TREX/qmckl

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Francois Coppens c07553480c Pedagogical Naive kernel works.

2023-02-15 11:46:48 +01:00

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Sherman-Morrison-Woodbury

Headers
Naïve Sherman-Morrison
End of files

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

#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;

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

Naïve Sherman-Morrison

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.

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

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.

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(lds, nupdates) :: Updates
  real*8    , intent(out) , dimension(dim, lds)      :: Inverse

  integer*8                                          :: i, j

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

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

subroutine copy_back(Inverse, s_inv, lds, dim)
  implicit none
  integer*8 , intent(in)                       :: lds, dim
  real*8    , intent(in) , dimension(dim, lds) :: 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, lds
      s_inv((i - 1) * lds + j) = Inverse(i, j)
    end do
  end do
end subroutine copy_back

integer function qmckl_sherman_morrison_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(lds, nupdates) :: Updates
  real*8 , dimension(dim, lds)      :: Inverse
  real*8 , dimension(dim)           :: C
  real*8 , dimension(lds)           :: 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

  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(Inverse, s_inv, lds, dim)

  info = QMCKL_SUCCESS

end function qmckl_sherman_morrison_naive_doc_f

C interface to the pedagogical kernel (not directly exposed)

The following Fortran function qmckl_sherman_morrison_naive_doc makes sure that the pedagogical kernel qmckl_sherman_morrison_naive_doc_f, written in Fortran, can be called from C using the ISO_C_BINDING. The Fortran function qmckl_sherman_morrison_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.

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

C headers (exposed in qmckl.h)

qmckl_exit_code qmckl_sherman_morrison_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 );

qmckl_exit_code qmckl_sherman_morrison_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 );

qmckl_exit_code qmckl_sherman_morrison_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 );

C sources

Common includes and macros used by all the Sherman-Morrison-Woodbury kernels.

#include <stdbool.h>
#include <math.h>
#include "qmckl.h"
#include "config.h"

// Order important because
// __GNUC__ also set in ICC, ICX and CLANG
// __clang__ also set in ICX
#if defined(__INTEL_COMPILER)
    #define IVDEP _Pragma("ivdep")
    #define ALIGNED _Pragma("vector aligned") 
#elif defined(__INTEL_LLVM_COMPILER)
    #define IVDEP _Pragma("ivdep")
    #define ALIGNED _Pragma("vector aligned") 
#elif defined(__clang__)
    #define IVDEP _Pragma("clang loop vectorize(enable)")
    #define ALIGNED
#elif defined(__GNUC__)
    #define IVDEP _Pragma("GCC ivdep")
    #define ALIGNED
#endif

qmckl_sherman_morrison_naive_hpc is a high performance variation of qmckl_sherman_morrison_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_LEGTH.

qmckl_exit_code qmckl_sherman_morrison_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_sherman_morrison_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;
}

qmckl_exit_code qmckl_sherman_morrison_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

  static inline qmckl_exit_code qmckl_sherman_morrison_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_sherman_morrison_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;
  }

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

Python script that generated C switch cases that call individual kernel instances. #+NAME:naive_switch-case_generator

qmckl_sherman_morrison_naive is the general 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.

<<naive_kernel_generator()>>

qmckl_exit_code qmckl_sherman_morrison_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_sherman_morrison_naive",
                          NULL);
  }

  #ifdef HAVE_HPC
  if (LDS == (1+(Dim-1)/SIMD_LENGTH)*SIMD_LENGTH) { // Most cases
    switch (Dim) {
      <<naive_switch-case_generator()>>
    }
  }
  else { // When SIMD_LENGTH > 1, called with LDS == Dim AND Dim != (1+(Dim-1)/SIMD_LENGTH)*SIMD_LENGTH)
    return qmckl_sherman_morrison_naive_hpc(context,
                                      LDS,
                                      Dim,
    	                                N_updates,
    	                                Updates,
    	                                Updates_index,
    	                                breakdown,
    	                                Slater_inv,
    	                                determinant);

  }
  #else
  return qmckl_sherman_morrison_naive_doc(context,
                                       LDS,
                                       Dim,
    	                                 N_updates,
                                       Updates,
    	                                 Updates_index,
    	                                 breakdown,
    	                                 Slater_inv,
    	                                 determinant);
  #endif
  
  return QMCKL_FAILURE;
}

Fortran interfaces (exposed in qmckl_f.F90)

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.

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_sherman_morrison_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 of files

assert (qmckl_context_destroy(context) == QMCKL_SUCCESS);
return 0;

}

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