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536 lines
18 KiB
C++
536 lines
18 KiB
C++
#include <cmath>
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// Sherman-Morrison-Woodbury break-down threshold
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#ifndef THRESHOLD
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#define THRESHOLD 1e-3
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#endif
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double threshold();
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// Naïve Sherman Morrison
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void sherman_morrison(double *Slater_inv, unsigned int Dim, unsigned int N_updates,
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double *Updates, unsigned int *Updates_index);
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// Woodbury 2x2 kernel
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bool woodbury_2(double *Slater_inv, const unsigned int Dim, double *Updates,
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const unsigned int *Updates_index);
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// Woodbury 3x3 kernel
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bool woodbury_3(double *Slater_inv, const unsigned int Dim, double *Updates,
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const unsigned int *Updates_index);
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// Sherman Morrison, with J. Slagel splitting (caller function)
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void sherman_morrison_splitting_caller(double *Slater_inv, unsigned int Dim, unsigned int N_updates,
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double *Updates, unsigned int *Updates_index);
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// Sherman Morrison, with J. Slagel splitting
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// http://hdl.handle.net/10919/52966
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void sherman_morrison_splitting(double *Slater_inv, unsigned int Dim, unsigned int N_updates,
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double *Updates, unsigned int *Updates_index,
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double *later_updates, unsigned int *later_index,
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unsigned int *later);
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// Mixed Sherman-Morrison-Woodbury kernel using
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// Woodbury 2x2 and Sherman-Morrison with update-splitting
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void sherman_morrison_woodbury_2(double *Slater_inv, const unsigned int Dim,
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const unsigned int N_updates, double *Updates,
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unsigned int *Updates_index);
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// Mixed Sherman-Morrison-Woodbury kernel using
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// Woodbury 3x3, Woodbury 2x2 and Sherman-Morrison with update-splitting
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void sherman_morrison_woodbury_3(double *Slater_inv, const unsigned int Dim,
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const unsigned int N_updates, double *Updates,
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unsigned int *Updates_index);
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// Sherman-Morrison-Woodbury break-down threshold
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double threshold() {
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const double threshold = THRESHOLD;
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#ifdef DEBUG2
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std::cerr << "Break-down threshold set to: " << threshold << std::endl;
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#endif
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return threshold;
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}
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// Naïve Sherman Morrison
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void sherman_morrison(double *Slater_inv, unsigned int Dim, unsigned int N_updates,
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double *Updates, unsigned int *Updates_index) {
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#ifdef DEBUG1
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std::cerr << "Called sherman_morrison with " << N_updates << " updates" << std::endl;
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#endif
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double C[Dim];
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double D[Dim];
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unsigned int l = 0;
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// For each update
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while (l < N_updates) {
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// C = A^{-1} x U_l
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for (unsigned int i = 0; i < Dim; i++) {
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C[i] = 0;
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for (unsigned int j = 0; j < Dim; j++) {
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C[i] += Slater_inv[i * Dim + j] * Updates[l * Dim + j];
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}
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}
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// Denominator
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double den = 1 + C[Updates_index[l] - 1];
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if (std::fabs(den) < threshold()) {
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#ifdef DEBUG1
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std::cerr << "Breakdown condition triggered at " << Updates_index[l]
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<< std::endl;
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#endif
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}
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double iden = 1 / den;
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// D = v^T x A^{-1}
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for (unsigned int j = 0; j < Dim; j++) {
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D[j] = Slater_inv[(Updates_index[l] - 1) * Dim + j];
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}
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// A^{-1} = A^{-1} - C x D / den
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for (unsigned int i = 0; i < Dim; i++) {
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for (unsigned int j = 0; j < Dim; j++) {
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double update = C[i] * D[j] * iden;
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Slater_inv[i * Dim + j] -= update;
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}
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}
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l += 1;
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}
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}
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// Woodbury 2x2 kernel
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bool woodbury_2(double *Slater_inv, const unsigned int Dim, double *Updates,
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const unsigned int *Updates_index) {
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/*
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C := S^{-1} * U, dim x 2
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B := 1 + V * C, 2 x 2
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D := V * S^{-1}, 2 x dim
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*/
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#ifdef DEBUG1
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std::cerr << "Called Woodbury 2x2 kernel" << std::endl;
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#endif
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const unsigned int row1 = (Updates_index[0] - 1);
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const unsigned int row2 = (Updates_index[1] - 1);
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// Compute C = S_inv * U !! NON-STANDARD MATRIX MULTIPLICATION BECAUSE
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// OF LAYOUT OF 'Updates' !!
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double C[2 * Dim];
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for (unsigned int i = 0; i < Dim; i++) {
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for (unsigned int j = 0; j < 2; j++) {
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C[i * 2 + j] = 0;
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for (unsigned int k = 0; k < Dim; k++) {
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C[i * 2 + j] += Slater_inv[i * Dim + k] * Updates[Dim * j + k];
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}
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}
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}
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// Compute B = 1 + V * C
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const double B0 = C[row1 * 2] + 1;
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const double B1 = C[row1 * 2 + 1];
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const double B2 = C[row2 * 2];
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const double B3 = C[row2 * 2 + 1] + 1;
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// Check if determinant of inverted matrix is not zero
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double det = B0 * B3 - B1 * B2;
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if (std::fabs(det) < threshold()) {
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#ifdef DEBUG1
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std::cerr << "Determinant too close to zero! No inverse found."
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<< std::endl;
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std::cerr << "Determinant = " << det << std::endl;
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#endif
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return false;
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}
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// Compute B^{-1} with explicit formula for 2x2 inversion
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double Binv[4], idet = 1.0 / det;
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Binv[0] = idet * B3;
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Binv[1] = -1.0 * idet * B1;
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Binv[2] = -1.0 * idet * B2;
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Binv[3] = idet * B0;
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// Compute tmp = B^{-1} x (V.S^{-1})
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double tmp[2 * Dim];
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for (unsigned int i = 0; i < 2; i++) {
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for (unsigned int j = 0; j < Dim; j++) {
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tmp[i * Dim + j] = Binv[i * 2] * Slater_inv[row1 * Dim + j];
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tmp[i * Dim + j] += Binv[i * 2 + 1] * Slater_inv[row2 * Dim + j];
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}
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}
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// Compute (S + U V)^{-1} = S^{-1} - C x tmp
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for (unsigned int i = 0; i < Dim; i++) {
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for (unsigned int j = 0; j < Dim; j++) {
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Slater_inv[i * Dim + j] -= C[i * 2] * tmp[j];
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Slater_inv[i * Dim + j] -= C[i * 2 + 1] * tmp[Dim + j];
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}
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}
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return true;
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}
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// Woodbury 3x3 kernel
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bool woodbury_3(double *Slater_inv, const unsigned int Dim, double *Updates,
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const unsigned int *Updates_index) {
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/*
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C := S^{-1} * U, dim x 3
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B := 1 + V * C, 3 x 3
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D := V * S^{-1}, 3 x dim
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*/
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#ifdef DEBUG1
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std::cerr << "Called Woodbury 3x3 kernel" << std::endl;
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#endif
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const unsigned int row1 = (Updates_index[0] - 1);
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const unsigned int row2 = (Updates_index[1] - 1);
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const unsigned int row3 = (Updates_index[2] - 1);
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// Compute C = S_inv * U !! NON-STANDARD MATRIX MULTIPLICATION BECAUSE
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// OF LAYOUT OF 'Updates' !!
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double C[3 * Dim];
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for (unsigned int i = 0; i < Dim; i++) {
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for (unsigned int j = 0; j < 3; j++) {
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C[i * 3 + j] = 0;
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for (unsigned int k = 0; k < Dim; k++) {
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C[i * 3 + j] += Slater_inv[i * Dim + k] * Updates[Dim * j + k];
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}
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}
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}
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#ifdef DEBUG2
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showMatrix2(C, Dim, 3, "C = S_inv * U");
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showMatrix2(D, 3, Dim, "D = V * S_inv");
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#endif
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// Compute B = 1 + V.C
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const double B0 = C[row1 * 3] + 1;
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const double B1 = C[row1 * 3 + 1];
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const double B2 = C[row1 * 3 + 2];
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const double B3 = C[row2 * 3];
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const double B4 = C[row2 * 3 + 1] + 1;
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const double B5 = C[row2 * 3 + 2];
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const double B6 = C[row3 * 3];
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const double B7 = C[row3 * 3 + 1];
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const double B8 = C[row3 * 3 + 2] + 1;
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#ifdef DEBUG2
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showMatrix2(B, 3, 3, "B = 1 + V * C");
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#endif
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// Check if determinant of B is not too close to zero
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double det;
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det = B0 * (B4 * B8 - B5 * B7) - B1 * (B3 * B8 - B5 * B6) +
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B2 * (B3 * B7 - B4 * B6);
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#ifdef DEBUG2
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std::cerr << "Determinant of B = " << det << std::endl;
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#endif
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if (std::fabs(det) < threshold()) {
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#ifdef DEBUG1
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std::cerr << "Determinant too close to zero! No inverse found."
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<< std::endl;
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std::cerr << "Determinant = " << det << std::endl;
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#endif
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return false;
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}
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// Compute B^{-1} with explicit formula for 3x3 inversion
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double Binv[9], idet = 1.0 / det;
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Binv[0] = (B4 * B8 - B7 * B5) * idet;
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Binv[1] = -(B1 * B8 - B7 * B2) * idet;
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Binv[2] = (B1 * B5 - B4 * B2) * idet;
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Binv[3] = -(B3 * B8 - B6 * B5) * idet;
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Binv[4] = (B0 * B8 - B6 * B2) * idet;
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Binv[5] = -(B0 * B5 - B3 * B2) * idet;
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Binv[6] = (B3 * B7 - B6 * B4) * idet;
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Binv[7] = -(B0 * B7 - B6 * B1) * idet;
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Binv[8] = (B0 * B4 - B3 * B1) * idet;
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#ifdef DEBUG2
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std::cerr << "Conditioning number of B = " << condition1(B, Binv, 3)
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<< std::endl;
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showMatrix2(Binv, 3, 3, "Binv");
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#endif
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// Compute tmp = B^{-1} x (V.S^{-1})
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double tmp[3 * Dim];
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for (unsigned int i = 0; i < 3; i++) {
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for (unsigned int j = 0; j < Dim; j++) {
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tmp[i * Dim + j] = Binv[i * 3] * Slater_inv[row1 * Dim + j];
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tmp[i * Dim + j] += Binv[i * 3 + 1] * Slater_inv[row2 * Dim + j];
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tmp[i * Dim + j] += Binv[i * 3 + 2] * Slater_inv[row3 * Dim + j];
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}
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}
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#ifdef DEBUG2
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showMatrix2(tmp, 3, Dim, "tmp = Binv * D");
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#endif
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// Compute (S + U V)^{-1} = S^{-1} - C x tmp
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for (unsigned int i = 0; i < Dim; i++) {
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for (unsigned int j = 0; j < Dim; j++) {
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Slater_inv[i * Dim + j] -= C[i * 3] * tmp[j];
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Slater_inv[i * Dim + j] -= C[i * 3 + 1] * tmp[Dim + j];
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Slater_inv[i * Dim + j] -= C[i * 3 + 2] * tmp[2 * Dim + j];
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}
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}
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#ifdef DEBUG2
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showMatrix2(Slater_inv, Dim, Dim, "Slater_inv AFTER update");
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#endif
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return true;
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}
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// Sherman Morrison, with J. Slagel splitting (caller function)
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// http://hdl.handle.net/10919/52966
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void sherman_morrison_splitting_caller(double *Slater_inv, unsigned int Dim, unsigned int N_updates,
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double *Updates, unsigned int *Updates_index) {
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#ifdef DEBUG1
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std::cerr << "Called sherman_morrison_splitting with " << N_updates << " updates" << std::endl;
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#endif
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double later_updates[Dim * N_updates];
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unsigned int later_index[N_updates];
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unsigned int later = 0;
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sherman_morrison_splitting(Slater_inv, Dim, N_updates, Updates, Updates_index, later_updates,
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later_index, &later);
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if (later > 0) {
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sherman_morrison_splitting_caller(Slater_inv, Dim, later, later_updates, later_index);
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}
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}
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// Sherman Morrison, with J. Slagel splitting
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// http://hdl.handle.net/10919/52966
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void sherman_morrison_splitting(double *Slater_inv, unsigned int Dim, unsigned int N_updates,
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double *Updates, unsigned int *Updates_index,
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double *later_updates, unsigned int *later_index,
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unsigned int *later) {
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#ifdef DEBUG1
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std::cerr << "Called sherman_morrison_splitting* with " << N_updates << " updates" << std::endl;
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#endif
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double C[Dim];
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double D[Dim];
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unsigned int l = 0;
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// For each update
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while (l < N_updates) {
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// C = S^{-1} x U_l
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for (unsigned int i = 0; i < Dim; i++) {
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C[i] = 0;
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for (unsigned int j = 0; j < Dim; j++) {
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C[i] += Slater_inv[i * Dim + j] * Updates[l * Dim + j];
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}
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}
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// Denominator
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double den = 1 + C[Updates_index[l] - 1];
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if (std::fabs(den) < threshold()) {
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#ifdef DEBUG1
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std::cerr << "Breakdown condition triggered at " << Updates_index[l]
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<< std::endl;
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std::cerr << "Denominator = " << den << std::endl;
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#endif
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// U_l = U_l / 2 (do the split)
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for (unsigned int i = 0; i < Dim; i++) {
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later_updates[*later * Dim + i] = Updates[l * Dim + i] / 2.0;
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C[i] /= 2.0;
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}
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later_index[*later] = Updates_index[l];
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(*later)++;
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den = 1 + C[Updates_index[l] - 1];
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}
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double iden = 1 / den;
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// D = v^T x S^{-1}
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for (unsigned int j = 0; j < Dim; j++) {
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D[j] = Slater_inv[(Updates_index[l] - 1) * Dim + j];
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}
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// S^{-1} = S^{-1} - C x D / den
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for (unsigned int i = 0; i < Dim; i++) {
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for (unsigned int j = 0; j < Dim; j++) {
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double update = C[i] * D[j] * iden;
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Slater_inv[i * Dim + j] -= update;
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}
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}
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l += 1;
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}
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}
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// Sherman-Morrison-Woodbury kernel 2
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// woodbury_2, sherman_morrison_splitting mixing scheme
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void sherman_morrison_woodbury_2(double *Slater_inv, const unsigned int Dim,
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const unsigned int N_updates, double *Updates,
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unsigned int *Updates_index) {
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#ifdef DEBUG2
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std::cerr << "Called Sherman-Morrison-Woodbury kernel 1 with " << N_updates
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<< " updates" << std::endl;
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showMatrix2(Updates_index, 1, N_updates, "Updates_index");
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showMatrix2(Updates, N_updates, Dim, "Updates");
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#endif
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unsigned int n_of_2blocks = N_updates / 2;
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unsigned int remainder = N_updates % 2;
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unsigned int length_2block = 2 * Dim;
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unsigned int length_1block = 1 * Dim;
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// Apply first 2*n_of_2blocks updates in n_of_2blocks blocks of 2 updates with
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// Woodbury 2x2 kernel
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double later_updates[Dim * N_updates];
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unsigned int later_index[N_updates];
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unsigned int later = 0;
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if (n_of_2blocks > 0) {
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for (unsigned int i = 0; i < n_of_2blocks; i++) {
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double *Updates_2block = &Updates[i * length_2block];
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unsigned int *Updates_index_2block = &Updates_index[i * 2];
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bool ok;
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ok = woodbury_2(Slater_inv, Dim, Updates_2block, Updates_index_2block);
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if (!ok) { // Send the entire block to sherman_morrison_splitting
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#ifdef DEBUG1
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std::cerr << "Woodbury 2x2 kernel failed! Sending block to sherman_morrison_splitting"
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<< std::endl;
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#endif
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#ifdef DEBUG2
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showMatrix2(Updates_2block, 2, Dim, "Updates_2block");
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showMatrix2(Updates_index_2block, 1, 2, "Updates_index_2block");
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#endif
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unsigned int l = 0;
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sherman_morrison_splitting(Slater_inv, Dim, 2, Updates_2block, Updates_index_2block,
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later_updates + (Dim * later), later_index + later, &l);
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later = later + l;
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}
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}
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}
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if (remainder == 1) { // Apply last remaining update with sherman_morrison_splitting
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double *Updates_1block = &Updates[n_of_2blocks * length_2block];
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unsigned int *Updates_index_1block = &Updates_index[2 * n_of_2blocks];
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unsigned int l = 0;
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sherman_morrison_splitting(Slater_inv, Dim, 1, Updates_1block, Updates_index_1block,
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later_updates + (Dim * later), later_index + later, &l);
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later = later + l;
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}
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if (later > 0) {
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sherman_morrison_splitting_caller(Slater_inv, Dim, later, later_updates, later_index);
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}
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}
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// Sherman-Morrison-Woodbury kernel 3
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// woodbury_2, woodbury_3, sherman_morrison_splitting mixing scheme
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void sherman_morrison_woodbury_3(double *Slater_inv, const unsigned int Dim,
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const unsigned int N_updates, double *Updates,
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unsigned int *Updates_index) {
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#ifdef DEBUG2
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std::cerr << "Called Sherman-Morrison-Woodbury kernel 1 with " << N_updates
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<< " updates" << std::endl;
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showMatrix2(Updates_index, 1, N_updates, "Updates_index");
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showMatrix2(Updates, N_updates, Dim, "Updates");
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#endif
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unsigned int n_of_3blocks = N_updates / 3;
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unsigned int remainder = N_updates % 3;
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unsigned int length_3block = 3 * Dim;
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unsigned int length_2block = 2 * Dim;
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unsigned int length_1block = 1 * Dim;
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// Apply first 3*n_of_3blocks updates in n_of_3blocks blocks of 3 updates with
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// Woodbury 3x3 kernel
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double later_updates[Dim * N_updates];
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unsigned int later_index[N_updates];
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unsigned int later = 0;
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if (n_of_3blocks > 0) {
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for (unsigned int i = 0; i < n_of_3blocks; i++) {
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double *Updates_3block = &Updates[i * length_3block];
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unsigned int *Updates_index_3block = &Updates_index[i * 3];
|
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bool ok;
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ok = woodbury_3(Slater_inv, Dim, Updates_3block, Updates_index_3block);
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if (!ok) { // Send the entire block to sherman_morrison_splitting
|
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#ifdef DEBUG2
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|
std::cerr << "Woodbury 3x3 kernel failed! Sending block to sherman_morrison_splitting"
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|
<< std::endl;
|
|
showMatrix2(Updates_3block, 3, Dim, "Updates_3block");
|
|
showMatrix2(Updates_index_3block, 1, 3, "Updates_index_3block");
|
|
#endif
|
|
unsigned int l = 0;
|
|
sherman_morrison_splitting(Slater_inv, Dim, 3, Updates_3block, Updates_index_3block,
|
|
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
|
|
double *Updates_2block = &Updates[n_of_3blocks * length_3block];
|
|
unsigned int *Updates_index_2block = &Updates_index[3 * n_of_3blocks];
|
|
bool ok;
|
|
ok = woodbury_2(Slater_inv, Dim, Updates_2block, Updates_index_2block);
|
|
if (!ok) { // Send the entire block to sherman_morrison_splitting
|
|
#ifdef DEBUG2
|
|
std::cerr << "Woodbury 2x2 kernel failed! Sending block to sherman_morrison_splitting"
|
|
<< std::endl;
|
|
#endif
|
|
unsigned int l = 0;
|
|
sherman_morrison_splitting(Slater_inv, Dim, 2, Updates_2block, Updates_index_2block,
|
|
later_updates + (Dim * later), later_index + later, &l);
|
|
later = later + l;
|
|
}
|
|
}
|
|
else if (remainder == 1) { // Apply last remaining update with sherman_morrison_splitting
|
|
double *Updates_1block = &Updates[n_of_3blocks * length_3block];
|
|
unsigned int *Updates_index_1block = &Updates_index[3 * n_of_3blocks];
|
|
unsigned int l = 0;
|
|
sherman_morrison_splitting(Slater_inv, Dim, 1, Updates_1block, Updates_index_1block,
|
|
later_updates + (Dim * later), later_index + later, &l);
|
|
later = later + l;
|
|
}
|
|
|
|
if (later > 0) {
|
|
sherman_morrison_splitting_caller(Slater_inv, Dim, later, later_updates, later_index);
|
|
}
|
|
}
|
|
|
|
|
|
|
|
extern "C" {
|
|
void sherman_morrison_f(double **linSlater_inv, unsigned int *Dim, unsigned int *N_updates,
|
|
double **linUpdates, unsigned int **Updates_index) {
|
|
sherman_morrison(*linSlater_inv, *Dim, *N_updates, *linUpdates, *Updates_index);
|
|
}
|
|
|
|
bool woodbury_2_f(double **linSlater_inv, unsigned int *Dim, double **linUpdates,
|
|
unsigned int **Updates_index) {
|
|
bool ok;
|
|
ok = woodbury_2(*linSlater_inv, *Dim, *linUpdates, *Updates_index);
|
|
return ok;
|
|
}
|
|
|
|
bool woodbury_3_f(double **linSlater_inv, unsigned int *Dim, double **linUpdates,
|
|
unsigned int **Updates_index) {
|
|
bool ok;
|
|
ok = woodbury_3(*linSlater_inv, *Dim, *linUpdates, *Updates_index);
|
|
return ok;
|
|
}
|
|
|
|
void sherman_morrison_splitting_caller_f(double **linSlater_inv, unsigned int *Dim, unsigned int *N_updates,
|
|
double **linUpdates, unsigned int **Updates_index) {
|
|
sherman_morrison_splitting_caller(*linSlater_inv, *Dim, *N_updates, *linUpdates, *Updates_index);
|
|
}
|
|
|
|
void sherman_morrison_woodbury_3_f(double **linSlater_inv, unsigned int *Dim, unsigned int *N_updates,
|
|
double **linUpdates, unsigned int **Updates_index) {
|
|
sherman_morrison_woodbury_3(*linSlater_inv, *Dim, *N_updates, *linUpdates, *Updates_index);
|
|
}
|
|
|
|
void sherman_morrison_woodbury_2_f(double **linSlater_inv, unsigned int *Dim, unsigned int *N_updates,
|
|
double **linUpdates, unsigned int **Updates_index) {
|
|
sherman_morrison_woodbury_2(*linSlater_inv, *Dim, *N_updates, *linUpdates, *Updates_index);
|
|
}
|
|
}
|