim$ elements
- ~Updates_index~ is allocated with $2$ elements
- ~breakdown~ is a small number such that $0 < breakdown << 1$
- ~Slater_inv~ is allocated with $Dim \times Dim$ elements

*** Pedagogical kernel source (in Fortran)
The following source code written in Fortran is inteded to illustrate how the kernel works. Even though the kernel is
able to do numerically correct computations, it does not do it in the most efficient way possible. It should therefore
not be used in real workloads.

#+begin_src f90 :tangle (eval f)
integer function qmckl_woodbury_2x2_doc_f(&
    context, &
    lds, dim, &
    upds, &
    updates_index, &
    breakdown, &
    s_inv, &
    determinant) result(info)

  use qmckl
  implicit none
  integer*8 , intent(in)            :: context
  integer*8 , intent(in)            :: lds, dim
  integer*8 , intent(in)            :: updates_index(2)
  real*8    , intent(in)            :: upds(2 * lds)
  real*8    , intent(in)            :: breakdown
  real*8    , intent(inout)         :: s_inv(dim * lds)
  real*8    , intent(inout)         :: determinant

  integer*8 , dimension(2, dim)     :: V
  integer*8 , dimension(2, 2)       :: Id
  real*8    , dimension(dim, dim)   :: Inverse
  real*8    , dimension(dim, 2)     :: Updates, C
  real*8    , dimension(2, 2)       :: D, invD
  real*8    , dimension(2, dim)     :: E, F
  
  real*8                            :: detD, idenominator, update
  integer*8                         :: i, j, k, l

  info = QMCKL_FAILURE

  if (context == QMCKL_NULL_CONTEXT) then
    info = QMCKL_INVALID_CONTEXT
    return
  endif

  ! Construct V(2, dim) matrix
  V = 0
  V(1, updates_index(1)) = 1
  V(2, updates_index(2)) = 1

  ! Construct Id(2, 2) matrix
  Id = 0
  Id(1, 1) = 1
  Id(2, 2) = 1

  ! Convert 'upds' and 's_inv' into the more easily readable Fortran
  ! matrices 'Updates' and 'Inverse'.
  call convert(upds, s_inv, Updates, Inverse, int(2,8), lds, dim)

  ! Compute C(dim, 2) = Inverse(dim, dim) x Updates(dim, 2)
  C = 0
  do i = 1, dim
    do j = 1, 2
      do k = 1, dim
        C(i, j) = C(i, j) + Inverse(i, k) * Updates(k, j)
      end do
    end do
  end do

  ! Construct matrix D(2, 2) := I(2, 2) + V(2, dim) x C(dim, 2)
  D = 0
  do i = 1, 2 
     do j = 1, 2
        do k = 2, dim
           D(i, j) = D(i, j) + V(i, k) * C(k, j)
        end do
      end do
  end do
  D = Id + D
  
  ! Compute determinant := det(D) explicitly
  detD = D(1,1) * D(2,2) - D(1,2) * D(2,1)

    ! Return early if det(D) is too small
  if (abs(detD) < breakdown) return

  ! Update det(S)
  determinant = determinant * detD

  ! Compute inv(D) explicitly
  invD(1,1) =   D(2,2)
  invD(1,2) = - D(1,2)
  invD(2,1) = - D(2,1)
  invD(2,2) =   D(1,1)
  invD = invD / detD

  ! Compute E(2, dim) := V(2, dim) x Inverse(dim, dim)
  E = 0
  do i = 1, 2 
    do j = 1, dim
      do k = 1, dim
        E(i, j) = E(i, j) + V(i, k) * Inverse(k, j)
      end do
    end do
  end do

  ! Compute F(2, dim) := invD(2, 2) x E(2, dim)
  F = 0
  do i = 1, 2
    do j = 1, dim
      do k = 1, 2
        F(i, j) = F(i, j) + invD(i, k) * E(k, j)
      end do
    end do
  end do

  ! Compute Inverse(dim, dim) := Inverse(dim, dim) - C(dim, 2) x F(2, dim)
  do i = 1, dim
    do j = 1, dim
      do k = 1, 2
        Inverse(i, j) = Inverse(i, j) - C(i, k) * F(k, j)
      end do
    end do
  end do

  ! Copy updated inverse and later updates
  ! back to s_inv and later_upds
  call copy_back_inv(Inverse, s_inv, lds, dim)

  info = QMCKL_SUCCESS

end function qmckl_woodbury_2x2_doc_f
#+end_src

**** C interface (not directly exposed)
The function ~qmckl_sm_splitting_core_doc~ makes sure that
~qmckl_sm_splitting_core_doc_f~ can be called from C using the
~ISO_C_BINDING~. Function ~qmckl_sm_splitting_core_doc~ will be
exposed in ~qmckl.h~ and ~qmckl_f.F90~, but
~qmckl_sm_splitting_core_doc_f~ will not.

#+CALL: generate_c_interface(table=qmckl_woodbury_2x2_args,rettyp=get_value("CRetType"),fname="qmckl_woodbury_2x2_doc")

#+RESULTS:
#+begin_src f90 :tangle (eval f) :comments org :exports none
integer(c_int32_t) function qmckl_woodbury_2x2_doc &
    (context, LDS, Dim, Updates, Updates_index, breakdown, Slater_inv, determinant) &
    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 :: LDS
  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)  , value :: breakdown
  real    (c_double ) , intent(inout)        :: Slater_inv(LDS*Dim)
  real    (c_double ) , intent(inout)        :: determinant

  integer(c_int32_t), external :: qmckl_woodbury_2x2_doc_f
  info = qmckl_woodbury_2x2_doc_f &
         (context, LDS, Dim, Updates, Updates_index, breakdown, Slater_inv, determinant)

end function qmckl_woodbury_2x2_doc
#+end_src

*** C headers (exposed in qmckl.h)
#+CALL: generate_c_header(table=qmckl_woodbury_2x2_args,rettyp=get_value("CRetType"),fname=get_value("Name"))

#+RESULTS:
#+begin_src c :tangle (eval h_func) :comments org
qmckl_exit_code qmckl_woodbury_2x2 (
      const qmckl_context context,
      const uint64_t LDS,
      const uint64_t Dim,
      const double* Updates,
      const uint64_t* Updates_index,
      const double breakdown,
      double* Slater_inv,
      double* determinant ); 
#+end_src

#+CALL: generate_c_header(table=qmckl_woodbury_2x2_args,rettyp=get_value("CRetType"),fname="qmckl_woodbury_2x2_hpc")

#+RESULTS:
#+begin_src c :tangle (eval h_func) :comments org
qmckl_exit_code qmckl_woodbury_2x2_hpc (
      const qmckl_context context,
      const uint64_t LDS,
      const uint64_t Dim,
      const double* Updates,
      const uint64_t* Updates_index,
      const double breakdown,
      double* Slater_inv,
      double* determinant ); 
#+end_src

#+CALL: generate_c_header(table=qmckl_woodbury_2x2_args,rettyp=get_value("CRetType"),fname="qmckl_woodbury_2x2_doc")

#+RESULTS:
#+begin_src c :tangle (eval h_func) :comments org
qmckl_exit_code qmckl_woodbury_2x2_doc (
      const qmckl_context context,
      const uint64_t LDS,
      const uint64_t Dim,
      const double* Updates,
      const uint64_t* Updates_index,
      const double breakdown,
      double* Slater_inv,
      double* determinant ); 
#+end_src

*** C sources
#+begin_src c :tangle (eval c) :comments org
qmckl_exit_code qmckl_woodbury_2x2_hpc(const qmckl_context context,
    const uint64_t LDS,
    const uint64_t Dim,
    const double* restrict Updates,
    const uint64_t* restrict Updates_index,
    const double breakdown,
    double* restrict Slater_inv,
    double* restrict determinant) {
/*
    C := S^{-1} * U,    dim x 2
    B := 1 + V * C,     2 x 2
    D := V * S^{-1},    2 x dim
*/

  if (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
    return qmckl_failwith(context,
                          QMCKL_NULL_CONTEXT,
                          "qmckl_woodbury_2x2_hpc",
                          NULL);
  }

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

  // Compute C = (S^T)^{-1}U : Dim x 2
  double __attribute__((aligned(8))) C[2 * Dim];
  for (uint64_t i = 0; i < Dim; i++) {
      C[i * 2] = 0;
      C[i * 2 + 1] = 0;
      for (uint64_t k = 0; k < LDS; k++) {
          C[i * 2] += Slater_inv[i * LDS + k] * Updates[k];
          C[i * 2 + 1] += Slater_inv[i * LDS + k] * Updates[LDS + k];
      }
  }

  // Compute B = 1 + VC : 2 x 2
  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;
  }

  // Update det(S) when passed
  if (determinant)
      *determinant *= det;

  // Compute B^{-1} with explicit formula for 2 x 2 inversion
  double __attribute__((aligned(8))) 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;

  // tmp = B^{-1}D : 2 x LDS
  double __attribute__((aligned(8))) tmp[2 * LDS];
  double* r1dim = &(Slater_inv[row1 * LDS]);
  double* r2dim = &(Slater_inv[row2 * LDS]);
  for (uint64_t j = 0; j < LDS; j++) {
      tmp[j] = Binv[0] * r1dim[j] + Binv[1] * r2dim[j];
      tmp[LDS + j] = Binv[2] * r1dim[j] + Binv[3] * r2dim[j];
  }

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

  return QMCKL_SUCCESS;
}
#+end_src

#+NAME:woodbury_2x2_kernel_template
#+begin_src c
static inline qmckl_exit_code qmckl_woodbury_2x2_{Dim}(
    const qmckl_context context,
    const double* restrict Updates,
    const uint64_t* restrict Updates_index,
    const double breakdown,
    double* restrict Slater_inv,
    double* restrict determinant) {
/*
    C := S^{-1} * U,    dim x 2
    B := 1 + V * C,     2 x 2
    D := V * S^{-1},    2 x dim
*/

  if (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
    return qmckl_failwith(context,
                          QMCKL_NULL_CONTEXT,
                          "qmckl_woodbury_2x2_{Dim}",
                          NULL);
  }

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

  // Compute C = (S^T)^{-1}U : {Dim} x 2
  double __attribute__((aligned(8))) C[2 * {Dim}];
  for (uint64_t i = 0; i < {Dim}; i++) {
      C[i * 2] = 0;
      C[i * 2 + 1] = 0;
      IVDEP
      ALIGNED
      for (uint64_t k = 0; k < D{Dim}_P; k++) {
          C[i * 2] += Slater_inv[i * D{Dim}_P + k] * Updates[k];
          C[i * 2 + 1] += Slater_inv[i * D{Dim}_P + k] * Updates[D{Dim}_P + k];
      }
  }

  // Compute B = 1 + VC : 2 x 2
  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;
  }

  // Update det(S) when passed
  if (determinant)
      *determinant *= det;

  // Compute B^{-1} with explicit formula for 2 x 2 inversion
  double __attribute__((aligned(8))) 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;

  // tmp = B^{-1}D : 2 x D{Dim}_P
  double __attribute__((aligned(8))) tmp[2 * D{Dim}_P];
  double* r1dim = &(Slater_inv[row1 * D{Dim}_P]);
  double* r2dim = &(Slater_inv[row2 * D{Dim}_P]);
  IVDEP
  ALIGNED
  for (uint64_t j = 0; j < D{Dim}_P; j++) {
      tmp[j] = Binv[0] * r1dim[j] + Binv[1] * r2dim[j];
      tmp[D{Dim}_P + j] = Binv[2] * r1dim[j] + Binv[3] * r2dim[j];
  }

  // Compute (S^T)^{-1} - C * tmp : {Dim} x D{Dim}_P
  for (uint64_t i = 0; i < {Dim}; i++) {
      IVDEP
      ALIGNED
      for (uint64_t j = 0; j < D{Dim}_P; j++) {
          Slater_inv[i * D{Dim}_P + j] -= C[i * 2] * tmp[j];
          Slater_inv[i * D{Dim}_P + j] -= C[i * 2 + 1] * tmp[D{Dim}_P + j];
      }
  }

  return QMCKL_SUCCESS;
}
#+end_src

#+NAME:woodbury_2x2_kernel_generator
#+begin_src python :noweb yes :exports none
text="""
<<woodbury_2x2_kernel_template>>
"""
result = []
for Dim in <<kernel_generator_range>>:
  Dim=str(Dim)
  result.append(text.replace("{Dim}",Dim) )

return ''.join(result)
#+end_src

#+NAME:woodbury_2x2_switch-case_generator
#+begin_src python :noweb yes :exports none
text="""
case {Dim}:  
  return qmckl_woodbury_2x2_{Dim}(context,
    Updates,
    Updates_index,
    breakdown,
    Slater_inv,
    determinant);"""
result = []
for Dim in <<kernel_generator_range>>:
  Dim=str(Dim)
  result.append(text.replace("{Dim}",Dim) )

return ''.join(result)
#+end_src

#+begin_src c :tangle (eval c) :comments org :noweb yes
<<woodbury_2x2_kernel_generator()>>
#+end_src

#+begin_src c :tangle (eval c) :comments org :noweb yes
qmckl_exit_code qmckl_woodbury_2x2(const qmckl_context context,
    const uint64_t LDS,
    const uint64_t Dim,
    const double* Updates,
    const uint64_t* Updates_index,
    const double breakdown,
    double* Slater_inv,
    double* determinant) {

  if (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
    return qmckl_failwith(
        context,
        QMCKL_NULL_CONTEXT,
        "qmckl_woodbury_2x2",
        NULL);
  }

  #ifdef HAVE_HPC__BROKEN_WITH_CRAY
  if (LDS == (1+(Dim-1)/SIMD_LENGTH)*SIMD_LENGTH) { // Most cases
    switch (Dim) {
      <<woodbury_2x2_switch-case_generator()>>
    }
  }
  else { // Updating smaller sub-matrix
    return qmckl_woodbury_2x2_hpc(
        context,
        LDS,
        Dim,
        Updates,
        Updates_index,
        breakdown,
        Slater_inv,
        determinant);
  }
  #else
  return qmckl_woodbury_2x2_doc(
      context,
      LDS,
      Dim,
      Updates,
      Updates_index,
      breakdown,
      Slater_inv,
      determinant);
  // return qmckl_woodbury_2x2_hpc(
  //     context,
  //     LDS,
  //     Dim,
  //     Updates,
  //     Updates_index,
  //     breakdown,
  //     Slater_inv,
  //     determinant);
  #endif

  return QMCKL_FAILURE;
}
#+end_src

*** Fortran interfaces (exposed in qmckl_f.F90)
:PROPERTIES:
:Name:     qmckl_woodbury_2x2
:CRetType: qmckl_exit_code
:FRetType: qmckl_exit_code
:END:

#+CALL: generate_f_interface(table=qmckl_woodbury_2x2_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_2x2 &
      (context, LDS, Dim, Updates, Updates_index, breakdown, Slater_inv, determinant) &
      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 :: LDS
    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)  , value :: breakdown
    real    (c_double ) , intent(inout)        :: Slater_inv(LDS*Dim)
    real    (c_double ) , intent(inout)        :: determinant

  end function qmckl_woodbury_2x2
end interface
#+end_src

#+CALL: generate_f_interface(table=qmckl_woodbury_2x2_args,rettyp=get_value("FRetType"),fname="qmckl_woodbury_2x2_hpc")

#+RESULTS:
#+begin_src f90 :tangle (eval fh_func) :comments org :exports none
interface
  integer(c_int32_t) function qmckl_woodbury_2x2_hpc &
      (context, LDS, Dim, Updates, Updates_index, breakdown, Slater_inv, determinant) &
      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 :: LDS
    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)  , value :: breakdown
    real    (c_double ) , intent(inout)        :: Slater_inv(LDS*Dim)
    real    (c_double ) , intent(inout)        :: determinant

  end function qmckl_woodbury_2x2_hpc
end interface
#+end_src

#+CALL: generate_f_interface(table=qmckl_woodbury_2x2_args,rettyp=get_value("FRetType"),fname="qmckl_woodbury_2x2_doc")

#+RESULTS:
#+begin_src f90 :tangle (eval fh_func) :comments org :exports none
interface
  integer(c_int32_t) function qmckl_woodbury_2x2_doc &
      (context, LDS, Dim, Updates, Updates_index, breakdown, Slater_inv, determinant) &
      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 :: LDS
    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)  , value :: breakdown
    real    (c_double ) , intent(inout)        :: Slater_inv(LDS*Dim)
    real    (c_double ) , intent(inout)        :: determinant

  end function qmckl_woodbury_2x2_doc
end interface
#+end_src

*** Performance
This function is most efficient when used in cases where there are only 2 rank-1 updates and
it is sure they will not result in a singular matrix.

*** Tests
#+begin_src c :tangle (eval c_test)
assert(Updates2 != NULL);
assert(Updates_index2 != NULL);
assert(Slater_inv2 != NULL);
det = -1.4432116661319376e-11;
rc = qmckl_woodbury_2x2(context, LDS, Dim, Updates2, Updates_index2, breakdown, Slater_inv2, &det);
assert(fabs(det-2.367058141251457e-10) < 1e-15);
for (unsigned int i = 0; i < Dim; i++) {
  for (unsigned int j = 0; j < Dim; j++) {
    res[i * Dim + j] = 0;
    for (unsigned int k = 0; k < Dim; k++) {
      res[i * Dim + j] += Slater2[i * Dim + k] * Slater_inv2[k * LDS + j];
    }
  }
}
rc = QMCKL_SUCCESS;
for (unsigned int i = 0; i < Dim; i++) {
  for (unsigned int j = 0; j < Dim; j++) {
    if (i == j && fabs(res[i * Dim + j] - 1) > tolerance) {
      rc = QMCKL_FAILURE;
    }
    if (i != j && fabs(res[i * Dim + j]) > tolerance) {
      rc = QMCKL_FAILURE;
    }
  }
}
assert(rc == QMCKL_SUCCESS);
#+end_src


* Woodbury 3x3
** ~qmckl_woodbury_3x3~
:PROPERTIES:
:Name:     qmckl_woodbury_3x3
:CRetType: qmckl_exit_code
:FRetType: qmckl_exit_code
:END:

*** Introduction
The Woodbury 3x3 kernel. It is used to apply two rank-1 updates at once. The formula used in
this algorithm is called the Woodbury Matrix Id
\[
(S + U V)^{-1}  =  S^{-1} - C B^{-1} D
\]
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 $\times 3$ matrix
$B := 1 + VC$, the $3 \times 3$ matrix that is going to be inverted
$D := VS^{-1}$, a $3 \times Dim$ matrix

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.

*** API
#+NAME: qmckl_woodbury_3x3_args
| Variable        | Type                | In/Out | Description                                  |
|-----------------+-------------------+-------+-------------------------------------------------|
| ~context~       | ~qmckl_context~   | in    | Global state                                    |
| ~LDS~           | ~uint64_t~        | in    | Leading dimension of Slater_inv                 |
| ~Dim~           | ~uint64_t~        | in    | Dimension of Slater_inv                         |
| ~Updates~       | ~double[3*Dim]~   | in    | Array containing the updates                    |
| ~Updates_index~ | ~uint64_t[3]~     | in    | Array containing the rank-1 updates             |
| ~breakdown~     | ~double~          | in    | Break-down parameter on which to fail or not    |
| ~Slater_inv~    | ~double[LDS*Dim]~ | inout | Array containing the inverse of a Slater-matrix |
| ~determinant~   | ~double~          | inout | Determinant of Slater-matrix                    |

*** Requirements
- ~context~ is not ~qmckl_null_context~
- ~LDS >= 3~
- ~Dim >= 3~
- ~Updates~ is allocated with $3 \times Dim$ elements
- ~Updates_index~ is allocated with $3$ elements
- ~breakdown~ is a small number such that $0 < breakdown << 1$
- ~Slater_inv~ is allocated with $Dim \times Dim$ elements

*** Pedagogical kernel source (in Fortran)
The following source code written in Fortran is inteded to illustrate how the kernel works. Even though the kernel is
able to do numerically correct computations, it does not do it in the most efficient way possible. It should therefore
not be used in real workloads.

#+begin_src f90 :tangle (eval f)
integer function qmckl_woodbury_3x3_doc_f(&
    context, &
    lds, dim, &
    upds, &
    updates_index, &
    breakdown, &
    s_inv, &
    determinant) result(info)

  use qmckl
  implicit none
  integer*8 , intent(in)            :: context
  integer*8 , intent(in)            :: lds, dim
  integer*8 , intent(in)            :: updates_index(3)
  real*8    , intent(in)            :: upds(3 * lds)
  real*8    , intent(in)            :: breakdown
  real*8    , intent(inout)         :: s_inv(dim * lds)
  real*8    , intent(inout)         :: determinant

  integer*8 , dimension(3, dim)     :: V
  integer*8 , dimension(3, 3)       :: Id
  real*8    , dimension(dim, dim)   :: Inverse
  real*8    , dimension(dim, 3)     :: Updates, C
  real*8    , dimension(3, 3)       :: D, invD
  real*8    , dimension(3, dim)     :: E, F
  
  real*8                            :: detD, idetD, idenominator, update
  integer*8                         :: i, j, k, l

  info = QMCKL_FAILURE

  if (context == QMCKL_NULL_CONTEXT) then
    info = QMCKL_INVALID_CONTEXT
    return
  endif

  ! Construct V(3, dim) matrix
  V = 0
  V(1, updates_index(1)) = 1
  V(2, updates_index(2)) = 1
  V(3, updates_index(3)) = 1

  ! Construct Id(3, 3) matrix
  Id = 0
  Id(1, 1) = 1
  Id(2, 2) = 1
  Id(3, 3) = 1

  ! Convert 'upds' and 's_inv' into the more easily readable Fortran
  ! matrices 'Updates' and 'Inverse'.
  call convert(upds, s_inv, Updates, Inverse, int(3,8), lds, dim)

  ! Compute C(dim, 3) = Inverse(dim, dim) x Updates(dim, 3)
  C = 0
  do i = 1, dim
    do j = 1, 3
      do k = 1, dim
        C(i, j) = C(i, j) + Inverse(i, k) * Updates(k, j)
      end do
    end do
  end do

  ! Construct matrix D(3, 3) := I(3, 3) + V(3, dim) x C(dim, 3)
  D = 0
  do i = 1, 3 
     do j = 1, 3
        do k = 3, dim
           D(i, j) = D(i, j) + V(i, k) * C(k, j)
        end do
      end do
  end do
  D = Id + D
  
  ! Compute determinant := det(D) explicitly
  detD = D(1,1) * (D(2,2) * D(3,3) - D(2,3) * D(3,2)) - &
         D(1,2) * (D(2,1) * D(3,3) - D(2,3) * D(3,1)) + &
         D(1,3) * (D(2,1) * D(3,2) - D(2,2) * D(3,1))
         
  ! Return early if det(D) is too small
  if (abs(detD) < breakdown) return

  ! Update det(S)
  determinant = determinant * detD

  idetD = 1.0d0 / detD
  ! Compute inv(D) explicitly
  invD(1,1) =  (D(2,2) * D(3,3) - D(3,2) * D(2,3)) * idetD
  invD(1,2) = -(D(1,2) * D(3,3) - D(3,2) * D(1,3)) * idetD
  invD(1,3) =  (D(1,2) * D(2,3) - D(2,2) * D(1,3)) * idetD
  invD(2,1) = -(D(2,1) * D(3,3) - D(3,1) * D(2,3)) * idetD
  invD(2,2) =  (D(1,1) * D(3,3) - D(3,1) * D(1,3)) * idetD
  invD(2,3) = -(D(1,1) * D(2,3) - D(2,1) * D(1,3)) * idetD
  invD(3,1) =  (D(2,1) * D(3,2) - D(3,1) * D(2,2)) * idetD
  invD(3,2) = -(D(1,1) * D(3,2) - D(3,1) * D(1,2)) * idetD
  invD(3,3) =  (D(1,1) * D(2,2) - D(2,1) * D(1,2)) * idetD
  
  ! Compute E(3, dim) := V(3, dim) x Inverse(dim, dim)
  E = 0
  do i = 1, 3 
    do j = 1, dim
      do k = 1, dim
        E(i, j) = E(i, j) + V(i, k) * Inverse(k, j)
      end do
    end do
  end do

  ! Compute F(3, dim) := invD(3, 3) x E(3, dim)
  F = 0
  do i = 1, 3
    do j = 1, dim
      do k = 1, 3
        F(i, j) = F(i, j) + invD(i, k) * E(k, j)
      end do
    end do
  end do

  ! Compute Inverse(dim, dim) := Inverse(dim, dim) - C(dim, 3) x F(3, dim)
  do i = 1, dim
    do j = 1, dim
      do k = 1, 3
        Inverse(i, j) = Inverse(i, j) - C(i, k) * F(k, j)
      end do
    end do
  end do

  ! Copy updated inverse and later updates
  ! back to s_inv and later_upds
  call copy_back_inv(Inverse, s_inv, lds, dim)

  info = QMCKL_SUCCESS

end function qmckl_woodbury_3x3_doc_f
#+end_src

**** C interface (not directly exposed)
The function ~qmckl_sm_splitting_core_doc~ makes sure that
~qmckl_sm_splitting_core_doc_f~ can be called from C using the
~ISO_C_BINDING~. Function ~qmckl_sm_splitting_core_doc~ will be
exposed in ~qmckl.h~ and ~qmckl_f.F90~, but
~qmckl_sm_splitting_core_doc_f~ will not.

#+CALL: generate_c_interface(table=qmckl_woodbury_3x3_args,rettyp=get_value("CRetType"),fname="qmckl_woodbury_3x3_doc")

#+RESULTS:
#+begin_src f90 :tangle (eval f) :comments org :exports none
integer(c_int32_t) function qmckl_woodbury_3x3_doc &
    (context, LDS, Dim, Updates, Updates_index, breakdown, Slater_inv, determinant) &
    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 :: LDS
  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(LDS*Dim)
  real    (c_double ) , intent(inout)        :: determinant

  integer(c_int32_t), external :: qmckl_woodbury_3x3_doc_f
  info = qmckl_woodbury_3x3_doc_f &
         (context, LDS, Dim, Updates, Updates_index, breakdown, Slater_inv, determinant)

end function qmckl_woodbury_3x3_doc
#+end_src

*** C headers (exposed in qmckl.h)
#+CALL: generate_c_header(table=qmckl_woodbury_3x3_args,rettyp=get_value("CRetType"),fname=get_value("Name"))

#+RESULTS:
#+begin_src c :tangle (eval h_func) :comments org
qmckl_exit_code qmckl_woodbury_3x3 (
      const qmckl_context context,
      const uint64_t LDS,
      const uint64_t Dim,
      const double* Updates,
      const uint64_t* Updates_index,
      const double breakdown,
      double* Slater_inv,
      double* determinant ); 
#+end_src

#+CALL: generate_c_header(table=qmckl_woodbury_3x3_args,rettyp=get_value("CRetType"),fname="qmckl_woodbury_3x3_hpc")

#+RESULTS:
#+begin_src c :tangle (eval h_func) :comments org
qmckl_exit_code qmckl_woodbury_3x3_hpc (
      const qmckl_context context,
      const uint64_t LDS,
      const uint64_t Dim,
      const double* Updates,
      const uint64_t* Updates_index,
      const double breakdown,
      double* Slater_inv,
      double* determinant ); 
#+end_src

#+CALL: generate_c_header(table=qmckl_woodbury_3x3_args,rettyp=get_value("CRetType"),fname="qmckl_woodbury_3x3_doc")

#+RESULTS:
#+begin_src c :tangle (eval h_func) :comments org
qmckl_exit_code qmckl_woodbury_3x3_doc (
      const qmckl_context context,
      const uint64_t LDS,
      const uint64_t Dim,
      const double* Updates,
      const uint64_t* Updates_index,
      const double breakdown,
      double* Slater_inv,
      double* determinant ); 
#+end_src

*** C sources
#+begin_src c :tangle (eval c) :comments org
qmckl_exit_code qmckl_woodbury_3x3_hpc(const qmckl_context context,
    const uint64_t LDS,
    const uint64_t Dim,
    const double* restrict Updates,
    const uint64_t* restrict Updates_index,
    const double breakdown,
    double* restrict Slater_inv,
    double* restrict determinant) {
/*
    C := S^{-1} * U,    dim x 3
    B := 1 + V * C,     3 x 3
    D := V * S^{-1},    3 x dim
*/

  if (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
    return qmckl_failwith(context,
                          QMCKL_NULL_CONTEXT,
                          "qmckl_woodbury_3x3_hpc",
                          NULL);
  }

  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^T)^{-1}U : Dim x 3
  double __attribute__((aligned(8))) C[3 * Dim];
  for (uint64_t i = 0; i < Dim; i++) {
      C[i * 3] = 0;
      C[i * 3 + 1] = 0;
      C[i * 3 + 2] = 0;
      IVDEP
      ALIGNED
      for (uint64_t k = 0; k < LDS; k++) {
          C[i * 3]     += Slater_inv[i * LDS + k] * Updates[k];
          C[i * 3 + 1] += Slater_inv[i * LDS + k] * Updates[LDS + k];
          C[i * 3 + 2] += Slater_inv[i * LDS + k] * Updates[2 * LDS + k];
      }
  }

  // Compute B = 1 + VC : 3 x 3
  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 inverted matrix is not 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;
  }

  // Update det(S) when passed
  if (determinant)
      *determinant *= det;

  // Compute B^{-1} with explicit formula for 2 x 2 inversion
  double __attribute__((aligned(8))) 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;

  // tmp = B^{-1}D : 2 x LDS
  double __attribute__((aligned(8))) tmp[3 * LDS];
  double* r1dim = &(Slater_inv[row1 * LDS]);
  double* r2dim = &(Slater_inv[row2 * LDS]);
  double* r3dim = &(Slater_inv[row3 * LDS]);
  IVDEP
  ALIGNED
  for (uint64_t j = 0; j < LDS; j++) {
      tmp[j] = Binv[0] * r1dim[j] + Binv[1] * r2dim[j] + Binv[2] * r3dim[j];
      tmp[LDS + j] =
          Binv[3] * r1dim[j] + Binv[4] * r2dim[j] + Binv[5] * r3dim[j];
      tmp[2 * LDS + j] =
          Binv[6] * r1dim[j] + Binv[7] * r2dim[j] + Binv[8] * r3dim[j];
  }

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

  return QMCKL_SUCCESS;
}
#+end_src

#+NAME:woodbury_3x3_kernel_template
#+begin_src c
static inline qmckl_exit_code qmckl_woodbury_3x3_{Dim}(
    const qmckl_context context,
    const double* restrict Updates,
    const uint64_t* restrict Updates_index,
    const double breakdown,
    double* restrict Slater_inv,
    double* restrict determinant) {
/*
    C := S^{-1} * U,    dim x 3
    B := 1 + V * C,     3 x 3
    D := V * S^{-1},    3 x dim
,*/

  if (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
    return qmckl_failwith(context,
                          QMCKL_NULL_CONTEXT,
                          "qmckl_woodbury_3x3_{Dim}",
                          NULL);
  }

  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^T)^{-1}U : {Dim} x 3
  double __attribute__((aligned(8))) C[3 * {Dim}];
  for (uint64_t i = 0; i < {Dim}; i++) {
      C[i * 3] = 0;
      C[i * 3 + 1] = 0;
      C[i * 3 + 2] = 0;
      IVDEP
      ALIGNED
      for (uint64_t k = 0; k < D{Dim}_P; k++) {
          C[i * 3] += Slater_inv[i * D{Dim}_P + k] * Updates[k];
          C[i * 3 + 1] += Slater_inv[i * D{Dim}_P + k] * Updates[D{Dim}_P + k];
          C[i * 3 + 2] += Slater_inv[i * D{Dim}_P + k] * Updates[2 * D{Dim}_P + k];
      }
  }

  // Compute B = 1 + VC : 3 x 3
  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;
  }

  // Update det(Slater) if passed
  if (determinant)
      *determinant *= det;

  // Compute B^{-1} with explicit formula for 3 x 3 inversion
  double __attribute__((aligned(8))) 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;

  // tmp = B^{-1}D : 3 x D{Dim}_P
  double __attribute__((aligned(8))) tmp[3 * D{Dim}_P];
  double* r1dim = &(Slater_inv[row1 * D{Dim}_P]);
  double* r2dim = &(Slater_inv[row2 * D{Dim}_P]);
  double* r3dim = &(Slater_inv[row3 * D{Dim}_P]);
      IVDEP
      ALIGNED
  for (uint64_t j = 0; j < D{Dim}_P; j++) {
      tmp[j] = Binv[0] * r1dim[j] + Binv[1] * r2dim[j] + Binv[2] * r3dim[j];
      tmp[D{Dim}_P + j] =
          Binv[3] * r1dim[j] + Binv[4] * r2dim[j] + Binv[5] * r3dim[j];
      tmp[2 * D{Dim}_P + j] =
          Binv[6] * r1dim[j] + Binv[7] * r2dim[j] + Binv[8] * r3dim[j];
  }

  // Compute (S^T)^{-1} - C * tmp : {Dim} x D{Dim}_P
  for (uint64_t i = 0; i < {Dim}; i++) {
      IVDEP
      ALIGNED
      for (uint64_t j = 0; j < D{Dim}_P; j++) {
          Slater_inv[i * D{Dim}_P + j] -= C[i * 3] * tmp[j];
          Slater_inv[i * D{Dim}_P + j] -= C[i * 3 + 1] * tmp[D{Dim}_P + j];
          Slater_inv[i * D{Dim}_P + j] -= C[i * 3 + 2] * tmp[2 * D{Dim}_P + j];
      }
  }

  return QMCKL_SUCCESS;
}
#+end_src

#+NAME:woodbury_3x3_kernel_generator
#+begin_src python :noweb yes :exports none
text="""
<<woodbury_3x3_kernel_template>>
"""
result = []
for Dim in <<kernel_generator_range>>:
  Dim=str(Dim)
  result.append(text.replace("{Dim}",Dim) )

return ''.join(result)
#+end_src

#+NAME:woodbury_3x3_switch-case_generator
#+begin_src python :noweb yes :exports none
text="""
case {Dim}:  
  return qmckl_woodbury_3x3_{Dim}(context,
    Updates,
    Updates_index,
    breakdown,
    Slater_inv,
    determinant);"""
result = []
for Dim in <<kernel_generator_range>>:
  Dim=str(Dim)
  result.append(text.replace("{Dim}",Dim) )

return ''.join(result)
#+end_src

#+begin_src c :tangle (eval c) :comments org :noweb yes
<<woodbury_3x3_kernel_generator()>>
#+end_src

#+begin_src c :tangle (eval c) :comments org :noweb yes
qmckl_exit_code qmckl_woodbury_3x3(const qmckl_context context,
    const uint64_t LDS,
    const uint64_t Dim,
    const double* Updates,
    const uint64_t* Updates_index,
    const double breakdown,
    double* Slater_inv,
    double* determinant) {

  if (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
    return qmckl_failwith(
        context,
        QMCKL_NULL_CONTEXT,
        "qmckl_woodbury_3x3",
        NULL);
  }

  #ifdef HAVE_HPC__BROKEN_WITH_CRAY
  if (LDS == (1+(Dim-1)/SIMD_LENGTH)*SIMD_LENGTH) { // Most cases
    switch (Dim) {
      <<woodbury_3x3_switch-case_generator()>>
    }
  }
  else { // Updating smaller sub-matrix
    return qmckl_woodbury_3x3_hpc(
        context,
        LDS,
        Dim,
        Updates,
        Updates_index,
        breakdown,
        Slater_inv,
        determinant);
  }
  #else
  return qmckl_woodbury_3x3_doc(
      context,
      LDS,
      Dim,
      Updates,
      Updates_index,
      breakdown,
      Slater_inv,
      determinant);
  // return qmckl_woodbury_3x3_hpc(
  //     context,
  //     LDS,
  //     Dim,
  //     Updates,
  //     Updates_index,
  //     breakdown,
  //     Slater_inv,
  //     determinant);
  #endif

  return QMCKL_FAILURE;
}
#+end_src

*** Fortran interfaces (exposed in qmckl_f.F90)
:PROPERTIES:
:Name:     qmckl_woodbury_3x3
:CRetType: qmckl_exit_code
:FRetType: qmckl_exit_code
:END:

#+CALL: generate_f_interface(table=qmckl_woodbury_3x3_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_3x3 &
      (context, LDS, Dim, Updates, Updates_index, breakdown, Slater_inv, determinant) &
      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 :: LDS
    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(LDS*Dim)
    real    (c_double ) , intent(inout)        :: determinant

  end function qmckl_woodbury_3x3
end interface
#+end_src

#+CALL: generate_f_interface(table=qmckl_woodbury_3x3_args,rettyp=get_value("FRetType"),fname="qmckl_woodbury_3x3_hpc")

#+RESULTS:
#+begin_src f90 :tangle (eval fh_func) :comments org :exports none
interface
  integer(c_int32_t) function qmckl_woodbury_3x3_hpc &
      (context, LDS, Dim, Updates, Updates_index, breakdown, Slater_inv, determinant) &
      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 :: LDS
    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(LDS*Dim)
    real    (c_double ) , intent(inout)        :: determinant

  end function qmckl_woodbury_3x3_hpc
end interface
#+end_src

#+CALL: generate_f_interface(table=qmckl_woodbury_3x3_args,rettyp=get_value("FRetType"),fname="qmckl_woodbury_3x3_doc")

#+RESULTS:
#+begin_src f90 :tangle (eval fh_func) :comments org :exports none
interface
  integer(c_int32_t) function qmckl_woodbury_3x3_doc &
      (context, LDS, Dim, Updates, Updates_index, breakdown, Slater_inv, determinant) &
      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 :: LDS
    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(LDS*Dim)
    real    (c_double ) , intent(inout)        :: determinant

  end function qmckl_woodbury_3x3_doc
end interface
#+end_src

*** Performance
This function is most efficient when used in cases where there are only 3 rank-1 updates and
it is sure they will not result in a singular matrix.

*** Tests
#+begin_src c :tangle (eval c_test)
assert(Updates3 != NULL);
assert(Updates_index3 != NULL);
assert(Slater_inv3_1 != NULL);
det = -1.23743195512859e-09;
rc = qmckl_woodbury_3x3(context, LDS, Dim, Updates3, Updates_index3, breakdown, Slater_inv3_1, &det);
assert(fabs(det - 1.602708950725074e-10) < 1e-15);
for (unsigned int i = 0; i < Dim; i++) {
  for (unsigned int j = 0; j < Dim; j++) {
    res[i * Dim + j] = 0;
    for (unsigned int k = 0; k < Dim; k++) {
      res[i * Dim + j] += Slater3[i * Dim + k] * Slater_inv3_1[k * LDS + j];
    }
  }
}
rc = QMCKL_SUCCESS;
for (unsigned int i = 0; i < Dim; i++) {
  for (unsigned int j = 0; j < Dim; j++) {
    if (i == j && fabs(res[i * Dim + j] - 1) > tolerance) {
      rc = QMCKL_FAILURE;
    }
    if (i != j && fabs(res[i * Dim + j]) > tolerance) {
      rc = QMCKL_FAILURE;
    }
  }
}
assert(rc == QMCKL_SUCCESS);
#+end_src


* Sherman-Morrison with Slagel Splitting
** ~qmckl_sm_splitting~
:PROPERTIES:
:Name:     qmckl_sm_splitting
:CRetType: qmckl_exit_code
:FRetType: qmckl_exit_code
:END:

*** Introduction
This is a variation on the 'Naive' Sherman-Morrison kernel. Whenever the denominator $1+v_j^T S^{-1} u_j$ in
the Sherman-Morrison formula is deemed to be too close to zero, the update $u_j$ is split in half:
$u_j \rightarrow \frac{1}{2} u_j$. One half is applied immediately --necessarily increasing the value of the
denominator because of the split-- while the other halve is put in a queue that will be applied when all the
remaining updates have been treated.

The kernel is executed recursively until the queue is eiter empty and all
updates are applied successfully, or the size of the queue equals the number of initial updates. In the last
case the Slater-matrix that would have resulted from applying the updates is singular and therefore the
kernel exits with an exit code.

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.

*** API
#+NAME: qmckl_sm_splitting_args
| Variable      | Type                  | In/Out | Description                                                   |
|---------------+-----------------------+--------+------------------------------------------------------|
| context       | qmckl_context         | in     | Global state                                         |
| LDS           | uint64_t              | in     | Leading dimension of Slater_inv                      |
| Dim           | uint64_t              | in     | Dimension of Slater_inv                              |
| N_updates     | uint64_t              | in     | Number of rank-1 updates to be applied to Slater_inv |
| Updates       | double[N_updates*LDS] | in     | Array containing the updates                         |
| Updates_index | uint64_t[N_updates]   | in     | Array containing the rank-1 updates                  |
| breakdown     | double                | in     | Break-down parameter on which to fail or not         |
| Slater_inv    | double[Dim*LDS]       | inout  | Array containing the inverse of a Slater-matrix      |
| determinant   | double                | inout  | Determinant of the Slater-matrix                     |

*** Requirements
  * ~context~ is not ~QMCKL_NULL_CONTEXT~
  * ~LDS >= 2~
  * ~Dim >= 2~
  * ~N_updates >= 1~
  * ~Updates~ is allocated with $N_updates \times Dim$ elements
  * ~Updates_index~ is allocated with $N_updates$ elements
  * ~breakdown~ is a small number such that $0 < breakdown << 1$
  * ~Slater_inv~ is allocated with $Dim \times Dim$ elements

*** Pedagogical kernel source (in Fortran)
The following source code written in Fortran is inteded to illustrate how the kernel works. Even though the kernel is
able to do numerically correct computations, it does not do it in the most efficient way possible. It should therefore
not be used in real workloads.

#+begin_src f90 :tangle (eval f)
integer recursive function qmckl_sm_splitting_doc_f( &
    context, &
    lds, dim, &
    nupdates, &
    upds, &
    updates_index, &
    breakdown, &
    s_inv, &
    determinant) result(info)

  use qmckl
  implicit none
  integer*8 , intent(in)                :: context
  integer*8 , intent(in)                :: lds, dim
  integer*8 , intent(in)                :: nupdates
  integer*8 , intent(in)                :: updates_index(nupdates)
  real*8    , intent(in)                :: upds(lds * nupdates)
  real*8    , intent(in)                :: breakdown
  real*8    , intent(inout)             :: s_inv(dim * lds)
  real*8    , intent(inout)             :: determinant

  integer   , external                  :: qmckl_sm_splitting_core_doc_f
  
  integer*8                             :: Later
  integer*8 , dimension(nupdates)       :: Later_index
  real*8    , dimension(lds * nupdates) :: Later_updates

  info = QMCKL_FAILURE

  if (context == QMCKL_NULL_CONTEXT) then
    info = QMCKL_INVALID_CONTEXT
    return
  endif

  Later = 0
  Later_index = 0
  Later_updates = 0

  info = qmckl_sm_splitting_core_doc_f( &
    context, &
    lds, dim, &
    nupdates, &
    upds, &
    updates_index, &
    breakdown, &
    s_inv, &
    Later_updates, &
    Later_index, &
    Later, &
    determinant)
  
  if (Later > 0) then
    info = qmckl_sm_splitting_doc_f( &
      context, &
      lds, dim, &
      Later, &
      Later_updates, &
      Later_index, &
      breakdown, &
      s_inv, &
      determinant)
  end if

  info = QMCKL_SUCCESS

end function qmckl_sm_splitting_doc_f
#+end_src

**** C interface to the pedagogical kernel (not directly exposed)
The following Fortran function ~qmckl_sm_splitting_core_doc~ makes sure
that the pedagogical kernel ~qmckl_sm_splitting_core_doc_f~, written in
Fortran, can be called from C using the ~ISO_C_BINDING~. The Fortran function
~qmckl_sm_splitting_core_doc~ will be exposed in the header file 'qmckl.h'
for C users and in the module file 'qmckl_f.F90' for Fortran users.

#+CALL: generate_c_interface(table=qmckl_sm_splitting_args,rettyp=get_value("CRetType"),fname="qmckl_sm_splitting_doc")

#+RESULTS:
#+begin_src f90 :tangle (eval f) :comments org :exports none
integer(c_int32_t) function qmckl_sm_splitting_doc &
    (context, LDS, Dim, N_updates, Updates, Updates_index, breakdown, Slater_inv, determinant) &
    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 :: LDS
  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*LDS)
  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*LDS)
  real    (c_double ) , intent(inout)        :: determinant

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

end function qmckl_sm_splitting_doc
#+end_src

*** C headers (exposed in qmckl.h)

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

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

#+CALL: generate_private_c_header(table=qmckl_sm_splitting_args,rettyp=get_value("CRetType"),fname="qmckl_sm_splitting_hpc")

#+RESULTS:
#+begin_src c :tangle (eval h_private_func) :comments org
qmckl_exit_code qmckl_sm_splitting_hpc (
      const qmckl_context context,
      const uint64_t LDS,
      const uint64_t Dim,
      const uint64_t N_updates,
      const double* Updates,
      const uint64_t* Updates_index,
      const double breakdown,
      double* Slater_inv,
      double* determinant ); 
#+end_src

#+CALL: generate_c_header(table=qmckl_sm_splitting_args,rettyp=get_value("CRetType"),fname="qmckl_sm_splitting_doc")

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

*** C source
#+NAME:splitting_switch-case_generator
#+begin_src python :noweb yes
text="""
case {Dim}: {
  rc = qmckl_sm_splitting_core_{Dim}(
      context,
      N_updates,
      Updates,
      Updates_index,
      breakdown,
      Slater_inv,
      later_updates,
      later_index, &later, determinant);
  break;
}
"""
result = []
for Dim in <<kernel_generator_range>>:
    Dim=str(Dim)
    result.append(text.replace("{Dim}",Dim) )

return '\n'.join(result)
#+end_src

#+begin_src c :tangle (eval c) :comments org :noweb yes
qmckl_exit_code qmckl_sm_splitting_hpc(
    const qmckl_context context,
    const uint64_t LDS,
    const uint64_t Dim,
    const uint64_t N_updates,
    const double* Updates,
    const uint64_t* Updates_index,
    const double breakdown,
    double* Slater_inv,
    double* determinant) {

  if (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
    return qmckl_failwith(
        context,
        QMCKL_NULL_CONTEXT,
        "qmckl_sm_splitting_hpc",
        NULL);
  }

  double __attribute__((aligned(8))) later_updates[LDS * N_updates];
  uint64_t later_index[N_updates];
  uint64_t later = 0;

  qmckl_exit_code rc;
  if (LDS == (1+(Dim-1)/SIMD_LENGTH)*SIMD_LENGTH) {
    switch (Dim) {
      <<splitting_switch-case_generator()>>
      default: {
        assert(0 == 1 && "TEMPLATE NOT IMPLEMENTED!");
        break;
      }
    }
  } else {
    rc = qmckl_sm_splitting_core_hpc(
            context, LDS, Dim, N_updates, Updates, Updates_index,
            breakdown, Slater_inv, later_updates,
            later_index, &later, determinant);
  }
  if (rc != QMCKL_SUCCESS) return QMCKL_FAILURE;

  if (later > 0) {
    qmckl_exit_code rc = qmckl_sm_splitting_hpc(
                            context, LDS, Dim, later,
                            later_updates, later_index,
                            breakdown, Slater_inv, determinant);
    if (rc != QMCKL_SUCCESS) return QMCKL_FAILURE;
  }

  return QMCKL_SUCCESS;
}
#+end_src

#+begin_src c :tangle (eval c) :comment org
qmckl_exit_code qmckl_sm_splitting(
    const qmckl_context context,
    const uint64_t LDS,
    const uint64_t Dim,
    const uint64_t N_updates,
    const double* Updates,
    const uint64_t* Updates_index,
    const double breakdown,
    double* Slater_inv,
    double* determinant) {

  if (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
    return qmckl_failwith(
        context,
        QMCKL_NULL_CONTEXT,
        "qmckl_sm_splitting",
        NULL);
  }
  #ifdef HAVE_HPC__BROKEN_WITH_CRAY
    return qmckl_sm_splitting_hpc(
        context,
        LDS,
        Dim,
        N_updates,
        Updates,
        Updates_index,
        breakdown,
        Slater_inv,
        determinant);
  #else
    return qmckl_sm_splitting_doc(
        context,
        LDS,
        Dim,
        N_updates,
        Updates,
        Updates_index,
        breakdown,
        Slater_inv,
        determinant);
  #endif 

}
#+end_src

*** Fortran interfaces (exposed in qmckl_f.F90)
   :PROPERTIES:
   :Name:     qmckl_sm_naive
   :CRetType: qmckl_exit_code
   :FRetType: qmckl_exit_code
   :END:

#+CALL: generate_f_interface(table=qmckl_sm_splitting_args,rettyp=get_value("FRetType"),fname="qmckl_sm_splitting_hpc")

#+RESULTS:
#+begin_src f90 :tangle (eval fh_func) :comments org :exports none
interface
  integer(c_int32_t) function qmckl_sm_splitting_hpc &
      (context, LDS, Dim, N_updates, Updates, Updates_index, breakdown, Slater_inv, determinant) &
      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 :: LDS
    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*LDS)
    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*LDS)
    real    (c_double ) , intent(inout)        :: determinant

  end function qmckl_sm_splitting_hpc
end interface
#+end_src

#+CALL: generate_f_interface(table=qmckl_sm_splitting_args,rettyp=get_value("FRetType"),fname="qmckl_sm_splitting_doc")

#+RESULTS:
#+begin_src f90 :tangle (eval fh_func) :comments org :exports none
interface
  integer(c_int32_t) function qmckl_sm_splitting_doc &
      (context, LDS, Dim, N_updates, Updates, Updates_index, breakdown, Slater_inv, determinant) &
      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 :: LDS
    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*LDS)
    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*LDS)
    real    (c_double ) , intent(inout)        :: determinant

  end function qmckl_sm_splitting_doc
end interface
#+end_src

#+CALL: generate_f_interface(table=qmckl_sm_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_sm_splitting &
      (context, LDS, Dim, N_updates, Updates, Updates_index, breakdown, Slater_inv, determinant) &
      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 :: LDS
    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*LDS)
    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*LDS)
    real    (c_double ) , intent(inout)        :: determinant

  end function qmckl_sm_splitting
end interface
#+end_src

*** Performance...
This kernel performs best when there are 2 or more rank-1 update cycles and fail-rate is high.

*** Test
#+begin_src c :tangle (eval c_test)
assert(Updates3 != NULL);
assert(Updates_index3 != NULL);
assert(Slater_inv3_2 != NULL);
det = -1.23743195512859e-09;
rc = qmckl_sm_splitting(context, LDS, Dim, N_updates3, Updates3, Updates_index3, breakdown, Slater_inv3_2, &det);
assert(fabs(det - 1.602708950725074e-10) < 1e-15);
for (unsigned int i = 0; i < Dim; i++) {
  for (unsigned int j = 0; j < Dim; j++) {
    res[i * Dim + j] = 0;
    for (unsigned int k = 0; k < Dim; k++) {
      res[i * Dim + j] += Slater3[i * Dim + k] * Slater_inv3_2[k * LDS + j];
    }
  }
}
rc = QMCKL_SUCCESS;
for (unsigned int i = 0; i < Dim; i++) {
  for (unsigned int j = 0; j < Dim; j++) {
    if (i == j && fabs(res[i * Dim + j] - 1) > tolerance) {
      rc = QMCKL_FAILURE;
    }
    if (i != j && fabs(res[i * Dim + j]) > tolerance) {
      rc = QMCKL_FAILURE;
    }
  }
}
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
