# Determinants for Continuous-Time Monte-Carlo¶

TRIQS comes with a class called det_manip to easily perform operations on a special type of matrices (see here). This library, among others, allows to easily add or remove lines or columns to the matrix, to calculate the determinant and the inverse. Here are a couple of simple examples showing the basic use of this class.

## Creation of an empty det_manip class¶

#include <triqs/det_manip/det_manip.hpp>

struct fun {

typedef double result_type;
typedef double argument_type;

// gives the coefficients of the matrix (function F of the documentation)
double operator()(double x, double y) const { return (x - y); }
};

int main() {

fun f;
int init_size = 100; // maximum size of the matrix before a resize

// creation of a class det_manip
triqs::det_manip::det_manip<fun> D(f, init_size);

// the initial matrix is empty:
std::cout << std::endl << "After construction: D.matrix()=" << D.matrix() << std::endl << std::endl;
}


## Creation of a non empty det_manip class¶

#include <triqs/det_manip/det_manip.hpp>

struct fun {

typedef double result_type;
typedef double argument_type;

// gives the coefficients of the matrix (function F of the documentation)
double operator()(double x, double y) const { return (x - y); }
};

int main() {

fun f;
std::vector<double> initial_x{1, 2}, initial_y{3, 4};

// creation of a class det_manip with a 2 by 2 matrix
triqs::det_manip::det_manip<fun> D(f, initial_x, initial_y);

// the initial matrix:
std::cout << std::endl << "After construction: D.matrix()=" << D.matrix() << std::endl << std::endl;
}


## Get information about a det_manip class¶

#include <triqs/det_manip/det_manip.hpp>

struct fun {
typedef double result_type;
typedef double argument_type;
double operator()(double x, double y) const { return (x - y); }
};

int main() {
fun f;
int i = 0, j = 1;
std::vector<double> initial_x{1, 2}, initial_y{3, 4};
triqs::det_manip::det_manip<fun> D(f, initial_x, initial_y);
std::cout << std::endl << "D.matrix()=" << D.matrix() << std::endl << std::endl;
std::cout << "The size of the matrix is " << D.size() << std::endl << std::endl;
std::cout << "The determinant is " << D.determinant() << std::endl << std::endl;
std::cout << "The inverse matrix is" << D.inverse_matrix() << std::endl << std::endl;
std::cout << "The value of the parameters for coefficient (i,j)=(" << i << "," << j << ") is (x,y)=(" << D.get_x(i) << "," << D.get_y(j) << ")"
<< std::endl
<< std::endl;
}


## Add a line and a column¶

#include <triqs/det_manip/det_manip.hpp>

struct fun {
typedef double result_type;
typedef double argument_type;
double operator()(double x, double y) const { return (exp(x) - y * y); }
};

int main() {
triqs::det_manip::det_manip<fun> D(fun(), std::vector<double>{1, 2, 2.5}, std::vector<double>{3, 4, 9});
std::cout << std::endl << "After construction, D.matrix()=" << D.matrix() << std::endl << std::endl;
double x0 = 2.1, y0 = 7;
int i = 2, j = 0; // number of the added line and column
std::cout << "We want to add a line and a column for i=" << i << ", j=" << j << ", x=" << x0 << ", y=" << y0 << "." << std::endl;
// (try of) insertion of a line and a column at position (3,1) in the matrix
// with x[i]=x0, y[j]=y0.
double detratio = D.try_insert(i, j, x0, y0); // the ratio between new and old determinants
// while the operation is not complete, the matrix stays unchanged
std::cout << "After try_insert, D.matrix()=" << D.matrix() << std::endl;
// here we validate the insertion: the (inverse) matrix and determinant are updated
D.complete_operation();
std::cout << "After complete_operation, D.matrix()=" << D.matrix() << std::endl << std::endl;
}


## Remove a line and a column¶

#include <triqs/det_manip/det_manip.hpp>

struct fun {
typedef double result_type;
typedef double argument_type;
double operator()(double x, double y) const { return (exp(x) - y * y); }
};

int main() {
triqs::det_manip::det_manip<fun> D(fun(), std::vector<double>{1, 2, 2.5}, std::vector<double>{3, 4, 9});
std::cout << std::endl << "After construction, D.matrix()=" << D.matrix() << std::endl << std::endl;
int i = 1, j = 0; // number of the removed line and column
std::cout << "We want to remove a line and a column for i=" << i << ", j=" << j << "." << std::endl;
// (try of) removal of a line and a column at position (1,0) in the matrix.
double detratio = D.try_remove(i, j); // the ratio between new and old determinants
// while the operation is not complete, the matrix stays unchanged
std::cout << "After try_remove, D.matrix()=" << D.matrix() << std::endl;
// here we validate the removal: the (inverse) matrix and determinant are updated
D.complete_operation();
std::cout << "After complete_operation, D.matrix()=" << D.matrix() << std::endl << std::endl;
}


## Add two lines and two columns¶

#include <triqs/det_manip/det_manip.hpp>

struct fun {
typedef double result_type;
typedef double argument_type;
double operator()(double x, double y) const { return (exp(x) - y * y); }
};

int main() {
triqs::det_manip::det_manip<fun> D(fun(), std::vector<double>{1, 2, 2.5}, std::vector<double>{3, 4, 9});
std::cout << std::endl << "After construction, D.matrix()=" << D.matrix() << std::endl << std::endl;
double x0 = 2.1, y0 = 7, x1 = 3.5, y1 = 5;
int i0 = 2, i1 = 1, j0 = 0, j1 = 3; // number of the added lines and columns
std::cout << "We want to add a line and a column for i0=" << i0 << ", j0=" << j0 << ", i1=" << i1 << ", j1=" << j1 << ", x0=" << x0 << ", y0=" << y0
<< ", x1=" << x1 << ", y1=" << y1 << ")." << std::endl;
// (try of) insertion of 2 lines and 2 columns in the matrix
double detratio = D.try_insert2(i0, i1, j0, j1, x0, x1, y0, y1); // the ratio between new and old determinants
// while the operation is not complete, the matrix stays unchanged
std::cout << "After try_insert2, D.matrix()=" << D.matrix() << std::endl;
// here we validate the insertion: the (inverse) matrix and determinant are updated
D.complete_operation();
std::cout << "After complete_operation, D.matrix()=" << D.matrix() << std::endl << std::endl;
}


## Remove two lines and two columns¶

#include <triqs/det_manip/det_manip.hpp>

struct fun {
typedef double result_type;
typedef double argument_type;
double operator()(double x, double y) const { return (exp(x) - y * y); }
};

int main() {
triqs::det_manip::det_manip<fun> D(fun(), std::vector<double>{1, 2, 2.5}, std::vector<double>{3, 4, 9});
std::cout << std::endl << "After construction, D.matrix()=" << D.matrix() << std::endl << std::endl;
int i0 = 2, i1 = 1, j0 = 0, j1 = 1; // number of the removed lines and columns
std::cout << "We want to remove 2 lines and 2 columns for i0=" << i0 << ", j0=" << j0 << ", i1=" << i1 << ", j1=" << j1 << "." << std::endl;
// (try of) removal of a line and a column at position (1,0) in the matrix.
double detratio = D.try_remove2(i0, i1, j0, j1); // the ratio between new and old determinants
// while the operation is not complete, the matrix stays unchanged
std::cout << "After try_remove2, D.matrix()=" << D.matrix() << std::endl;
// here we validate the removal: the (inverse) matrix and determinant are updated
D.complete_operation();
std::cout << "After complete_operation, D.matrix()=" << D.matrix() << std::endl << std::endl;
}


## Remove/add the one/two last lines and columns¶

#include <triqs/det_manip/det_manip.hpp>

struct fun {
typedef double result_type;
typedef double argument_type;
double operator()(double x, double y) const { return (exp(x) - y * y); }
};

int main() {
double detratio;
triqs::det_manip::det_manip<fun> D(fun(), std::vector<double>{1, 2, 2.5}, std::vector<double>{3, 4, 9});
std::cout << std::endl << "After construction, D.matrix()=" << D.matrix() << std::endl << std::endl;
int i0 = 2, i1 = 1, j0 = 0, j1 = 1; // number of the removed lines and columns
std::cout << "We want to add two lines and columns at the end." << std::endl;
// (try of) removal of a line and a column at position (1,0) in the matrix.
detratio = D.insert2_at_end(2.3, 5.4, 4.6, 7.5); // the ratio between new and old determinants
std::cout << "After insert2_at_end, D.matrix()=" << D.matrix() << std::endl << std::endl;
std::cout << "We want to remove two lines and columns at the end." << std::endl;
detratio = D.remove2_at_end(); // the ratio between new and old determinants
std::cout << "After remove2_at_end, D.matrix()=" << D.matrix() << std::endl << std::endl;
std::cout << "We want to add one line and column at the end." << std::endl;
detratio = D.insert_at_end(1.1, 3.3); // the ratio between new and old determinants
std::cout << "After remove_at_end, D.matrix()=" << D.matrix() << std::endl << std::endl;
std::cout << "We want to remove one line and column at the end." << std::endl;
detratio = D.remove_at_end(); // the ratio between new and old determinants
std::cout << "After remove_at_end, D.matrix()=" << D.matrix() << std::endl;
}


## Replace one line and column¶

#include <triqs/det_manip/det_manip.hpp>

struct fun {
typedef double result_type;
typedef double argument_type;
double operator()(double x, double y) const { return (exp(x) - y * y); }
};

int main() {

triqs::det_manip::det_manip<fun> D(fun(), std::vector<double>{1, 2, 2.5}, std::vector<double>{3, 4, 9});
std::cout << std::endl << "After construction, D.matrix()=" << D.matrix() << std::endl << std::endl;
std::cout << "We want to change a row and a column." << std::endl;
D.change_one_row_and_one_col(0, 1, 2.1, 1.2);
std::cout << "After change_one_row_and_one_col(0,1, 2.1, 1.2), D.matrix()=" << D.matrix() << std::endl << std::endl;

triqs::det_manip::det_manip<fun> D2(fun(), std::vector<double>{2, 2.5}, std::vector<double>{4, 9});
std::cout << std::endl << "After construction, D2.matrix()=" << D2.matrix() << std::endl << std::endl;
std::cout << "We want to change a row and a column." << std::endl;
D2.change_one_row_and_one_col(0, 1, 2.1, 1.2);
std::cout << "After change_one_row_and_one_col(0,1, 2.1, 1.2), D2.matrix()=" << D2.matrix() << std::endl << std::endl;
}