qmckl/org/qmckl_sherman_morrison_woodbury.org

Sherman-Morrison-Woodbury

• Headers
• Naïve Sherman-Morrison
  • qmckl_sherman_morrison
    • Requirements
    • C header
    • Source
    • 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

(org-babel-lob-ingest "../tools/lib.org")

#include "qmckl.h"
#include "assert.h"
#ifdef HAVE_CONFIG_H
#include "config.h"
#endif
#include <cmath>
#ifndef THRESHOLD
#define THRESHOLD 1e-3
#endif

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

Naïve Sherman-Morrison

qmckl_sherman_morrison

The Sherman-Morrison formula

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

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

Requirements

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

C header

qmckl_exit_code qmckl_sherman_morrison (
      const qmckl_context context,
      double *Slater_inv,
      unsigned int Dim,
      unsigned int N_updates,
      double *Updates,
      unsigned int *Updates_index );

Source

bool qmckl_sherman_morrison(double *Slater_inv, unsigned int Dim, unsigned int N_updates,
     double *Updates, unsigned int *Updates_index) {
#ifdef DEBUG
std::cerr << "Called qmckl_sherman_morrison with " << N_updates << " updates" << std::endl;
#endif

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

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

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

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

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

l += 1;
}
return true;
}

Performance

End of files

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