som: Main Python module of SOM

som.Som

class som.Som(rhs: Gf, errors: Gf, kind: str = 'FermionGf', norms=None, *, filtering_levels=None)

Implementation of the Stochastic Optimization Method.

__init__(rhs: Gf, errors: Gf, kind: str = 'FermionGf', norms=None, *, filtering_levels=None)

Parameters:

• rhs (triqs.gf.gf.Gf) – Right hand side of the integral equation to be solved, defined on triqs.gf.meshes.MeshImTime, triqs.gf.meshes.MeshImFreq or triqs.gf.meshes.MeshLegendre. The target shape of rhs must be \(M{\times}M\). If \(M>1\), only its diagonal matrix elements will be considered and used as input data for \(M\) independent continuation problems.

• errors – Either error bars \(\sigma_n\) (GF container of the same type and target shape as rhs) or a packed list of covariance matrices (Gf(mesh=MeshProduct(rhs.mesh, rhs.mesh), target_shape=[M]) with each target element corresponding to one covariance matrix).

• kind (str, optional) – Selection of the observable kind (and its respective integral kernel) to be used. One of FermionGf, FermionGfSymm, BosonCorr, BosonAutoCorr, ZeroTemp. Defaults to FermionGf.

• norms (float or list [float]) –

  Requested solution norms either as a list of \(M\) real numbers or as a single real number for all \(M\) continuation problems. For observable kinds FermionGf, FermionGfSymm and ZeroTemp this parameter is optional and defaults to 1.0. For BosonCorr and BosonAutoCorr it must be provided by the user.

  See also

  When unknown a priori in the latter case, the norms can be approximately estimated from rhs using estimate_boson_corr_spectrum_norms().

• filtering_levels (float or list [float], optional) – Filtering levels for covariance matrices either as a list of \(M\) real numbers or as a single real number for all \(M\) continuation problems. Can only be specified when the covariance matrices are used. Defaults to 0.

accumulate(**kwargs)

Accumulate particular solutions. This method can be called multiple times to accumulate the particular solutions incrementally.

Parameters:

**kwargs – See below.

Main parameters

Parameter Name	Type	Default	Documentation
energy_window	(float,float)	–	Estimated lower and upper bounds of the spectrum.
max_time	int	-1 = infinite	Maximum runtime in seconds, use -1 to set infinite.
verbosity	int	2 on MPI rank 0, 0 otherwise.	Verbosity level (max level - 3).
t	int	50	Number of elementary updates per global update (\(T\)). Bigger values make the algorithm more ergodic.
cc_update	bool	False	Enable Consistent Constraints updates.
f	int	100	Number of global updates (\(F\)). Bigger values make the algorithm more ergodic.
l	int	2000	Number of particular solutions to accumulate (\(L\)); ignored if adjust_l=True. Bigger values reduce noise in the final solution / make it smoother.
adjust_l	bool	False	Automatically adjust the number of particular solutions to accumulate (\(L\)).
make_histograms	bool	False	Accumulate histograms of \(\chi\), where \(\chi^2\) is the objective function.

Fine tuning options

Parameter Name	Type	Default	Documentation
random_seed	int	34788 + 928374 * MPI.rank	Seed for random number generator.
random_name	str	“”	Name of random number generator (MT19937 by default).
max_rects	int	60	Maximum number of rectangles in a particular solution (\(K_{max}\)).
min_rect_width	float	1e-3	Minimal width of a rectangle, in units of the energy window width.
min_rect_weight	float	1e-3	Minimal weight of a rectangle, in units of the requested solution norm.
t1	int	-1	Number of elementary updates in the first stage of a global update. When set to -1, the number of elementary updates will be chosen randomly for each global update from the \([1; T[\) range.
distrib_d_max	float	2.0	Maximal parameter of the power-law distribution function for the Metropolis algorithm.
gamma	float	2.0	Proposal probability parameter \(\gamma\).
cc_update_cycle_length	int	10	CC update: Number of proposed elementary updates between two successive CC updates (only during the first stage of a global update).
cc_update_max_iter	int	30	CC update: Maximum allowed number of height adjustment iterations.
cc_update_rect_norm_variation_tol	float	1e-3	CC update: The height adjustment procedure stops when variation of every rectangle’s norm between two consecutive iterations is below this value. This parameter is measured in units of the requested solution norm.
cc_update_height_penalty_max	float	1e3	CC update: Maximum value of the regularization parameters \(Q_0(k)\) that penalize negative heights. Measured in units of (energy window width) / (solution norm).
cc_update_height_penalty_divisor	float	10.0	CC update: Divisor used to reduce the regularization parameters \(Q_0(k)\) that penalize negative heights.
cc_update_der_penalty_init	float	1.0	CC update: Initial value of the regularization parameters \(Q_1(k)\) and \(Q_2(k)\) that penalize large derivatives of a solution. Measured in units of (energy window width)^2 / (solution norm) for \(Q_1(k)\) and in units of (energy window width)^3 / (solution norm) for \(Q_2(k)\).
cc_update_der_penalty_threshold	float	0.1	CC update: Sets the threshold value of the products \(|Q_1(k) A'(k)|\) and \(|Q_2(k) A''(k)|\), above which derivative regularization parameters \(Q_1(k)\) and \(Q_2(k)\) need to be reduced.
cc_update_der_penalty_increase_coeff	float	2.0	CC update: Coefficient used to increase the regularization parameters \(Q_1(k)\) and \(Q_2(k)\) that penalize large derivatives of a solution.
cc_update_der_penalty_limiter	float	1e3	CC update: Coefficient that limits growth of the regularization parameters \(Q_1(k)\) and \(Q_2(k)\) penalizing large derivatives of a solution.
adjust_l_range	(int,int)	(100,2000)	Automatic \(L\) adjustment: Search range for the number of particular solutions to accumulate.
adjust_l_good_chi	float	2.0	Automatic \(L\) adjustment: Maximal ratio \(\chi/\chi_\mathrm{min}\) for a particular solution to be considered good.
adjust_l_verygood_chi	float	4/3	Automatic \(L\) adjustment: Maximal ratio \(\chi/\chi_\mathrm{min}\) for a particular solution to be considered very good.
adjust_l_ratio	float	0.95	Automatic \(L\) adjustment: Critical ratio \(N_\mathrm{very good}/N_\mathrm{good}\) to stop the \(L\)-adjustment procedure.
hist_max	float	2.0	Right boundary of the histograms, in units of \(\chi_\mathrm{min}\) (left boundary is always set to \(\chi_\mathrm{min}\)).
hist_n_bins	int	100	Number of bins for the histograms.

last_accumulate_parameters

Set of parameters used in the last call to accumulate().

Type:

dict [str, object]

accumulate_status

Status code of accumulate() on exit.

• 0, if the accumulation has run until the end.

• 1, if it has run out of time.

• 2, if it has been stopped by receiving a signal.

Type:

int

adjust_f(**kwrags)

Automatically adjust the number of global updates \(F\).

This function scans a range of \(F\) (f_range), accumulates l particular solutions for each \(F\) and evaluates the following functional for each solution,

\[\kappa = \frac{1}{N-1}\sum_{n=2}^N \frac{1}{2}\left[1 - \frac{\Re[\Delta(n)\Delta^*(n-1)]}{|\Delta(n)\Delta^*(n-1)|} \right],\]

where \(\Delta(n)\) are discrepancies

\[\Delta(n) = \int_{\epsilon_\mathrm{min}}^{\epsilon_\mathrm{max}} K(n, \epsilon) A(\epsilon) d\epsilon - G(n).\]

\(\kappa\) characterizes anti-correlation between adjacent discrepancies and can serve as a measure of ergodicity in Markov chains. If \(\kappa\) exceeds the value of parameter kappa for at least a half of accumulated solutions, \(F\) is considered sufficient.

Deprecated since version 2.0: The automatic adjustment algorithm does not support continuation problems with full covariance matrices. If the Som object has been constructed using a covariance matrix, adjust_f() will raise a RuntimeError.

Parameters:

**kwargs – See below.

Returns:

Selected value of \(F\).

Return type:

int

Main parameters

Parameter Name	Type	Default	Documentation
energy_window	(float,float)	–	Estimated lower and upper bounds of the spectrum.
max_time	int	-1 = infinite	Maximum runtime in seconds, use -1 to set infinite.
verbosity	int	2 on MPI rank 0, 0 otherwise.	Verbosity level (max level - 3).
t	int	50	Number of elementary updates per global update (\(T\)). Bigger values make the algorithm more ergodic.
cc_update	bool	False	Enable Consistent Constraints updates.
f_range	(int,int)	(100,5000)	Search range for the number of global updates.

Fine tuning options

Parameter Name	Type	Default	Documentation
random_seed	int	34788 + 928374 * MPI.rank	Seed for random number generator.
random_name	str	“”	Name of random number generator (MT19937 by default).
max_rects	int	60	Maximum number of rectangles in a particular solution (\(K_{max}\)).
min_rect_width	float	1e-3	Minimal width of a rectangle, in units of the energy window width.
min_rect_weight	float	1e-3	Minimal weight of a rectangle, in units of the requested solution norm.
t1	int	-1	Number of elementary updates in the first stage of a global update. When set to -1, the number of elementary updates will be chosen randomly for each global update from the \([1; T[\) range.
distrib_d_max	float	2.0	Maximal parameter of the power-law distribution function for the Metropolis algorithm.
gamma	float	2.0	Proposal probability parameter \(\gamma\).
cc_update_cycle_length	int	10	CC update: Number of proposed elementary updates between two successive CC updates (only during the first stage of a global update).
cc_update_max_iter	int	30	CC update: Maximum allowed number of height adjustment iterations.
cc_update_rect_norm_variation_tol	float	1e-3	CC update: The height adjustment procedure stops when variation of every rectangle’s norm between two consecutive iterations is below this value. This parameter is measured in units of the requested solution norm.
cc_update_height_penalty_max	float	1e3	CC update: Maximum value of the regularization parameters \(Q_0(k)\) that penalize negative heights. Measured in units of (energy window width) / (solution norm).
cc_update_height_penalty_divisor	float	10.0	CC update: Divisor used to reduce the regularization parameters \(Q_0(k)\) that penalize negative heights.
cc_update_der_penalty_init	float	1.0	CC update: Initial value of the regularization parameters \(Q_1(k)\) and \(Q_2(k)\) that penalize large derivatives of a solution. Measured in units of (energy window width)^2 / (solution norm) for \(Q_1(k)\) and in units of (energy window width)^3 / (solution norm) for \(Q_2(k)\).
cc_update_der_penalty_threshold	float	0.1	CC update: Sets the threshold value of the products \(|Q_1(k) A'(k)|\) and \(|Q_2(k) A''(k)|\), above which derivative regularization parameters \(Q_1(k)\) and \(Q_2(k)\) need to be reduced.
cc_update_der_penalty_increase_coeff	float	2.0	CC update: Coefficient used to increase the regularization parameters \(Q_1(k)\) and \(Q_2(k)\) that penalize large derivatives of a solution.
cc_update_der_penalty_limiter	float	1e3	CC update: Coefficient that limits growth of the regularization parameters \(Q_1(k)\) and \(Q_2(k)\) penalizing large derivatives of a solution.
l	int	20	Number of particular solutions used to adjust \(F\).
kappa	float	0.25	Limiting value of \(\kappa\) used to adjust \(F\).

compute_final_solution(good_chi_rel: float = 2.0, good_chi_abs: float = cmath.inf, verbosity: int = 0)

Signature : (float good_chi_rel = 2.0, float good_chi_abs = HUGE_VAL, int verbosity = 0) -> list[float]

Select particular solutions according to the standard SOM criterion and compute the final solution.

Selected solutions must satisfy \(\chi[A_j] \leq \chi_c^\mathrm{abs}\) and \(\chi[A_j] \leq \min_{j'}(\chi[A_{j'}]) \chi_c^\mathrm{rel}\).

Parameters:

• good_chi_rel (float) – Solution selection parameter \(\chi_c^\mathrm{rel}\).

• good_chi_abs (float) – Solution selection parameter \(\chi_c^\mathrm{abs}\).

• verbosity (int) – When positive, print extra information.

Returns:

A length-\(M\) list of values of \(\chi^2\), one element per continuation problem.

Return type:

list [float]

compute_final_solution_cc(**kwargs)

Select particular solutions and compute the final solution using the SOCC protocol.

Selected solutions must satisfy \(\chi[A_j] \leq \chi_c^\mathrm{abs}\) and \(\chi[A_j] \leq \min_{j'}(\chi[A_{j'}]) \chi_c^\mathrm{rel}\).

Parameters:

**kwargs – See below.

Returns:

A length-\(M\) list of values of \(\chi^2\), one element per continuation problem.

Return type:

list [float]

Main parameters

Parameter Name	Type	Default	Documentation
refreq_mesh	MeshReFreq or real 1D array_like	–	Grid of energy points \(\epsilon_k\) used in CC regularization procedure. Either a TRIQS real-frequency mesh object or a strictly ordered list of points.
verbosity	int	1 on MPI rank 0, 0 otherwise.	Verbosity level (max level - 2).
good_chi_rel	float	2.0	Maximal ratio \(\chi/\chi_\mathrm{min}\) for a particular solution to be selected. This criterion must be fulfilled together with the one set by good_chi_abs.
good_chi_abs	float	infinity	Maximal value of \(\chi\) for a particular solution to be selected. This criterion must be fulfilled together with the one set by good_chi_rel.
default_model	Real 1D array	[]	Optional default model of the spectral function evaluated at energy points of refreq_mesh (\(A_T(\epsilon_k)\)).
default_model_weights	Real 1D array	[]	Weights determining how much deviations from default_model are penalized at each energy point (\(T_k\)).
max_iter	int	50	Maximum allowed number of parameter adjustment iterations.
max_sum_abs_c	float	2.0	Stop parameter adjustment iterations when expansion coefficients \(c_j\) make the sum \(\sum_j |c_j|\) exceed this value.

Fine tuning options

Parameter Name	Type	Default	Documentation
ew_penalty_coeff	float	1.0	Coefficient of the term that penalizes large deviations from the equal-weight superposition.
amp_penalty_max	float	1e3	Maximum value of the regularization parameters that penalize negative values of the spectral function (\(\mathcal{Q}_k\)).
amp_penalty_divisor	float	10.0	Divisor used to reduce the regularization parameters that penalize negative values of the spectral function (\(\mathcal{Q}_k\)).
der_penalty_init	float	0.1	Initial value of the regularization parameters that penalize large derivatives of the solution (\(\mathcal{D}\)). Before this parameter is used, it is divided by the number of selected particular solutions.
der_penalty_coeff	float	2.0	Coefficient used to increase the regularization parameters that penalize large derivatives of the solution (\(f\)).
monitor	function	None	Monitor function called at each parameter adjustment iteration. It takes 4 arguments, Current list of expansion coefficients \(c_j\); Amplitudes of the spectrum and respective regularization parameters as a \((A_k, Q_k)\) pair; Derivatives of the spectrum and respective regularization parameters as a \((A'_k, D_k)\) pair; Second derivatives of the spectrum and respective regularization parameters as a \((A''_k, B_k)\) pair. Returning True from the function stops the adjustment procedure.

run(**kwargs)

Accumulate particular solutions and compute the final solution using the standard SOM criterion.

Calling this method is equivalent to calling accumulate() and compute_final_solution() with good_chi_rel=adjust_l_good_chi.

Deprecated since version 2.0: Use accumulate() and compute_final_solution() instead.

Parameters:

**kwargs – See below.

Main parameters

Parameter Name	Type	Default	Documentation
energy_window	(float,float)	–	Estimated lower and upper bounds of the spectrum.
max_time	int	-1 = infinite	Maximum runtime in seconds, use -1 to set infinite.
verbosity	int	2 on MPI rank 0, 0 otherwise.	Verbosity level (max level - 3).
t	int	50	Number of elementary updates per global update (\(T\)). Bigger values make the algorithm more ergodic.
cc_update	bool	False	Enable Consistent Constraints updates.
f	int	100	Number of global updates (\(F\)). Bigger values make the algorithm more ergodic.
l	int	2000	Number of particular solutions to accumulate (\(L\)); ignored if adjust_l=True. Bigger values reduce noise in the final solution / make it smoother.
adjust_l	bool	False	Automatically adjust the number of particular solutions to accumulate (\(L\)).
make_histograms	bool	False	Accumulate histograms of \(\chi\), where \(\chi^2\) is the objective function.

Fine tuning options

Parameter Name	Type	Default	Documentation
random_seed	int	34788 + 928374 * MPI.rank	Seed for random number generator.
random_name	str	“”	Name of random number generator (MT19937 by default).
max_rects	int	60	Maximum number of rectangles in a particular solution (\(K_{max}\)).
min_rect_width	float	1e-3	Minimal width of a rectangle, in units of the energy window width.
min_rect_weight	float	1e-3	Minimal weight of a rectangle, in units of the requested solution norm.
t1	int	-1	Number of elementary updates in the first stage of a global update. When set to -1, the number of elementary updates will be chosen randomly for each global update from the \([1; T[\) range.
distrib_d_max	float	2.0	Maximal parameter of the power-law distribution function for the Metropolis algorithm.
gamma	float	2.0	Proposal probability parameter \(\gamma\).
cc_update_cycle_length	int	10	CC update: Number of proposed elementary updates between two successive CC updates (only during the first stage of a global update).
cc_update_max_iter	int	30	CC update: Maximum allowed number of height adjustment iterations.
cc_update_rect_norm_variation_tol	float	1e-3	CC update: The height adjustment procedure stops when variation of every rectangle’s norm between two consecutive iterations is below this value. This parameter is measured in units of the requested solution norm.
cc_update_height_penalty_max	float	1e3	CC update: Maximum value of the regularization parameters \(Q_0(k)\) that penalize negative heights. Measured in units of (energy window width) / (solution norm).
cc_update_height_penalty_divisor	float	10.0	CC update: Divisor used to reduce the regularization parameters \(Q_0(k)\) that penalize negative heights.
cc_update_der_penalty_init	float	1.0	CC update: Initial value of the regularization parameters \(Q_1(k)\) and \(Q_2(k)\) that penalize large derivatives of a solution. Measured in units of (energy window width)^2 / (solution norm) for \(Q_1(k)\) and in units of (energy window width)^3 / (solution norm) for \(Q_2(k)\).
cc_update_der_penalty_threshold	float	0.1	CC update: Sets the threshold value of the products \(|Q_1(k) A'(k)|\) and \(|Q_2(k) A''(k)|\), above which derivative regularization parameters \(Q_1(k)\) and \(Q_2(k)\) need to be reduced.
cc_update_der_penalty_increase_coeff	float	2.0	CC update: Coefficient used to increase the regularization parameters \(Q_1(k)\) and \(Q_2(k)\) that penalize large derivatives of a solution.
cc_update_der_penalty_limiter	float	1e3	CC update: Coefficient that limits growth of the regularization parameters \(Q_1(k)\) and \(Q_2(k)\) penalizing large derivatives of a solution.
adjust_l_range	(int,int)	(100,2000)	Automatic \(L\) adjustment: Search range for the number of particular solutions to accumulate.
adjust_l_good_chi	float	2.0	Automatic \(L\) adjustment: Maximal ratio \(\chi/\chi_\mathrm{min}\) for a particular solution to be considered good.
adjust_l_verygood_chi	float	4/3	Automatic \(L\) adjustment: Maximal ratio \(\chi/\chi_\mathrm{min}\) for a particular solution to be considered very good.
adjust_l_ratio	float	0.95	Automatic \(L\) adjustment: Critical ratio \(N_\mathrm{very good}/N_\mathrm{good}\) to stop the \(L\)-adjustment procedure.
hist_max	float	2.0	Right boundary of the histograms, in units of \(\chi_\mathrm{min}\) (left boundary is always set to \(\chi_\mathrm{min}\)).
hist_n_bins	int	100	Number of bins for the histograms.

clear()

Signature : () -> None

Discard all accumulated particular solutions, histograms and final solutions.

observable_kind

Kind of the physical observable selected upon construction, one of FermionGf, FermionGfSymm, BosonCorr, BosonAutoCorr, ZeroTemp.

Type:

str

dim

Number \(M\) of diagonal matrix elements in the observable.

Type:

int

particular_solutions(i: int)

Signature : (int i) -> list[std::pair<configuration, double>]

Particular solutions and their respective values of the objective function \(\chi^2\) for the i-th diagonal element of the observable accumulated on this MPI rank.

Parameters:

i (int) – Index of the diagonal matrix element of the observable.

Return type:

list [tuple [Configuration, float)]]

objf_min

Minimum of the objective function \(\chi^2\) over all accumulated particular solutions (one value per diagonal matrix element of the observable).

Type:

float

solution(i: int)

Signature : (int i) -> Configuration

Final solution for the i-th diagonal element of the observable.

Parameters:

i (int) – Index of the diagonal matrix element of the observable.

Return type:

Configuration

solutions

Final solutions, one per diagonal element of the observable.

Type:

list [Configuration]

objf(i: int)

Signature : (int i) -> float

Value of the objective function \(\chi^2\) of the final solution for the i-th diagonal element of the observable.

Parameters:

i (int) – Index of the diagonal matrix element of the observable.

Return type:

float

objf_list

Values of the objective function \(\chi^2\) of the final solutions, one value per diagonal element of the observable.

Type:

list [float]

histogram(i: int)

Signature : (int i) -> std::optional<histogram>

Accumulated \(\chi\) histogram for the i-th diagonal element of the observable. Returns None if histograms have not been accumulated.

Parameters:

i (int) – Index of the diagonal matrix element of the observable.

Return type:

triqs.stat.histograms.Histogram or None

histograms

Accumulated \(\chi\) histograms, one per diagonal element of the observable. None if histograms have not been accumulated.

Type:

list [triqs.stat.histograms.Histogram] or None

Free functions

som.fill_refreq()

Signature : (triqs::gfs::gf_view<refreq> g_w, SomCore cont, bool with_binning = true) -> None Fill a Green’s function object with the real-frequency version of the continued observable from a computed SOM solution.

Parameters:

G_w:

triqs.gf.gf.Gf defined on triqs.gf.meshes.MeshReFreq, target Green’s function object.

Cont:

Som, Analytic continuation object.

With_binning:

bool, Use binning while projecting onto the real-frequency mesh.

som.compute_tail()

Signature : (int max_order, SomCore cont) -> nda::array<dcomplex, 3> Extract high-frequency expansion coefficients from a computed SOM solution.

Parameters:

Max_order:

int, Maximum tail expansion order.

Cont:

Som, Analytic continuation object.

Returns: 3D complex NumPy array of shape (max_order+1, cont.dim, cont.dim) with the high-frequency expansion coefficients.

som.reconstruct()

Signature : (triqs::gfs::gf_view<imtime> g, SomCore cont) -> None Reconstruct a physical observable in the imaginary-time representation from a computed SOM solution.

Parameters:

G:

triqs.gf.gf.Gf defined on triqs.gf.meshes.MeshImTime, target Green’s function object.

Cont:

Som, Analytic continuation object.

som.estimate_boson_corr_spectrum_norms(chi: Gf) → List[float]

Given a correlator \(\chi\) of bosonic or boson-like (fermion-number-conserving) operators, estimates the corresponding spectrum normalization constants \(\mathcal{N}\).

Depending on the mesh \(\chi\) is defined on, one of the following expressions is used.

• Imaginary frequencies: \(\mathcal{N} = \pi\chi(i\Omega=0)\).

• Imaginary time: \(\mathcal{N} = \pi\int_0^\beta d\tau \chi(\tau)\).

• Legendre polynomial basis coefficients: \(\mathcal{N} = \pi\chi(\ell=0)\).

Parameters:

chi (triqs.gf.gf.Gf) – The correlator \(\chi\).

Returns:

A list of estimated spectrum normalization constants, one constant per diagonal matrix element of \(\chi\).

Return type:

list [float]

som.count_good_solutions(hist: Histogram, good_chi_rel: float = 2.0, good_chi_abs: float = inf) → int

Given a histogram of values \(\chi\) for the objective function \(\chi^2\), counts the number of such solutions that \(\chi \leq\) good_chi_abs and \(\chi/\chi_\mathrm{min} \leq\) good_chi_rel.

Parameters:

• hist (triqs.stat.histograms.Histogram) – Histogram to analyze.

• good_chi_rel (float, optional) – \(\chi/\chi_\mathrm{min}\) threshold for good solutions.

• good_chi_abs (float, optional) – \(\chi\) threshold for good solutions.

Return type:

int

Auxiliary types

class som.Rectangle

The rectangular function of energy, basis element of a spectral function.

center

float: Center of the rectangle \(c\).

width

float: Width of the rectangle \(w\).

height

float: Height of the rectangle \(h\).

norm

float: Norm (area) of the rectangle \(hw\).

__call__(epsilon: float)

Call self as a function.

class som.Configuration

A SOM configuration (spectral function) as a list of Rectangle’s, and a function of energy, which is the sum of the rectangles.

__len__()

Return len(self).

__getitem__(key, /)

Return self[key].

__iter__()

Implement iter(self).

__call__(epsilon: float)

Call self as a function.