diff --git a/ANN_Data_Generator.py b/ANN_Data_Generator.py
index d821b03ae3f436376547019226cfa576f4d40192..81461f99e435d03ce1a2b653027d2e9820f89d8f 100644
--- a/ANN_Data_Generator.py
+++ b/ANN_Data_Generator.py
@@ -13,7 +13,7 @@ import DG_Approximation
class TrainingDataGenerator(object):
"""Class for training data generator.
- Generate random training data for given initial conditions.
+ Generates random training data for given initial conditions.
Attributes
----------
@@ -26,7 +26,7 @@ class TrainingDataGenerator(object):
Methods
-------
- build_training_data()
+ build_training_data(num_samples)
Builds random training data.
"""
@@ -151,14 +151,14 @@ class TrainingDataGenerator(object):
Number of training data samples to generate.
initial_conditions : list
List of names of initial conditions for training.
- is_smooth : boolean
+ is_smooth : bool
Flag whether initial conditions are smooth.
Returns
-------
- input_data : np.array
+ input_data : ndarray
Array containing input data.
- output_data : np.array
+ output_data : ndarray
Array containing output data.
"""
@@ -215,11 +215,11 @@ class TrainingDataGenerator(object):
Returns
-------
- interval : np.array
+ interval : ndarray
List containing left and right bound of interval.
- stencil : np.array
+ stencil : ndarray
List of cell centers in stencil.
- grid_spacing: float
+ grid_spacing : float
Length of cell in grid.
"""
@@ -244,16 +244,16 @@ class TrainingDataGenerator(object):
@staticmethod
def _normalize_data(input_data):
- """Normalize data.
+ """Normalizes data.
Parameters
----------
- input_data : np.array
+ input_data : ndarray
Array containing input data.
Returns
-------
- np.array
+ ndarray
Array containing normalized input data.
"""
diff --git a/ANN_Model.py b/ANN_Model.py
index 350eab2d1e1047c6138b74a8055040bea9221cd7..e0140b520eee4fd365b7a61798ce16ccfe1845ba 100644
--- a/ANN_Model.py
+++ b/ANN_Model.py
@@ -17,17 +17,24 @@ class ThreeLayerReLu(torch.nn.Module):
Attributes
----------
- name: str
+ name : str
String containing name of model.
- input_linear: torch.nn.Module
+ input_linear : torch.nn.Module
Linear input layer.
- middle_linear: torch.nn.Module
+ middle_linear : torch.nn.Module
Linear middle layer.
- output_linear: torch.nn.Module
+ output_linear : torch.nn.Module
Linear output layer.
- output_layer: torch.nn.Module
+ output_layer : torch.nn.Module
Activation layer for output calculation.
+ Methods
+ -------
+ forward(input_data)
+ Executes forward propagation.
+ get_name()
+ Returns string of model name.
+
"""
def __init__(self, config):
"""Initializes ThreeLayerReLu.
@@ -64,12 +71,12 @@ class ThreeLayerReLu(torch.nn.Module):
Parameters
----------
- input_data: ndarray
+ input_data : ndarray
2D array containing input data.
Returns
-------
- prediction: ndarray
+ prediction : ndarray
Matrix containing predicted output data.
"""
diff --git a/ANN_Training.py b/ANN_Training.py
index 67f11cbbd4ba8c61806c442a1ed38b2274d6941a..e7628a204253885df68c743e6a19eb52431227ec 100644
--- a/ANN_Training.py
+++ b/ANN_Training.py
@@ -12,7 +12,8 @@ TODO: Write-protect all data and models
TODO: Put legend outside plot (bbox_to_anchor)
TODO: Put plotting into separate function
TODO: Reduce number of testing epochs to 50
-TODO: Adapt docstring to uniform standard
+TODO: Adapt docstring to uniform standard -> Done
+TODO: Change maximal line length to 79 (as advised by PEP8)
"""
import numpy as np
@@ -50,11 +51,11 @@ class ModelTrainer(object):
Methods
-------
- epoch_training()
+ epoch_training(dataset, num_epochs, verbose)
Trains model for a given number of epochs.
- test_model()
+ test_model(training_set, test_set)
Evaluates predictions of a model.
- save_model()
+ save_model(directory, model_name)
Saves state and validation loss of a model.
"""
@@ -121,7 +122,7 @@ class ModelTrainer(object):
dataset : torch.utils.data.dataset.TensorDataset
Training dataset.
num_epochs : int, optional
- Number of epochs for training. If None, set to instance value. Default: None.
+ Number of epochs for training. Default: None (i.e. instance variable).
verbose : bool, optional
Flag whether commentary in console is wanted. Default: False.
@@ -222,9 +223,9 @@ class ModelTrainer(object):
Parameters
----------
- directory: str
+ directory : str
Path to directory in which model is saved.
- model_name: str, optional
+ model_name : str, optional
Name of model for saving. Default: 'test_model'.
"""
@@ -244,9 +245,9 @@ def read_training_data(directory: str,
Parameters
----------
- directory: str
+ directory : str
Path to directory in which training data is saved.
- normalized: bool, optional
+ normalized : bool, optional
Flag whether normalized data should be used. Default: True.
Returns
@@ -267,15 +268,15 @@ def evaluate_models(models: dict, directory: str, num_iterations: int = 100, col
Parameters
----------
- models: dict
+ models : dict
Dictionary of models to evaluate.
- directory: str
+ directory : str
Path to directory for saving resulting plots.
- num_iterations: int, optional
+ num_iterations : int, optional
Number of iterations for evaluation. Default: 100.
- colors: dict, optional
+ colors : dict, optional
Dictionary containing plotting colors. If None, set to default colors. Default: None.
- compare_normalization: bool, optional
+ compare_normalization : bool, optional
Flag whether both normalized and raw data should be evaluated. Default: False.
"""
diff --git a/Basis_Function.py b/Basis_Function.py
index 0630198c0d29f1a61d124f31d5c6439fc6340f19..197be81bc29ea0ff6561635cd5770718f92537eb 100644
--- a/Basis_Function.py
+++ b/Basis_Function.py
@@ -17,9 +17,9 @@ class Vector(object):
Attributes
----------
- basis : np.array
+ basis : ndarray
Array of basis.
- wavelet : np.array
+ wavelet : ndarray
Array of wavelet.
Methods
@@ -61,7 +61,7 @@ class Vector(object):
Returns
-------
- np.array
+ ndarray
Vector containing basis evaluated at evaluation point.
"""
@@ -81,7 +81,7 @@ class Vector(object):
Returns
-------
- np.array
+ ndarray
Vector containing wavelet evaluated at evaluation point.
"""
@@ -108,7 +108,7 @@ class Legendre(Vector):
Returns
-------
- np.array
+ ndarray
Vector containing basis evaluated at evaluation point.
"""
@@ -124,7 +124,7 @@ class Legendre(Vector):
Returns
-------
- np.array
+ ndarray
Vector containing Legendre polynomial evaluated at evaluation point.
"""
@@ -163,7 +163,7 @@ class OrthonormalLegendre(Legendre):
Returns
-------
- np.array
+ ndarray
Vector containing basis evaluated at evaluation point.
"""
@@ -181,7 +181,7 @@ class OrthonormalLegendre(Legendre):
Returns
-------
- np.array
+ ndarray
Vector containing wavelet evaluated at evaluation point.
Notes
@@ -227,7 +227,7 @@ class OrthonormalLegendre(Legendre):
Returns
-------
- np.array
+ ndarray
Array containing the basis projection based on the integrals of the product
of two basis vectors for each degree combination.
@@ -248,7 +248,7 @@ class OrthonormalLegendre(Legendre):
Returns
-------
- np.array
+ ndarray
Matrix containing the integral of basis products.
"""
@@ -268,7 +268,7 @@ class OrthonormalLegendre(Legendre):
Returns
-------
- np.array
+ ndarray
Array containing the multiwavelet projection based on the integrals of the product
of a basis vector and a wavelet vector for each degree combination.
@@ -286,12 +286,12 @@ class OrthonormalLegendre(Legendre):
First parameter.
second_param : float
Second parameter.
- is_left_matrix : boolean
+ is_left_matrix : bool
Flag whether the left matrix is calculated.
Returns
-------
- np.array
+ ndarray
Matrix containing the integral of products of a basis and a wavelet vector.
"""
diff --git a/DG_Approximation.py b/DG_Approximation.py
index fef6c673d36a453f360ccb95585bed972eda0f48..3d24bd0caec0733849d3c9780c92a3c9b4a69d99 100644
--- a/DG_Approximation.py
+++ b/DG_Approximation.py
@@ -25,6 +25,7 @@ TODO: Add way of saving data (np.savez('data/' + name,
TODO: Outsource scripts into separate directory
TODO: Allow comparison between ANN training datasets
TODO: Add a default model state
+TODO: Add type annotations to function heads
"""
import os
@@ -59,9 +60,9 @@ class DGScheme(object):
Length of a cell in mesh.
basis : Basis object
Basis for calculation.
- mesh : array
+ mesh : ndarray
List of mesh valuation points.
- inv_mass : np.array
+ inv_mass : ndarray
Inverse mass matrix.
Methods
@@ -98,8 +99,8 @@ class DGScheme(object):
Left boundary of interval. Default: -1.
right_bound : float, optional
Right boundary of interval. Default: 1.
- verbose : boolean, optional
- Flag whether commentary in console is wanted. Default: False
+ verbose : bool, optional
+ Flag whether commentary in console is wanted. Default: False.
plot_dir : str, optional
Path to directory in which plots are saved. Default: 'test'.
history_threshold : float, optional
@@ -218,6 +219,11 @@ class DGScheme(object):
Sets plot directory, if not already existing, and saves plots generated during the last
approximation.
+ Parameters
+ ----------
+ plot_name : str
+ Name of plot.
+
"""
# Set paths for plot files if not existing already
if not os.path.exists(self._plot_dir):
@@ -273,7 +279,7 @@ class DGScheme(object):
Returns
-------
- np.array
+ ndarray
Matrix containing projection of size (N+2, p+1) with N being the number of grid cells
and p being the polynomial degree.
@@ -323,7 +329,7 @@ class DGScheme(object):
Returns
-------
- np.array
+ ndarray
Matrix containing cell averages and reconstructions for initial projection.
"""
diff --git a/Initial_Condition.py b/Initial_Condition.py
index 4015d2af17c32cf66ba2e149364a7fc97749b4d3..625164fd265c8f41d556fbbcc4812a804a940a00 100644
--- a/Initial_Condition.py
+++ b/Initial_Condition.py
@@ -2,8 +2,6 @@
"""
@author: Laura C. Kühle
-TODO: Add is_smooth() to DiscontinuousConstant class -> Done
-
"""
import numpy as np
diff --git a/Limiter.py b/Limiter.py
index 2734df71d9ff9c5559a0fe8d6ebcd5fc68c46c27..567b491491b4c972dc03907eb2997d994622d54c 100644
--- a/Limiter.py
+++ b/Limiter.py
@@ -47,14 +47,14 @@ class Limiter(object):
Parameters
----------
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
- cell
+ cell: int
Index of cell.
Returns
-------
- np.array
+ ndarray
Matrix of updated projection for each polynomial degree.
"""
@@ -117,14 +117,14 @@ class MinMod(Limiter):
Parameters
----------
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
cell : int
Index of cell.
Returns
-------
- adapted_projection : np.array
+ adapted_projection : ndarray
Matrix of updated projection for each polynomial degree.
"""
@@ -145,7 +145,7 @@ class MinMod(Limiter):
Parameters
----------
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
cell : int
Index of cell.
@@ -154,7 +154,7 @@ class MinMod(Limiter):
Returns
-------
- boolean
+ bool
Flag whether cell should be adjusted.
"""
@@ -170,7 +170,7 @@ class MinMod(Limiter):
Parameters
----------
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
cell : int
Index of cell.
@@ -237,7 +237,7 @@ class ModifiedMinMod(MinMod):
Parameters
----------
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
cell : int
Index of cell.
@@ -246,7 +246,7 @@ class ModifiedMinMod(MinMod):
Returns
-------
- boolean
+ bool
Flag whether cell should be adjusted.
"""
diff --git a/Plotting.py b/Plotting.py
index 2ce4ec9c53c829c196f78a98a655fc69839e3fad..0406f13f081a3d1fe22eabba5dd45614c2be8f0a 100644
--- a/Plotting.py
+++ b/Plotting.py
@@ -102,7 +102,7 @@ def plot_shock_tube(num_grid_cells: int, troubled_cell_history: list, time_histo
Number of cells in the mesh. Usually exponential of 2.
troubled_cell_history : list
List of detected troubled cells for each time step.
- time_history: list
+ time_history : list
List of value of each time step.
"""
@@ -223,9 +223,9 @@ def calculate_exact_solution(mesh: ndarray, cell_len: float, wave_speed: float,
Returns
-------
- grid: ndarray
+ grid : ndarray
Array containing evaluation grid for a function.
- exact: ndarray
+ exact : ndarray
Array containing exact evaluation of a function.
"""
@@ -258,9 +258,9 @@ def plot_classification_accuracy(evaluation_dict: dict, colors: dict) -> None:
Parameters
----------
- evaluation_dict: dict
+ evaluation_dict : dict
Dictionary containing classification evaluation data.
- colors: dict
+ colors : dict
Dictionary containing plotting colors.
"""
@@ -292,9 +292,9 @@ def plot_boxplot(evaluation_dict: dict, colors: dict) -> None:
Parameters
----------
- evaluation_dict: dict
+ evaluation_dict : dict
Dictionary containing classification evaluation data.
- colors: dict
+ colors : dict
Dictionary containing plotting colors.
"""
diff --git a/Quadrature.py b/Quadrature.py
index 9b4cc8922717d2f6b62d852c327f72bc575cbed7..c9920a461c2e9c43535d2d76acf653e0102267b1 100644
--- a/Quadrature.py
+++ b/Quadrature.py
@@ -15,9 +15,9 @@ class Quadrature(object):
----------
num_eval_points : int
Number of evaluation points per cell used for approximation.
- eval_points : np.array
+ eval_points : ndarray
Evaluation points per cell used for approximation.
- weights : np.array
+ weights : ndarray
Weights used for approximation calculation.
Methods
@@ -80,9 +80,9 @@ class Gauss(Quadrature):
----------
num_eval_points : int
Number of evaluation points per cell used for approximation.
- eval_points : np.array
+ eval_points : ndarray
Evaluation points per cell used for approximation.
- weights : np.array
+ weights : ndarray
Weights used for approximation calculation.
Methods
diff --git a/Troubled_Cell_Detector.py b/Troubled_Cell_Detector.py
index 0dd15ec5ab2eb6c17869fc9363814f1aa13837f2..e73d6d5d9629002dce37c7a357618bfea3bb88bd 100644
--- a/Troubled_Cell_Detector.py
+++ b/Troubled_Cell_Detector.py
@@ -53,7 +53,7 @@ class TroubledCellDetector(object):
Parameters
----------
- mesh : array
+ mesh : ndarray
List of mesh valuation points.
wave_speed : float
Speed of wave in rightward direction.
@@ -123,7 +123,7 @@ class TroubledCellDetector(object):
Parameters
----------
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
"""
@@ -137,14 +137,14 @@ class TroubledCellDetector(object):
Parameters
----------
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
stencil_length : int
Size of data array.
Returns
-------
- np.array
+ ndarray
Matrix containing cell averages and reconstructions for initial projection.
"""
@@ -164,11 +164,11 @@ class TroubledCellDetector(object):
Parameters
----------
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
troubled_cell_history : list
List of detected troubled cells for each time step.
- time_history:
+ time_history : list
List of value of each time step.
"""
@@ -184,12 +184,12 @@ class TroubledCellDetector(object):
Parameters
----------
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
Returns
-------
- max_error
+ max_error : float
Maximum error between exact and approximate solution.
"""
@@ -225,9 +225,14 @@ class NoDetection(TroubledCellDetector):
Parameters
----------
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
+ Returns
+ -------
+ list
+ List of indices for all detected troubled cells.
+
"""
return []
@@ -239,7 +244,7 @@ class ArtificialNeuralNetwork(TroubledCellDetector):
----------
stencil_length : int
Size of input data array.
- model : ANNModel object
+ model : torch.nn.Model
ANN model instance for evaluation.
Methods
@@ -280,7 +285,7 @@ class ArtificialNeuralNetwork(TroubledCellDetector):
Parameters
----------
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
Returns
@@ -345,7 +350,7 @@ class WaveletDetector(TroubledCellDetector):
Parameters
----------
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
Returns
@@ -362,12 +367,12 @@ class WaveletDetector(TroubledCellDetector):
Parameters
----------
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
Returns
-------
- np.array
+ ndarray
Matrix of wavelet coefficients.
"""
@@ -383,9 +388,9 @@ class WaveletDetector(TroubledCellDetector):
Parameters
----------
- multiwavelet_coeffs : np.array
+ multiwavelet_coeffs : ndarray
Matrix of multiwavelet coefficients.
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
Returns
@@ -403,11 +408,11 @@ class WaveletDetector(TroubledCellDetector):
Parameters
----------
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
troubled_cell_history : list
List of detected troubled cells for each time step.
- time_history:
+ time_history : list
List of value of each time step.
"""
@@ -424,12 +429,12 @@ class WaveletDetector(TroubledCellDetector):
Parameters
----------
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
Returns
-------
- np.array
+ ndarray
Matrix of projection on coarse grid for each polynomial degree.
"""
@@ -453,12 +458,12 @@ class WaveletDetector(TroubledCellDetector):
Parameters
----------
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
Returns
-------
- max_error
+ max_error : float
Maximum error between exact and approximate solution.
"""
@@ -487,7 +492,7 @@ class WaveletDetector(TroubledCellDetector):
Parameters
----------
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
"""
@@ -540,9 +545,9 @@ class Boxplot(WaveletDetector):
Parameters
----------
- multiwavelet_coeffs : np.array
+ multiwavelet_coeffs : ndarray
Matrix of multiwavelet coefficients.
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
Returns
@@ -619,9 +624,9 @@ class Theoretical(WaveletDetector):
Parameters
----------
- multiwavelet_coeffs : np.array
+ multiwavelet_coeffs : ndarray
Matrix of multiwavelet coefficients.
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
Returns
@@ -645,16 +650,16 @@ class Theoretical(WaveletDetector):
Parameters
----------
- multiwavelet_coeffs : np.array
+ multiwavelet_coeffs : ndarray
Matrix of multiwavelet coefficients.
cell : int
Index of cell.
- max_avg
+ max_avg : float
Maximum average of projection.
Returns
-------
- boolean
+ bool
Flag whether cell is troubled.
"""
diff --git a/Update_Scheme.py b/Update_Scheme.py
index 5d91109a0eb175634b76ebabb3f07c813dc077ef..8200949a898d702d668abba2a9cce4e1e8b2608b 100644
--- a/Update_Scheme.py
+++ b/Update_Scheme.py
@@ -16,9 +16,9 @@ class UpdateScheme(object):
Attributes
----------
- stiffness_matrix : np.array
+ stiffness_matrix : ndarray
Matrix
- boundary_matrix : np.array
+ boundary_matrix : ndarray
Matrix
Methods
@@ -85,14 +85,14 @@ class UpdateScheme(object):
Parameters
----------
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
cfl_number : float
CFL number to ensure stability.
Returns
-------
- current_projection : np.array
+ current_projection : ndarray
Matrix of projection of current update step for each polynomial degree.
troubled_cells : list
List of indices for all detected troubled cells.
@@ -107,14 +107,14 @@ class UpdateScheme(object):
Parameters
----------
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
cfl_number : float
CFL number to ensure stability.
Returns
-------
- current_projection : np.array
+ current_projection : ndarray
Matrix of projection of current update step for each polynomial degree.
troubled_cells : list
List of indices for all detected troubled cells.
@@ -127,12 +127,12 @@ class UpdateScheme(object):
Parameters
----------
- current_projection : np.array
+ current_projection : ndarray
Matrix of projection of current update step for each polynomial degree.
Returns
-------
- new_projection : np.array
+ new_projection : ndarray
Matrix of updated projection for each polynomial degree.
troubled_cells : list
List of indices for all detected troubled cells.
@@ -153,12 +153,12 @@ class UpdateScheme(object):
Parameters
----------
- current_projection : np.array
+ current_projection : ndarray
Matrix of projection of current update step for each polynomial degree.
Returns
-------
- current_projection : np.array
+ current_projection : ndarray
Matrix of projection of current update step for each polynomial degree.
"""
@@ -181,14 +181,14 @@ class SSPRK3(UpdateScheme):
Parameters
----------
- projection : np.array
+ projection : ndarray
Matrix of projection for each polynomial degree.
cfl_number : float
CFL number to ensure stability.
Returns
-------
- current_projection : np.array
+ current_projection : ndarray
Matrix of projection of current update step for each polynomial degree.
troubled_cells : list
List of indices for all detected troubled cells.
@@ -213,18 +213,18 @@ class SSPRK3(UpdateScheme):
return current_projection, troubled_cells
def _apply_first_step(self, original_projection, cfl_number):
- """Apply first step of SSPRK3.
+ """Applies first step of SSPRK3.
Parameters
----------
- original_projection : np.array
+ original_projection : ndarray
Matrix of original projection for each polynomial degree.
cfl_number : float
CFL number to ensure stability.
Returns
-------
- np.array
+ ndarray
Matrix of updated projection for each polynomial degree.
"""
@@ -232,20 +232,20 @@ class SSPRK3(UpdateScheme):
return original_projection + (cfl_number*right_hand_side)
def _apply_second_step(self, original_projection, current_projection, cfl_number):
- """Apply second step of SSPRK3.
+ """Applies second step of SSPRK3.
Parameters
----------
- original_projection : np.array
+ original_projection : ndarray
Matrix of original projection for each polynomial degree.
- current_projection : np.array
+ current_projection : ndarray
Matrix of projection of current update step for each polynomial degree.
cfl_number : float
CFL number to ensure stability.
Returns
-------
- np.array
+ ndarray
Matrix of updated projection for each polynomial degree.
"""
@@ -253,20 +253,20 @@ class SSPRK3(UpdateScheme):
return 1/4 * (3*original_projection + (current_projection + cfl_number*right_hand_side))
def _apply_third_step(self, original_projection, current_projection, cfl_number):
- """Apply third step of SSPRK3.
+ """Applies third step of SSPRK3.
Parameter
---------
- original_projection : np.array
+ original_projection : ndarray
Matrix of original projection for each polynomial degree.
- current_projection : np.array
+ current_projection : ndarray
Matrix of projection of current update step for each polynomial degree.
cfl_number : float
CFL number to ensure stability.
Returns
-------
- np.array
+ ndarray
Matrix of updated projection for each polynomial degree.
"""
@@ -274,16 +274,16 @@ class SSPRK3(UpdateScheme):
return 1/3 * (original_projection + 2*(current_projection + cfl_number*right_hand_side))
def _update_right_hand_side(self, current_projection):
- """Update right-hand side.
+ """Updates right-hand side.
Parameter
---------
- current_projection : np.array
+ current_projection : ndarray
Matrix of projection of current update step for each polynomial degree.
Returns
-------
- np.array
+ ndarray
Matrix of right-hand side.
"""