qmckl/org/qmckl_sherman_morrison_woodbury.org

Sherman-Morrison-Woodbury

• Headers
• Helper Functions
  • qmckl_slagel_splitting
    • Requirements
    • C header
    • Source Fortran
    • Source C
    • Performance
• Naïve Sherman-Morrison
  • qmckl_sherman_morrison
    • Requirements
    • C header
    • Source Fortran
    • Source C
    • Performance
• Woodbury 2x2
  • qmckl_woodbury_2
    • Requirements
    • C header
    • Source Fortran
    • Source C
    • Performance
• Woodbury 3x3
  • qmckl_woodbury_3
    • Requirements
    • C header
    • Source Fortran
    • Source C
    • Performance…
• Sherman-Morrison with update splitting
  • qmckl_sherman_morrison_splitting
    • Requirements
    • C header
    • Source Fortran
    • Source C
    • Performance…
• Woodbury 2x2 with Sherman-Morrison and update splitting
  • qmckl_sherman_morrison_smw2s
    • Requirements
    • C header
    • Source Fortran
    • Source C
    • Performance…
• Woodbury 3x3 with Sherman-Morrison and update splitting
  • qmckl_sherman_morrison_smw3s
    • Requirements
    • C header
    • Source Fortran
    • Source C
    • Performance…
• Woodbury 3x3 and 2x2 with Sherman-Morrison and update splitting
  • qmckl_sherman_morrison_smw32s
    • Requirements
    • C header
    • Source Fortran
    • Source C
    • Performance…
• 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-singualr 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;

Helper Functions

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

qmckl_slagel_splitting

qmckl_slagel_splitting is used internally to apply a list of rank-1 updates while splitting an update if necessary. In case of a split it applies the first half of the update while putting the second half in waiting queue to be applied at the end.

For a given update $u_j$ we define a threshold value $\epsilon_j$, which is the minimum value of $1+v_j^TS^{-1}u_j$ for a non-singular matrix $S$. If $1+v_j^TS^{-1}u_j \geq \epsilon_j$, the update is applied as usual. Otherwise, $u_j$ will be redefined as $\frac{u_j}{2}$, and the other half (to be applied at the end) will be defined using vectors $\frac{u_{j'}}{2}$ and $v_{j'}^T=v_{j'}^T$.

uint64_t	Dim	in	Leading dimension of Slater_inv
uint64_t	N_updates	in	Number of rank-1 updates to be applied to Slater_inv
double	Updates[N_updates*Dim]	in	Array containing the rank-1 updates
uint64_t	Updates_index[N_updates]	in	Array containing positions of the rank-1 updates
double	breakdown	in	Break-down parameter on which to fail or not
double	Slater_inv[Dim*Dim]	inout	Array containing the inverse Slater-matrix
double	later_updates[Dim * N_updates]	inout	Array containing the split updates for later
uint64_t	later_index[N_updates]	inout	Array containing the positions of the split updates for later
uint64_t	later	inout	Number of split updates for later

Requirements

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

C header

qmckl_exit_code qmckl_slagel_splitting_c (
const uint64_t Dim,
const uint64_t N_updates,
const double* Updates,
const uint64_t* Updates_index,
const double breakdown,
double* Slater_inv,
double* later_updates,
uint64_t* later_index,
uint64_t* later );

Source Fortran

Source C

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

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

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

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

// Denominator
double den = 1 + C[Updates_index[l] - 1];
if (fabs(den) < breakdown) {

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

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

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

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

return QMCKL_SUCCESS;
}

Performance

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

Naïve Sherman-Morrison

qmckl_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.

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

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

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

Requirements

• context is not QMCKL_NULL_CONTEXT
• Dim >= 2
• N_updates >= 1
• Updates is allocated with at least $1 \times 2 \times 8$ bytes
• Updates_index is allocated with at least $1 \times 8$ bytes
• breakdown is a small number such that $0 < breakdown << 1$
• Slater_inv is allocated with at least $Dim \times Dim \times 8$ bytes

C header

qmckl_exit_code qmckl_sherman_morrison_c (
const qmckl_context context,
const uint64_t Dim,
const uint64_t N_updates,
const double* Updates,
const uint64_t* Updates_index,
const double breakdown,
double* Slater_inv );

Source Fortran

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

Source C

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

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

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

uint64_t l = 0;
// For each update
while (l < N_updates) {
// C = A^{-1} x U_l
for (uint64_t i = 0; i < Dim; i++) {
  C[i] = 0;
  for (uint64_t j = 0; j < Dim; j++) {
    C[i] += Slater_inv[i * Dim + j] * Updates[l * Dim + j];
  }
}

// Denominator
double den = 1 + C[Updates_index[l] - 1];
if (fabs(den) < breakdown) {
  return QMCKL_FAILURE;
}
double iden = 1 / den;

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

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

l += 1;
}

return QMCKL_SUCCESS;
}

Performance

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

Woodbury 2x2

qmckl_woodbury_2

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

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

Requirements

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

C header

qmckl_exit_code qmckl_woodbury_2_c (
const qmckl_context context,
const uint64_t Dim,
const double* Updates,
const uint64_t* Updates_index,
const double breakdown,
double* Slater_inv );

Source Fortran

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

Source C

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

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

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

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

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

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

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

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

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

return QMCKL_SUCCESS;
}

Performance

This function is most efficient when used in cases where there are only 2 rank-1 updates.

Woodbury 3x3

qmckl_woodbury_3

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

$C:= S^{-1}U$, 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

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

Requirements

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

C header

qmckl_exit_code qmckl_woodbury_3_c (
const qmckl_context context,
const uint64_t Dim,
const double* Updates,
const uint64_t* Updates_index,
const double breakdown,
double* Slater_inv );

Source Fortran

integer function qmckl_woodbury_3_f(context, Dim,    &
     Updates, Updates_index, breakdown, Slater_inv) result(info) 
use qmckl
implicit none
integer(qmckl_context)  , intent(in)  :: context
integer*8  , intent(in), value  :: Dim
integer*8  , intent(in)  :: Updates_index(3)
real*8     , intent(in) :: Updates(3*Dim)
real*8     , intent(in) :: breakdown
real*8     , intent(inout)  :: Slater_inv(Dim*Dim)
!logical, external :: qmckl_woodbury_3_f
info = qmckl_woodbury_3(context, Dim, Updates, Updates_index, breakdown, Slater_inv)
end function qmckl_woodbury_3_f

Source C

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

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

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

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

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

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

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

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

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

return QMCKL_SUCCESS;
}

Performance…

This function is most efficient when used in cases where there are only 3 rank-1 updates.

Sherman-Morrison with update splitting

qmckl_sherman_morrison_splitting

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: $u_j \rightarrow \frac{1}{1} u_j$. 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 un-invertable and therefore the kernel exits with an exit code.

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

Requirements

• context is not QMCKL_NULL_CONTEXT
• Dim >= 2
• N_updates >= 1
• Updates is allocated with at least $1 \times 2 \times 8$ bytes
• Updates_index is allocated with at least $1 \times 8$ bytes
• breakdown is a small number such that $0 < breakdown << 1$
• Slater_inv is allocated with at least $Dim \times Dim \times 8$ bytes

C header

qmckl_exit_code qmckl_sherman_morrison_splitting_c (
const qmckl_context context,
const uint64_t Dim,
const uint64_t N_updates,
const double* Updates,
const uint64_t* Updates_index,
const double breakdown,
double* Slater_inv );

Source Fortran

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

Source C

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

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

qmckl_exit_code rc;

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

rc = qmckl_slagel_splitting_c(Dim, N_updates, Updates, Updates_index,
                        breakdown, Slater_inv, later_updates, later_index, &later);

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

return QMCKL_SUCCESS;
}

Performance…

This kernel performs best when there are only 1 rank-1 update cycles and/or when the fail-rate is high.

Woodbury 2x2 with Sherman-Morrison and update splitting

qmckl_sherman_morrison_smw2s

The Woodbury 2x2 kernel with Sherman-Morrison and update splitting combines the low-level Woodbury 2x2 kernel and Sherman-Morrison with update splitting. For a given number of updates $N$ it splits the number of updates in blocks of two updates. The blocks of two updates are then applied one by one using Woodbury 2x2. If a block of updates fails, both updates in the block are applied with Sherman-Morrison instead, split if necessary and with their second half put in a queue. After all blocks are processed the remaining one update –in case there was an odd number of updates to begin with– is also aplpied with Sherman-Morrison and split if necessary. The queue containing the collected second halves of all the processed updates is processed at the very end to avoid having many intermediate queues containing only a few updates that risks an increased probability of artificially created non-singular intermediate matrices, resulting from division up the total number of updates in blocks of three.

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

Requirements

• context is not QMCKL_NULL_CONTEXT
• Dim >= 2
• N_updates >= 1
• Updates is allocated with at least $1 \times 2 \times 8$ bytes
• Updates_index is allocated with at least $1 \times 8$ bytes
• breakdown is a small number such that $0 < breakdown << 1$
• Slater_inv is allocated with at least $Dim \times Dim \times 8$ bytes

C header

qmckl_exit_code qmckl_sherman_morrison_smw2s_c (
const qmckl_context context,
const uint64_t Dim,
const uint64_t N_updates,
const double* Updates,
const uint64_t* Updates_index,
const double breakdown,
double* Slater_inv );

Source Fortran

integer function qmckl_sherman_morrison_smw2s_f(context, Slater_inv, Dim, N_updates,    &
     Updates, Updates_index) result(info) 
use qmckl
implicit none
integer(qmckl_context)  , intent(in)  :: context
integer*8  , intent(in), value  :: Dim, N_updates
integer*8  , intent(in)  :: Updates_index(N_updates)
real*8     , intent(in) :: Updates(N_updates*Dim)
real*8     , intent(inout)  :: Slater_inv(Dim*Dim)
info = qmckl_sherman_morrison_smw2s (context, Dim, N_updates, Updates, Updates_index, Slater_inv)
end function qmckl_sherman_morrison_smw2s_f

Source C

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

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

qmckl_exit_code rc;

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

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

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

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

Performance…

This kernel performs best for the case of two rank-1 update and a low fail-rate.

Woodbury 3x3 with Sherman-Morrison and update splitting

qmckl_sherman_morrison_smw3s

The Woodbury 3x3 kernel with Sherman-Morrison and update splitting combines the low-level Woodbury 3x3 kernel and Sherman-Morrison with update splitting. It works the same as Woodbury 2x2 with Sherman-Morrison and update splitting, except that the updates are divided in blocks of three rank-1 updates instead of blocks of two rank-1 updates.

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

Requirements

• context is not QMCKL_NULL_CONTEXT
• Dim >= 2
• N_updates >= 1
• Updates is allocated with at least $1 \times 2 \times 8$ bytes
• Updates_index is allocated with at least $1 \times 8$ bytes
• breakdown is a small number such that $0 < breakdown << 1$
• Slater_inv is allocated with at least $Dim \times Dim \times 8$ bytes

C header

qmckl_exit_code qmckl_sherman_morrison_smw3s_c (
const qmckl_context context,
const uint64_t Dim,
const uint64_t N_updates,
const double* Updates,
const uint64_t* Updates_index,
const double breakdown,
double* Slater_inv );

Source Fortran

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

Source C

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

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

qmckl_exit_code rc;

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

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

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

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

Performance…

This kernel performs best for the case of three rank-1 update and a low fail-rate.

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

qmckl_sherman_morrison_smw32s

The Woodbury 3x3 and 2x2 kernel with Sherman-Morrison and update splitting combines the low-level Woodbury 3x3 kernel, the Woobury 2x2 kernel and Sherman-Morrison with update splitting. It works the almost the same as Woodbury 3x3 with Sherman-Morrison and update splitting, except that when there is a remainder of two rank-1 updates, it is first tried with Woodbury 2x2 instead of sending them all to Sherman-Morrison with update splitting. For example, in the case of 5 updates the updates are applied in 1 block of 3 updates end 1 block of 2 updates.

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

Requirements

• context is not QMCKL_NULL_CONTEXT
• Dim >= 2
• N_updates >= 1
• Updates is allocated with at least $1 \times 2 \times 8$ bytes
• Updates_index is allocated with at least $1 \times 8$ bytes
• breakdown is a small number such that $0 < breakdown << 1$
• Slater_inv is allocated with at least $Dim \times Dim \times 8$ bytes

C header

qmckl_exit_code qmckl_sherman_morrison_smw32s_c (
const qmckl_context context,
const uint64_t Dim,
const uint64_t N_updates,
const double* Updates,
const uint64_t* Updates_index,
const double breakdown,
double* Slater_inv );

Source Fortran

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

Source C

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

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

qmckl_exit_code rc;

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

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

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

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

Performance…

This kernel performs best when the number of rank-1 updates is larger than 3 and fail-rates are low.

End of files

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