""" The mathutils/function.py is a rocketpy module totally dedicated to function
operations, including interpolation, extrapolation, integration, differentiation
and more. This is a core class of our package, and should be maintained
carefully as it may impact all the rest of the project.
"""
import warnings
from collections.abc import Iterable
from copy import deepcopy
from inspect import signature
from pathlib import Path
import matplotlib.pyplot as plt
import numpy as np
from scipy import integrate, linalg, optimize
try:
from functools import cached_property
except ImportError:
from ..tools import cached_property
NUMERICAL_TYPES = (float, int, complex, np.ndarray, np.integer, np.floating)
[docs]
class Function:
"""Class converts a python function or a data sequence into an object
which can be handled more naturally, enabling easy interpolation,
extrapolation, plotting and algebra.
"""
# Arithmetic priority
__array_ufunc__ = None
[docs]
def __init__(
self,
source,
inputs=None,
outputs=None,
interpolation=None,
extrapolation=None,
title=None,
):
"""Convert source into a Function, to be used more naturally.
Set inputs, outputs, domain dimension, interpolation and extrapolation
method, and process the source.
Parameters
----------
source : callable, scalar, ndarray, string, or Function
The data source to be used for the function:
- Callable: Called for evaluation with input values. Must have the
desired inputs as arguments and return a single output value.
Input order is important. Example: Python functions, classes, and
methods.
- int or float: Treated as a constant value function.
- ndarray: Used for interpolation. Format as [(x0, y0, z0),
(x1, y1, z1), ..., (xn, yn, zn)], where 'x' and 'y' are inputs,
and 'z' is the output.
- string: Path to a CSV file. The file is read and converted into an
ndarray. The file can optionally contain a single header line, see
notes below for more information.
- Function: Copies the source of the provided Function object,
creating a new Function with adjusted inputs and outputs.
inputs : string, sequence of strings, optional
The name of the inputs of the function. Will be used for
representation and graphing (axis names). 'Scalar' is default.
If source is function, int or float and has multiple inputs,
this parameter must be given for correct operation.
outputs : string, sequence of strings, optional
The name of the outputs of the function. Will be used for
representation and graphing (axis names). Scalar is default.
interpolation : string, optional
Interpolation method to be used if source type is ndarray.
For 1-D functions, linear, polynomial, akima and spline are
supported. For N-D functions, only shepard is supported.
Default for 1-D functions is spline.
extrapolation : string, optional
Extrapolation method to be used if source type is ndarray.
Options are 'natural', which keeps interpolation, 'constant',
which returns the value of the function at the edge of the interval,
and 'zero', which returns zero for all points outside of source
range. Default for 1-D functions is constant.
title : string, optional
Title to be displayed in the plots' figures. If none, the title will
be constructed using the inputs and outputs arguments in the form
of "{inputs} x {outputs}".
Returns
-------
None
Notes
-----
(I) CSV files may include an optional single header line. If this
header line is present and contains names for each data column, those
names will be used to label the inputs and outputs unless specified
otherwise by the `inputs` and `outputs` arguments.
If the header is specified for only a few columns, it is ignored.
Commas in a header will be interpreted as a delimiter, which may cause
undesired input or output labeling. To avoid this, specify each input
and output name using the `inputs` and `outputs` arguments.
(II) Fields in CSV files may be enclosed in double quotes. If fields
are not quoted, double quotes should not appear inside them.
"""
if inputs is None:
inputs = ["Scalar"]
if outputs is None:
outputs = ["Scalar"]
inputs, outputs, interpolation, extrapolation = self._check_user_input(
source, inputs, outputs, interpolation, extrapolation
)
# initialize variables to avoid errors when being called by other methods
self.get_value_opt = None
self.__polynomial_coefficients__ = None
self.__akima_coefficients__ = None
self.__spline_coefficients__ = None
# store variables
self.set_inputs(inputs)
self.set_outputs(outputs)
self.__interpolation__ = interpolation
self.__extrapolation__ = extrapolation
self.set_source(source)
self.set_title(title)
# Define all set methods
[docs]
def set_outputs(self, outputs):
"""Set the name and number of the output of the Function.
Parameters
----------
outputs : string, sequence of strings
The name of the output of the function. Example: Distance (m).
Returns
-------
self : Function
"""
self.__outputs__ = [outputs] if isinstance(outputs, str) else list(outputs)
self.__img_dim__ = len(self.__outputs__)
return self
[docs]
def set_source(self, source):
"""Sets the data source for the function, defining how the function
produces output from a given input.
Parameters
----------
source : callable, scalar, ndarray, string, or Function
The data source to be used for the function:
- Callable: Called for evaluation with input values. Must have the
desired inputs as arguments and return a single output value.
Input order is important. Example: Python functions, classes, and
methods.
- int or float: Treated as a constant value function.
- ndarray: Used for interpolation. Format as [(x0, y0, z0),
(x1, y1, z1), ..., (xn, yn, zn)], where 'x' and 'y' are inputs,
and 'z' is the output.
- string: Path to a CSV file. The file is read and converted into an
ndarray. The file can optionally contain a single header line.
- Function: Copies the source of the provided Function object,
creating a new Function with adjusted inputs and outputs.
Notes
-----
(I) CSV files may include an optional single header line. If this
header line is present and contains names for each data column, those
names will be used to label the inputs and outputs unless specified
otherwise. If the header is specified for only a few columns, it is
ignored.
Commas in a header will be interpreted as a delimiter, which may cause
undesired input or output labeling. To avoid this, specify each input
and output name using the `inputs` and `outputs` arguments.
(II) Fields in CSV files may be enclosed in double quotes. If fields
are not quoted, double quotes should not appear inside them.
Returns
-------
self : Function
Returns the Function instance.
"""
*_, interpolation, extrapolation = self._check_user_input(
source,
self.__inputs__,
self.__outputs__,
self.__interpolation__,
self.__extrapolation__,
)
# If the source is a Function
if isinstance(source, Function):
source = source.get_source()
# Import CSV if source is a string or Path and convert values to ndarray
if isinstance(source, (str, Path)):
with open(source, "r") as file:
try:
source = np.loadtxt(file, delimiter=",", dtype=float)
except ValueError:
# If an error occurs, headers are present
source = np.loadtxt(source, delimiter=",", dtype=float, skiprows=1)
except Exception as e:
raise ValueError(
"The source file is not a valid csv or txt file."
) from e
# Convert to ndarray if source is a list
if isinstance(source, (list, tuple)):
source = np.array(source, dtype=np.float64)
# Convert number source into vectorized lambda function
if isinstance(source, (int, float)):
temp = 1 * source
def source_function(_):
return temp
source = source_function
# Handle callable source or number source
if callable(source):
# Set source
self.source = source
# Set get_value_opt
self.get_value_opt = source
# Set arguments name and domain dimensions
parameters = signature(source).parameters
self.__dom_dim__ = len(parameters)
if self.__inputs__ == ["Scalar"]:
self.__inputs__ = list(parameters)
# Set interpolation and extrapolation
self.__interpolation__ = None
self.__extrapolation__ = None
# Handle ndarray source
else:
# Check to see if dimensions match incoming data set
new_total_dim = len(source[0, :])
old_total_dim = self.__dom_dim__ + self.__img_dim__
# If they don't, update default values or throw error
if new_total_dim != old_total_dim:
# Update dimensions and inputs
self.__dom_dim__ = new_total_dim - 1
self.__inputs__ = self.__dom_dim__ * self.__inputs__
# Do things if domDim is 1
if self.__dom_dim__ == 1:
source = source[source[:, 0].argsort()]
self.x_array = source[:, 0]
self.x_initial, self.x_final = self.x_array[0], self.x_array[-1]
self.y_array = source[:, 1]
self.y_initial, self.y_final = self.y_array[0], self.y_array[-1]
# Finally set data source as source
self.source = source
# Do things if function is multivariate
else:
self.x_array = source[:, 0]
self.x_initial, self.x_final = self.x_array[0], self.x_array[-1]
self.y_array = source[:, 1]
self.y_initial, self.y_final = self.y_array[0], self.y_array[-1]
self.z_array = source[:, 2]
self.z_initial, self.z_final = self.z_array[0], self.z_array[-1]
# Finally set data source as source
self.source = source
# Update extrapolation method
if self.__extrapolation__ is None:
self.set_extrapolation(extrapolation)
# Set default interpolation for point source if it hasn't
if self.__interpolation__ is None:
self.set_interpolation(interpolation)
else:
# Updates interpolation coefficients
self.set_interpolation(self.__interpolation__)
return self
@cached_property
def min(self):
"""Get the minimum value of the Function y_array.
Raises an error if the Function is lambda based.
Returns
-------
minimum : float
"""
return self.y_array.min()
@cached_property
def max(self):
"""Get the maximum value of the Function y_array.
Raises an error if the Function is lambda based.
Returns
-------
maximum : float
"""
return self.y_array.max()
[docs]
def set_interpolation(self, method="spline"):
"""Set interpolation method and process data is method requires.
Parameters
----------
method : string, optional
Interpolation method to be used if source type is ndarray.
For 1-D functions, linear, polynomial, akima and spline is
supported. For N-D functions, only shepard is supported.
Default is 'spline'.
Returns
-------
self : Function
"""
# Set interpolation method
self.__interpolation__ = method
# Spline, akima and polynomial need data processing
# Shepard, and linear do not
if method == "spline":
self.__interpolate_spline__()
elif method == "polynomial":
self.__interpolate_polynomial__()
elif method == "akima":
self.__interpolate_akima__()
# Set get_value_opt
self.set_get_value_opt()
return self
[docs]
def set_get_value_opt(self):
"""Crates a method that evaluates interpolations rather quickly
when compared to other options available, such as just calling
the object instance or calling ``Function.get_value`` directly. See
``Function.get_value_opt`` for documentation.
Returns
-------
self : Function
"""
# Retrieve general info
x_data = self.x_array
y_data = self.y_array
x_min, x_max = self.x_initial, self.x_final
if self.__extrapolation__ == "zero":
extrapolation = 0 # Extrapolation is zero
elif self.__extrapolation__ == "natural":
extrapolation = 1 # Extrapolation is natural
elif self.__extrapolation__ == "constant":
extrapolation = 2 # Extrapolation is constant
else:
raise ValueError(f"Invalid extrapolation type {self.__extrapolation__}")
# Crete method to interpolate this info for each interpolation type
if self.__interpolation__ == "spline":
coeffs = self.__spline_coefficients__
def get_value_opt(x):
x_interval = np.searchsorted(x_data, x)
# Interval found... interpolate... or extrapolate
if x_min <= x <= x_max:
# Interpolate
x_interval = x_interval if x_interval != 0 else 1
a = coeffs[:, x_interval - 1]
x = x - x_data[x_interval - 1]
y = a[3] * x**3 + a[2] * x**2 + a[1] * x + a[0]
else:
# Extrapolate
if extrapolation == 0: # Extrapolation == zero
y = 0
elif extrapolation == 1: # Extrapolation == natural
a = coeffs[:, 0] if x < x_min else coeffs[:, -1]
x = x - x_data[0] if x < x_min else x - x_data[-2]
y = a[3] * x**3 + a[2] * x**2 + a[1] * x + a[0]
else: # Extrapolation is set to constant
y = y_data[0] if x < x_min else y_data[-1]
return y
elif self.__interpolation__ == "linear":
def get_value_opt(x):
x_interval = np.searchsorted(x_data, x)
# Interval found... interpolate... or extrapolate
if x_min <= x <= x_max:
# Interpolate
dx = float(x_data[x_interval] - x_data[x_interval - 1])
dy = float(y_data[x_interval] - y_data[x_interval - 1])
y = (x - x_data[x_interval - 1]) * (dy / dx) + y_data[
x_interval - 1
]
else:
# Extrapolate
if extrapolation == 0: # Extrapolation == zero
y = 0
elif extrapolation == 1: # Extrapolation == natural
x_interval = 1 if x < x_min else -1
dx = float(x_data[x_interval] - x_data[x_interval - 1])
dy = float(y_data[x_interval] - y_data[x_interval - 1])
y = (x - x_data[x_interval - 1]) * (dy / dx) + y_data[
x_interval - 1
]
else: # Extrapolation is set to constant
y = y_data[0] if x < x_min else y_data[-1]
return y
elif self.__interpolation__ == "akima":
coeffs = np.array(self.__akima_coefficients__)
def get_value_opt(x):
x_interval = np.searchsorted(x_data, x)
# Interval found... interpolate... or extrapolate
if x_min <= x <= x_max:
# Interpolate
x_interval = x_interval if x_interval != 0 else 1
a = coeffs[4 * x_interval - 4 : 4 * x_interval]
y = a[3] * x**3 + a[2] * x**2 + a[1] * x + a[0]
else:
# Extrapolate
if extrapolation == 0: # Extrapolation == zero
y = 0
elif extrapolation == 1: # Extrapolation == natural
a = coeffs[:4] if x < x_min else coeffs[-4:]
y = a[3] * x**3 + a[2] * x**2 + a[1] * x + a[0]
else: # Extrapolation is set to constant
y = y_data[0] if x < x_min else y_data[-1]
return y
elif self.__interpolation__ == "polynomial":
coeffs = self.__polynomial_coefficients__
def get_value_opt(x):
# Interpolate... or extrapolate
if x_min <= x <= x_max:
# Interpolate
y = 0
for i, coef in enumerate(coeffs):
y += coef * (x**i)
else:
# Extrapolate
if extrapolation == 0: # Extrapolation == zero
y = 0
elif extrapolation == 1: # Extrapolation == natural
y = 0
for i, coef in enumerate(coeffs):
y += coef * (x**i)
else: # Extrapolation is set to constant
y = y_data[0] if x < x_min else y_data[-1]
return y
elif self.__interpolation__ == "shepard":
# change the function's name to avoid mypy's error
def get_value_opt_multiple(*args):
return self.__interpolate_shepard__(args)
get_value_opt = get_value_opt_multiple
self.get_value_opt = get_value_opt
return self
[docs]
def set_discrete(
self,
lower=0,
upper=10,
samples=200,
interpolation="spline",
extrapolation="constant",
one_by_one=True,
mutate_self=True,
):
"""This method discretizes a 1-D or 2-D Function by evaluating it at
certain points (sampling range) and storing the results in a list,
which is converted into a Function and then returned. By default, the
original Function object is replaced by the new one, which can be
changed by the attribute `mutate_self`.
This method is specially useful to change a dataset sampling or to
convert a Function defined by a callable into a list based Function.
Parameters
----------
lower : scalar, optional
Value where sampling range will start. Default is 0.
upper : scalar, optional
Value where sampling range will end. Default is 10.
samples : int, optional
Number of samples to be taken from inside range. Default is 200.
interpolation : string
Interpolation method to be used if source type is ndarray.
For 1-D functions, linear, polynomial, akima and spline is
supported. For N-D functions, only shepard is supported.
Default is 'spline'.
extrapolation : string, optional
Extrapolation method to be used if source type is ndarray.
Options are 'natural', which keeps interpolation, 'constant',
which returns the value of the function at the edge of the interval,
and 'zero', which returns zero for all points outside of source
range. Default is 'constant'.
one_by_one : boolean, optional
If True, evaluate Function in each sample point separately. If
False, evaluates Function in vectorized form. Default is True.
mutate_self : boolean, optional
If True, the original Function object source will be replaced by
the new one. If False, the original Function object source will
remain unchanged, and the new one is simply returned.
Default is True.
Returns
-------
self : Function
Notes
-----
1. This method performs by default in place replacement of the original
Function object source. This can be changed by the attribute `mutate_self`.
2. Currently, this method only supports 1-D and 2-D Functions.
"""
func = deepcopy(self) if not mutate_self else self
if func.__dom_dim__ == 1:
xs = np.linspace(lower, upper, samples)
ys = func.get_value(xs.tolist()) if one_by_one else func.get_value(xs)
func.set_source(np.concatenate(([xs], [ys])).transpose())
func.set_interpolation(interpolation)
func.set_extrapolation(extrapolation)
elif func.__dom_dim__ == 2:
lower = 2 * [lower] if isinstance(lower, (int, float)) else lower
upper = 2 * [upper] if isinstance(upper, (int, float)) else upper
sam = 2 * [samples] if isinstance(samples, (int, float)) else samples
# Create nodes to evaluate function
xs = np.linspace(lower[0], upper[0], sam[0])
ys = np.linspace(lower[1], upper[1], sam[1])
xs, ys = np.array(np.meshgrid(xs, ys)).reshape(2, xs.size * ys.size)
# Evaluate function at all mesh nodes and convert it to matrix
zs = np.array(func.get_value(xs, ys))
func.set_source(np.concatenate(([xs], [ys], [zs])).transpose())
func.__interpolation__ = "shepard"
func.__extrapolation__ = "natural"
else:
raise ValueError(
"Discretization is only supported for 1-D and 2-D Functions."
)
return func
[docs]
def set_discrete_based_on_model(
self, model_function, one_by_one=True, keep_self=True, mutate_self=True
):
"""This method transforms the domain of a 1-D or 2-D Function instance
into a list of discrete points based on the domain of a model Function
instance. It does so by retrieving the domain, domain name,
interpolation method and extrapolation method of the model Function
instance. It then evaluates the original Function instance in all
points of the retrieved domain to generate the list of discrete points
that will be used for interpolation when this Function is called.
By default, the original Function object is replaced by the new one,
which can be changed by the attribute `mutate_self`.
Parameters
----------
model_function : Function
Function object that will be used to define the sampling points,
interpolation method and extrapolation method.
Must be a Function whose source attribute is a list (i.e. a list
based Function instance). Must have the same domain dimension as the
Function to be discretized.
one_by_one : boolean, optional
If True, evaluate Function in each sample point separately. If
False, evaluates Function in vectorized form. Default is True.
keep_self : boolean, optional
If True, the original Function interpolation and extrapolation
methods will be kept. If False, those are substituted by the ones
from the model Function. Default is True.
mutate_self : boolean, optional
If True, the original Function object source will be replaced by
the new one. If False, the original Function object source will
remain unchanged, and the new one is simply returned.
Returns
-------
self : Function
See also
--------
Function.set_discrete
Examples
--------
This method is particularly useful when algebraic operations are carried
out using Function instances defined by different discretized domains
(same range, but different mesh size). Once an algebraic operation is
done, it will not directly be applied between the list of discrete
points of the two Function instances. Instead, the result will be a
Function instance defined by a callable that calls both Function
instances and performs the operation. This makes the evaluation of the
resulting Function inefficient, due to extra function calling overhead
and multiple interpolations being carried out.
>>> from rocketpy import Function
>>> f = Function([(0, 0), (1, 1), (2, 4), (3, 9), (4, 16)])
>>> g = Function([(0, 0), (2, 2), (4, 4)])
>>> h = f * g
>>> type(h.source)
<class 'function'>
Therefore, it is good practice to make sure both Function instances are
defined by the same domain, i.e. by the same list of mesh points. This
way, the algebraic operation will be carried out directly between the
lists of discrete points, generating a new Function instance defined by
this result. When it is evaluated, there are no extra function calling
overheads neither multiple interpolations.
>>> g.set_discrete_based_on_model(f)
'Function from R1 to R1 : (Scalar) → (Scalar)'
>>> h = f * g
>>> h.source
array([[ 0., 0.],
[ 1., 1.],
[ 2., 8.],
[ 3., 27.],
[ 4., 64.]])
Notes
-----
1. This method performs by default in place replacement of the original
Function object source. This can be changed by the attribute `mutate_self`.
2. This method is similar to set_discrete, but it uses the domain of a
model Function to define the domain of the new Function instance.
3. Currently, this method only supports 1-D and 2-D Functions.
"""
if not isinstance(model_function.source, np.ndarray):
raise TypeError("model_function must be a list based Function.")
if model_function.__dom_dim__ != self.__dom_dim__:
raise ValueError("model_function must have the same domain dimension.")
func = deepcopy(self) if not mutate_self else self
if func.__dom_dim__ == 1:
xs = model_function.source[:, 0]
ys = func.get_value(xs.tolist()) if one_by_one else func.get_value(xs)
func.set_source(np.concatenate(([xs], [ys])).transpose())
elif func.__dom_dim__ == 2:
# Create nodes to evaluate function
xs = model_function.source[:, 0]
ys = model_function.source[:, 1]
# Evaluate function at all mesh nodes and convert it to matrix
zs = np.array(func.get_value(xs, ys))
func.set_source(np.concatenate(([xs], [ys], [zs])).transpose())
else:
raise ValueError(
"Discretization is only supported for 1-D and 2-D Functions."
)
interp = (
func.__interpolation__ if keep_self else model_function.__interpolation__
)
extrap = (
func.__extrapolation__ if keep_self else model_function.__extrapolation__
)
func.set_interpolation(interp)
func.set_extrapolation(extrap)
return func
[docs]
def reset(
self,
inputs=None,
outputs=None,
interpolation=None,
extrapolation=None,
title=None,
):
"""This method allows the user to reset the inputs, outputs,
interpolation and extrapolation settings of a Function object, all at
once, without having to call each of the corresponding methods.
Parameters
----------
inputs : string, sequence of strings, optional
List of input variable names. If None, the original inputs are kept.
See Function.set_inputs for more information.
outputs : string, sequence of strings, optional
List of output variable names. If None, the original outputs are
kept. See Function.set_outputs for more information.
interpolation : string, optional
Interpolation method to be used if source type is ndarray.
See Function.set_interpolation for more information.
extrapolation : string, optional
Extrapolation method to be used if source type is ndarray.
See Function.set_extrapolation for more information.
Examples
--------
A simple use case is to reset the inputs and outputs of a Function
object that has been defined by algebraic manipulation of other Function
objects.
>>> from rocketpy import Function
>>> v = Function(lambda t: (9.8*t**2)/2, inputs='t', outputs='v')
>>> mass = 10 # Mass
>>> kinetic_energy = mass * v**2 / 2
>>> v.get_inputs(), v.get_outputs()
(['t'], ['v'])
>>> kinetic_energy
'Function from R1 to R1 : (x) → (Scalar)'
>>> kinetic_energy.reset(inputs='t', outputs='Kinetic Energy');
'Function from R1 to R1 : (t) → (Kinetic Energy)'
Returns
-------
self : Function
"""
if inputs is not None:
self.set_inputs(inputs)
if outputs is not None:
self.set_outputs(outputs)
if interpolation is not None and interpolation != self.__interpolation__:
self.set_interpolation(interpolation)
if extrapolation is not None and extrapolation != self.__extrapolation__:
self.set_extrapolation(extrapolation)
self.set_title(title)
return self
# Define all get methods
[docs]
def get_outputs(self):
"Return tuple of outputs of the function."
return self.__outputs__
[docs]
def get_source(self):
"Return source list or function of the Function."
return self.source
[docs]
def get_image_dim(self):
"Return int describing dimension of the image space of the function."
return self.__img_dim__
[docs]
def get_domain_dim(self):
"Return int describing dimension of the domain space of the function."
return self.__dom_dim__
[docs]
def get_interpolation_method(self):
"Return string describing interpolation method used."
return self.__interpolation__
[docs]
def get_value(self, *args):
"""This method returns the value of the Function at the specified
point. See Function.get_value_opt for a faster, but limited,
implementation.
Parameters
----------
args : scalar, list
Value where the Function is to be evaluated. If the Function is
1-D, only one argument is expected, which may be an int, a float
or a list of ints or floats, in which case the Function will be
evaluated at all points in the list and a list of floats will be
returned. If the function is N-D, N arguments must be given, each
one being an scalar or list.
Returns
-------
ans : scalar, list
Value of the Function at the specified point(s).
Examples
--------
>>> from rocketpy import Function
Testing with callable source (1 dimension):
>>> f = Function(lambda x: x**2)
>>> f.get_value(2)
4
>>> f.get_value(2.5)
6.25
>>> f.get_value([1, 2, 3])
[1, 4, 9]
>>> f.get_value([1, 2.5, 4.0])
[1, 6.25, 16.0]
Testing with callable source (2 dimensions):
>>> f2 = Function(lambda x, y: x**2 + y**2)
>>> f2.get_value(1, 2)
5
>>> f2.get_value([1, 2, 3], [1, 2, 3])
[2, 8, 18]
>>> f2.get_value([5], [5])
[50]
Testing with ndarray source (1 dimension):
>>> f3 = Function(
... [(0, 0), (1, 1), (1.5, 2.25), (2, 4), (2.5, 6.25), (3, 9), (4, 16)]
... )
>>> f3.get_value(2)
4.0
>>> f3.get_value(2.5)
6.25
>>> f3.get_value([1, 2, 3])
[1.0, 4.0, 9.0]
>>> f3.get_value([1, 2.5, 4.0])
[1.0, 6.25, 16.0]
Testing with ndarray source (2 dimensions):
>>> f4 = Function(
... [(0, 0, 0), (1, 1, 1), (1, 2, 2), (2, 4, 8), (3, 9, 27)]
... )
>>> f4.get_value(1, 1)
1.0
>>> f4.get_value(2, 4)
8.0
>>> abs(f4.get_value(1, 1.5) - 1.5) < 1e-2 # the interpolation is not perfect
True
>>> f4.get_value(3, 9)
27.0
"""
if len(args) != self.__dom_dim__:
raise ValueError(
f"This Function takes {self.__dom_dim__} arguments, {len(args)} given."
)
# Return value for Function of function type
if callable(self.source):
# if the function is 1-D:
if self.__dom_dim__ == 1:
# if the args is a simple number (int or float)
if isinstance(args[0], (int, float)):
return self.source(args[0])
# if the arguments are iterable, we map and return a list
if isinstance(args[0], Iterable):
return list(map(self.source, args[0]))
# if the function is n-D:
else:
# if each arg is a simple number (int or float)
if all(isinstance(arg, (int, float)) for arg in args):
return self.source(*args)
# if each arg is iterable, we map and return a list
if all(isinstance(arg, Iterable) for arg in args):
return [self.source(*arg) for arg in zip(*args)]
# Returns value for shepard interpolation
elif self.__interpolation__ == "shepard":
return self.__interpolate_shepard__(args)
# Returns value for polynomial interpolation function type
elif self.__interpolation__ == "polynomial":
if isinstance(args[0], (int, float)):
args = [list(args)]
x = np.array(args[0])
x_data = self.x_array
y_data = self.y_array
x_min, x_max = self.x_initial, self.x_final
coeffs = self.__polynomial_coefficients__
matrix = np.zeros((len(args[0]), coeffs.shape[0]))
for i in range(coeffs.shape[0]):
matrix[:, i] = x**i
ans = matrix.dot(coeffs).tolist()
for i, _ in enumerate(x):
if not x_min <= x[i] <= x_max:
if self.__extrapolation__ == "constant":
ans[i] = y_data[0] if x[i] < x_min else y_data[-1]
elif self.__extrapolation__ == "zero":
ans[i] = 0
return ans if len(ans) > 1 else ans[0]
# Returns value for spline, akima or linear interpolation function type
elif self.__interpolation__ in ["spline", "akima", "linear"]:
if isinstance(args[0], (int, float, complex, np.integer)):
args = [list(args)]
x = list(args[0])
x_data = self.x_array
y_data = self.y_array
x_intervals = np.searchsorted(x_data, x)
x_min, x_max = self.x_initial, self.x_final
if self.__interpolation__ == "spline":
coeffs = self.__spline_coefficients__
for i, _ in enumerate(x):
if x[i] == x_min or x[i] == x_max:
x[i] = y_data[x_intervals[i]]
elif x_min < x[i] < x_max or (self.__extrapolation__ == "natural"):
if not x_min < x[i] < x_max:
a = coeffs[:, 0] if x[i] < x_min else coeffs[:, -1]
x[i] = (
x[i] - x_data[0] if x[i] < x_min else x[i] - x_data[-2]
)
else:
a = coeffs[:, x_intervals[i] - 1]
x[i] = x[i] - x_data[x_intervals[i] - 1]
x[i] = a[3] * x[i] ** 3 + a[2] * x[i] ** 2 + a[1] * x[i] + a[0]
else:
# Extrapolate
if self.__extrapolation__ == "zero":
x[i] = 0
else: # Extrapolation is set to constant
x[i] = y_data[0] if x[i] < x_min else y_data[-1]
elif self.__interpolation__ == "linear":
for i, _ in enumerate(x):
# Interval found... interpolate... or extrapolate
inter = x_intervals[i]
if x_min <= x[i] <= x_max:
# Interpolate
dx = float(x_data[inter] - x_data[inter - 1])
dy = float(y_data[inter] - y_data[inter - 1])
x[i] = (x[i] - x_data[inter - 1]) * (dy / dx) + y_data[
inter - 1
]
else:
# Extrapolate
if self.__extrapolation__ == "zero": # Extrapolation == zero
x[i] = 0
elif (
self.__extrapolation__ == "natural"
): # Extrapolation == natural
inter = 1 if x[i] < x_min else -1
dx = float(x_data[inter] - x_data[inter - 1])
dy = float(y_data[inter] - y_data[inter - 1])
x[i] = (x[i] - x_data[inter - 1]) * (dy / dx) + y_data[
inter - 1
]
else: # Extrapolation is set to constant
x[i] = y_data[0] if x[i] < x_min else y_data[-1]
else:
coeffs = self.__akima_coefficients__
for i, _ in enumerate(x):
if x[i] == x_min or x[i] == x_max:
x[i] = y_data[x_intervals[i]]
elif x_min < x[i] < x_max or (self.__extrapolation__ == "natural"):
if not x_min < x[i] < x_max:
a = coeffs[:4] if x[i] < x_min else coeffs[-4:]
else:
a = coeffs[4 * x_intervals[i] - 4 : 4 * x_intervals[i]]
x[i] = a[3] * x[i] ** 3 + a[2] * x[i] ** 2 + a[1] * x[i] + a[0]
else:
# Extrapolate
if self.__extrapolation__ == "zero":
x[i] = 0
else: # Extrapolation is set to constant
x[i] = y_data[0] if x[i] < x_min else y_data[-1]
if isinstance(args[0], np.ndarray):
return np.array(x)
else:
return x if len(x) > 1 else x[0]
[docs]
def __getitem__(self, args):
"""Returns item of the Function source. If the source is not an array,
an error will result.
Parameters
----------
args : int, float
Index of the item to be retrieved.
Returns
-------
self.source[args] : float, array
Item specified from Function.source.
"""
return self.source[args]
[docs]
def __len__(self):
"""Returns length of the Function source. If the source is not an
array, an error will result.
Returns
-------
len(self.source) : int
Length of Function.source.
"""
return len(self.source)
[docs]
def __bool__(self):
"""Returns true if self exists. This is to avoid getting into __len__
method in boolean statements.
Returns
-------
bool : bool
Always True.
"""
return True
# Define all conversion methods
[docs]
def to_frequency_domain(self, lower, upper, sampling_frequency, remove_dc=True):
"""Performs the conversion of the Function to the Frequency Domain and
returns the result. This is done by taking the Fourier transform of the
Function. The resulting frequency domain is symmetric, i.e., the
negative frequencies are included as well.
Parameters
----------
lower : float
Lower bound of the time range.
upper : float
Upper bound of the time range.
sampling_frequency : float
Sampling frequency at which to perform the Fourier transform.
remove_dc : bool, optional
If True, the DC component is removed from the Fourier transform.
Returns
-------
Function
The Function in the frequency domain.
Examples
--------
>>> from rocketpy import Function
>>> import numpy as np
>>> main_frequency = 10 # Hz
>>> time = np.linspace(0, 10, 1000)
>>> signal = np.sin(2 * np.pi * main_frequency * time)
>>> time_domain = Function(np.array([time, signal]).T)
>>> frequency_domain = time_domain.to_frequency_domain(
... lower=0, upper=10, sampling_frequency=100
... )
>>> peak_frequencies_index = np.where(frequency_domain[:, 1] > 0.001)
>>> peak_frequencies = frequency_domain[peak_frequencies_index, 0]
>>> print(peak_frequencies)
[[-10. 10.]]
"""
# Get the time domain data
sampling_time_step = 1.0 / sampling_frequency
sampling_range = np.arange(lower, upper, sampling_time_step)
number_of_samples = len(sampling_range)
sampled_points = self(sampling_range)
if remove_dc:
sampled_points -= np.mean(sampled_points)
fourier_amplitude = np.abs(np.fft.fft(sampled_points) / (number_of_samples / 2))
fourier_frequencies = np.fft.fftfreq(number_of_samples, sampling_time_step)
return Function(
source=np.array([fourier_frequencies, fourier_amplitude]).T,
inputs="Frequency (Hz)",
outputs="Amplitude",
interpolation="linear",
extrapolation="zero",
)
[docs]
def low_pass_filter(self, alpha, file_path=None):
"""Implements a low pass filter with a moving average filter. This does
not mutate the original Function object, but returns a new one with the
filtered source. The filtered source is also saved to a CSV file if a
file path is given.
Parameters
----------
alpha : float
Attenuation coefficient, 0 <= alpha <= 1
For a given dataset, the larger alpha is, the more closely the
filtered function returned will match the function the smaller
alpha is, the smoother the filtered function returned will be
(but with a phase shift)
file_path : string, optional
File path or file name of the CSV to save. Don't save any CSV if
if no argument is passed. Initiated to None.
Returns
-------
Function
The function with the incoming source filtered
"""
filtered_signal = np.zeros_like(self.source)
filtered_signal[0] = self.source[0]
for i in range(1, len(self.source)):
# for each point of our dataset, we apply a exponential smoothing
filtered_signal[i] = (
alpha * self.source[i] + (1 - alpha) * filtered_signal[i - 1]
)
if isinstance(file_path, str):
self.savetxt(file_path)
return Function(
source=filtered_signal,
inputs=self.__inputs__,
outputs=self.__outputs__,
interpolation=self.__interpolation__,
extrapolation=self.__extrapolation__,
title=self.title,
)
# Define all presentation methods
[docs]
def __call__(self, *args):
"""Plot the Function if no argument is given. If an
argument is given, return the value of the function at the desired
point.
Parameters
----------
args : scalar, list, optional
Value where the Function is to be evaluated. If the Function is
1-D, only one argument is expected, which may be an int, a float
or a list of ints or floats, in which case the Function will be
evaluated at all points in the list and a list of floats will be
returned. If the function is N-D, N arguments must be given, each
one being an scalar or list.
Returns
-------
ans : None, scalar, list
"""
if len(args) == 0:
return self.plot()
else:
return self.get_value(*args)
[docs]
def __str__(self):
"Return a string representation of the Function"
return str(
"Function from R"
+ str(self.__dom_dim__)
+ " to R"
+ str(self.__img_dim__)
+ " : ("
+ ", ".join(self.__inputs__)
+ ") → ("
+ ", ".join(self.__outputs__)
+ ")"
)
[docs]
def __repr__(self):
"Return a string representation of the Function"
return repr(
"Function from R"
+ str(self.__dom_dim__)
+ " to R"
+ str(self.__img_dim__)
+ " : ("
+ ", ".join(self.__inputs__)
+ ") → ("
+ ", ".join(self.__outputs__)
+ ")"
)
[docs]
def set_title(self, title):
"""Used to define the title of the Function object.
Parameters
----------
title : str
Title to be assigned to the Function.
"""
if title:
self.title = title
else:
if self.__dom_dim__ == 1:
self.title = (
self.__outputs__[0].title() + " x " + self.__inputs__[0].title()
)
elif self.__dom_dim__ == 2:
self.title = (
self.__outputs__[0].title()
+ " x "
+ self.__inputs__[0].title()
+ " x "
+ self.__inputs__[1].title()
)
[docs]
def plot(self, *args, **kwargs):
"""Call Function.plot_1d if Function is 1-Dimensional or call
Function.plot_2d if Function is 2-Dimensional and forward arguments
and key-word arguments."""
if isinstance(self, list):
# Compare multiple plots
Function.compare_plots(self)
else:
if self.__dom_dim__ == 1:
self.plot_1d(*args, **kwargs)
elif self.__dom_dim__ == 2:
self.plot_2d(*args, **kwargs)
else:
print("Error: Only functions with 1D or 2D domains are plottable!")
[docs]
def plot1D(self, *args, **kwargs):
"""Deprecated method, use Function.plot_1d instead."""
warnings.warn(
"The `Function.plot1D` method is set to be deprecated and fully "
+ "removed in rocketpy v2.0.0, use `Function.plot_1d` instead. "
+ "This method is calling `Function.plot_1d`.",
DeprecationWarning,
)
return self.plot_1d(*args, **kwargs)
[docs]
def plot_1d(
self,
lower=None,
upper=None,
samples=1000,
force_data=False,
force_points=False,
return_object=False,
equal_axis=False,
):
"""Plot 1-Dimensional Function, from a lower limit to an upper limit,
by sampling the Function several times in the interval. The title of
the graph is given by the name of the axes, which are taken from
the Function`s input and output names.
Parameters
----------
lower : scalar, optional
The lower limit of the interval in which the function is to be
plotted. The default value for function type Functions is 0. By
contrast, if the Function is given by a dataset, the default
value is the start of the dataset.
upper : scalar, optional
The upper limit of the interval in which the function is to be
plotted. The default value for function type Functions is 10. By
contrast, if the Function is given by a dataset, the default
value is the end of the dataset.
samples : int, optional
The number of samples in which the function will be evaluated for
plotting it, which draws lines between each evaluated point.
The default value is 1000.
force_data : Boolean, optional
If Function is given by an interpolated dataset, setting force_data
to True will plot all points, as a scatter, in the dataset.
Default value is False.
force_points : Boolean, optional
Setting force_points to True will plot all points, as a scatter, in
which the Function was evaluated in the dataset. Default value is
False.
Returns
-------
None
"""
# Define a mesh and y values at mesh nodes for plotting
fig = plt.figure()
ax = fig.axes
if callable(self.source):
# Determine boundaries
lower = 0 if lower is None else lower
upper = 10 if upper is None else upper
else:
# Determine boundaries
x_data = self.x_array
x_min, x_max = self.x_initial, self.x_final
lower = x_min if lower is None else lower
upper = x_max if upper is None else upper
# Plot data points if force_data = True
too_low = x_min >= lower
too_high = x_max <= upper
lo_ind = 0 if too_low else np.where(x_data >= lower)[0][0]
up_ind = len(x_data) - 1 if too_high else np.where(x_data <= upper)[0][0]
points = self.source[lo_ind : (up_ind + 1), :].T.tolist()
if force_data:
plt.scatter(points[0], points[1], marker="o")
# Calculate function at mesh nodes
x = np.linspace(lower, upper, samples)
y = self.get_value(x.tolist())
# Plots function
if force_points:
plt.scatter(x, y, marker="o")
if equal_axis:
plt.axis("equal")
plt.plot(x, y)
# Turn on grid and set title and axis
plt.grid(True)
plt.title(self.title)
plt.xlabel(self.__inputs__[0].title())
plt.ylabel(self.__outputs__[0].title())
plt.show()
if return_object:
return fig, ax
[docs]
def plot2D(self, *args, **kwargs):
"""Deprecated method, use Function.plot_2d instead."""
warnings.warn(
"The `Function.plot2D` method is set to be deprecated and fully "
+ "removed in rocketpy v2.0.0, use `Function.plot_2d` instead. "
+ "This method is calling `Function.plot_2d`.",
DeprecationWarning,
)
return self.plot_2d(*args, **kwargs)
[docs]
def plot_2d(
self,
lower=None,
upper=None,
samples=None,
force_data=True,
disp_type="surface",
alpha=0.6,
cmap="viridis",
):
"""Plot 2-Dimensional Function, from a lower limit to an upper limit,
by sampling the Function several times in the interval. The title of
the graph is given by the name of the axis, which are taken from
the Function`s inputs and output names.
Parameters
----------
lower : scalar, array of int or float, optional
The lower limits of the interval in which the function is to be
plotted, which can be an int or float, which is repeated for both
axis, or an array specifying the limit for each axis. The default
value for function type Functions is 0. By contrast, if the
Function is given by a dataset, the default value is the start of
the dataset for each axis.
upper : scalar, array of int or float, optional
The upper limits of the interval in which the function is to be
plotted, which can be an int or float, which is repeated for both
axis, or an array specifying the limit for each axis. The default
value for function type Functions is 0. By contrast, if the
Function is given by a dataset, the default value is the end of
the dataset for each axis.
samples : int, array of int, optional
The number of samples in which the function will be evaluated for
plotting it, which draws lines between each evaluated point.
The default value is 30 for each axis.
force_data : Boolean, optional
If Function is given by an interpolated dataset, setting force_data
to True will plot all points, as a scatter, in the dataset.
Default value is False.
disp_type : string, optional
Display type of plotted graph, which can be surface, wireframe,
contour, or contourf. Default value is surface.
alpha : float, optional
Transparency of plotted graph, which can be a value between 0 and
1. Default value is 0.6.
cmap : string, optional
Colormap of plotted graph, which can be any of the color maps
available in matplotlib. Default value is viridis.
Returns
-------
None
"""
if samples is None:
samples = [30, 30]
# Prepare plot
figure = plt.figure()
axes = figure.add_subplot(111, projection="3d")
# Define a mesh and f values at mesh nodes for plotting
if callable(self.source):
# Determine boundaries
lower = [0, 0] if lower is None else lower
lower = 2 * [lower] if isinstance(lower, (int, float)) else lower
upper = [10, 10] if upper is None else upper
upper = 2 * [upper] if isinstance(upper, (int, float)) else upper
else:
# Determine boundaries
x_data = self.x_array
y_data = self.y_array
x_min, x_max = x_data.min(), x_data.max()
y_min, y_max = y_data.min(), y_data.max()
lower = [x_min, y_min] if lower is None else lower
lower = 2 * [lower] if isinstance(lower, (int, float)) else lower
upper = [x_max, y_max] if upper is None else upper
upper = 2 * [upper] if isinstance(upper, (int, float)) else upper
# Plot data points if force_data = True
if force_data:
axes.scatter(x_data, y_data, self.source[:, -1])
# Create nodes to evaluate function
x = np.linspace(lower[0], upper[0], samples[0])
y = np.linspace(lower[1], upper[1], samples[1])
mesh_x, mesh_y = np.meshgrid(x, y)
# Evaluate function at all mesh nodes and convert it to matrix
z = np.array(self.get_value(mesh_x.flatten(), mesh_y.flatten())).reshape(
mesh_x.shape
)
z_min, z_max = z.min(), z.max()
color_map = plt.colormaps[cmap]
norm = plt.Normalize(z_min, z_max)
# Plot function
if disp_type == "surface":
surf = axes.plot_surface(
mesh_x,
mesh_y,
z,
rstride=1,
cstride=1,
cmap=color_map,
linewidth=0,
alpha=alpha,
vmin=z_min,
vmax=z_max,
)
figure.colorbar(surf)
elif disp_type == "wireframe":
axes.plot_wireframe(mesh_x, mesh_y, z, rstride=1, cstride=1)
elif disp_type == "contour":
figure.clf()
contour_set = plt.contour(mesh_x, mesh_y, z)
plt.clabel(contour_set, inline=1, fontsize=10)
elif disp_type == "contourf":
figure.clf()
contour_set = plt.contour(mesh_x, mesh_y, z)
plt.contourf(mesh_x, mesh_y, z)
plt.clabel(contour_set, inline=1, fontsize=10)
plt.title(self.title)
axes.set_xlabel(self.__inputs__[0].title())
axes.set_ylabel(self.__inputs__[1].title())
axes.set_zlabel(self.__outputs__[0].title())
plt.show()
[docs]
@staticmethod
def compare_plots(
plot_list,
lower=None,
upper=None,
samples=1000,
title="",
xlabel="",
ylabel="",
force_data=False,
force_points=False,
return_object=False,
):
"""Plots N 1-Dimensional Functions in the same plot, from a lower
limit to an upper limit, by sampling the Functions several times in
the interval.
Parameters
----------
plot_list : list
List of Functions or list of tuples in the format (Function,
label), where label is a string which will be displayed in the
legend.
lower : scalar, optional
The lower limit of the interval in which the Functions are to be
plotted. The default value for function type Functions is 0. By
contrast, if the Functions given are defined by a dataset, the
default value is the lowest value of the datasets.
upper : scalar, optional
The upper limit of the interval in which the Functions are to be
plotted. The default value for function type Functions is 10. By
contrast, if the Functions given are defined by a dataset, the
default value is the highest value of the datasets.
samples : int, optional
The number of samples in which the functions will be evaluated for
plotting it, which draws lines between each evaluated point.
The default value is 1000.
title : string, optional
Title of the plot. Default value is an empty string.
xlabel : string, optional
X-axis label. Default value is an empty string.
ylabel : string, optional
Y-axis label. Default value is an empty string.
force_data : Boolean, optional
If Function is given by an interpolated dataset, setting force_data
to True will plot all points, as a scatter, in the dataset.
Default value is False.
force_points : Boolean, optional
Setting force_points to True will plot all points, as a scatter, in
which the Function was evaluated to plot it. Default value is
False.
Returns
-------
None
"""
no_range_specified = lower is None and upper is None
# Convert to list of tuples if list of Function was given
plots = []
for plot in plot_list:
if isinstance(plot, (tuple, list)):
plots.append(plot)
else:
plots.append((plot, ""))
# Create plot figure
fig, ax = plt.subplots()
# Define a mesh and y values at mesh nodes for plotting
if lower is None:
lower = 0
for plot in plots:
if not callable(plot[0].source):
# Determine boundaries
x_min = plot[0].source[0, 0]
lower = x_min if x_min < lower else lower
if upper is None:
upper = 10
for plot in plots:
if not callable(plot[0].source):
# Determine boundaries
x_max = plot[0].source[-1, 0]
upper = x_max if x_max > upper else upper
x = np.linspace(lower, upper, samples)
# Iterate to plot all plots
for plot in plots:
# Deal with discrete data sets when no range is given
if no_range_specified and not callable(plot[0].source):
ax.plot(plot[0][:, 0], plot[0][:, 1], label=plot[1])
if force_points:
ax.scatter(plot[0][:, 0], plot[0][:, 1], marker="o")
else:
# Calculate function at mesh nodes
y = plot[0].get_value(x.tolist())
# Plots function
ax.plot(x, y, label=plot[1])
if force_points:
ax.scatter(x, y, marker="o")
# Plot data points if specified
if force_data:
for plot in plots:
if not callable(plot[0].source):
x_data = plot[0].source[:, 0]
x_min, x_max = x_data[0], x_data[-1]
too_low = x_min >= lower
too_high = x_max <= upper
lo_ind = 0 if too_low else np.where(x_data >= lower)[0][0]
up_ind = (
len(x_data) - 1 if too_high else np.where(x_data <= upper)[0][0]
)
points = plot[0].source[lo_ind : (up_ind + 1), :].T.tolist()
ax.scatter(points[0], points[1], marker="o")
# Setup legend
ax.legend(loc="best", shadow=True)
# Turn on grid and set title and axis
plt.grid(True)
plt.title(title)
plt.xlabel(xlabel)
plt.ylabel(ylabel)
plt.show()
if return_object:
return fig, ax
# Define all interpolation methods
[docs]
def __interpolate_polynomial__(self):
"""Calculate polynomial coefficients that fit the data exactly."""
# Find the degree of the polynomial interpolation
degree = self.source.shape[0] - 1
# Get x and y values for all supplied points.
x = self.x_array
y = self.y_array
# Check if interpolation requires large numbers
if np.amax(x) ** degree > 1e308:
warnings.warn(
"Polynomial interpolation of too many points can't be done."
" Once the degree is too high, numbers get too large."
" The process becomes inefficient. Using spline instead."
)
return self.set_interpolation("spline")
# Create coefficient matrix1
sys_coeffs = np.zeros((degree + 1, degree + 1))
for i in range(degree + 1):
sys_coeffs[:, i] = x**i
# Solve the system and store the resultant coefficients
self.__polynomial_coefficients__ = np.linalg.solve(sys_coeffs, y)
[docs]
def __interpolate_spline__(self):
"""Calculate natural spline coefficients that fit the data exactly."""
# Get x and y values for all supplied points
x, y = self.x_array, self.y_array
m_dim = len(x)
h = np.diff(x)
# Initialize the matrix
banded_matrix = np.zeros((3, m_dim))
banded_matrix[1, 0] = banded_matrix[1, m_dim - 1] = 1
# Construct the Ab banded matrix and B vector
vector_b = [0]
banded_matrix[2, :-2] = h[:-1]
banded_matrix[1, 1:-1] = 2 * (h[:-1] + h[1:])
banded_matrix[0, 2:] = h[1:]
vector_b.extend(3 * ((y[2:] - y[1:-1]) / h[1:] - (y[1:-1] - y[:-2]) / h[:-1]))
vector_b.append(0)
# Solve the system for c coefficients
c = linalg.solve_banded(
(1, 1), banded_matrix, vector_b, overwrite_ab=True, overwrite_b=True
)
# Calculate other coefficients
b = (y[1:] - y[:-1]) / h - h * (2 * c[:-1] + c[1:]) / 3
d = (c[1:] - c[:-1]) / (3 * h)
# Store coefficients
self.__spline_coefficients__ = np.vstack([y[:-1], b, c[:-1], d])
[docs]
def __interpolate_akima__(self):
"""Calculate akima spline coefficients that fit the data exactly"""
# Get x and y values for all supplied points
x, y = self.x_array, self.y_array
# Estimate derivatives at each point
d = [0] * len(x)
d[0] = (y[1] - y[0]) / (x[1] - x[0])
d[-1] = (y[-1] - y[-2]) / (x[-1] - x[-2])
for i in range(1, len(x) - 1):
w1, w2 = (x[i] - x[i - 1]), (x[i + 1] - x[i])
d1, d2 = ((y[i] - y[i - 1]) / w1), ((y[i + 1] - y[i]) / w2)
d[i] = (w1 * d2 + w2 * d1) / (w1 + w2)
# Calculate coefficients for each interval with system already solved
coeffs = [0] * 4 * (len(x) - 1)
for i in range(len(x) - 1):
xl, xr = x[i], x[i + 1]
yl, yr = y[i], y[i + 1]
dl, dr = d[i], d[i + 1]
matrix = np.array(
[
[1, xl, xl**2, xl**3],
[1, xr, xr**2, xr**3],
[0, 1, 2 * xl, 3 * xl**2],
[0, 1, 2 * xr, 3 * xr**2],
]
)
result = np.array([yl, yr, dl, dr]).T
coeffs[4 * i : 4 * i + 4] = np.linalg.solve(matrix, result)
self.__akima_coefficients__ = coeffs
[docs]
def __interpolate_shepard__(self, args):
"""Calculates the shepard interpolation from the given arguments.
The shepard interpolation is computed by a inverse distance weighting
in a vectorized manner.
Parameters
----------
args : scalar, list
Values where the Function is to be evaluated.
Returns
-------
result : scalar, list
The result of the interpolation.
"""
x_data = self.source[:, 0:-1] # Support for N-Dimensions
y_data = self.source[:, -1]
arg_stack = np.column_stack(args)
arg_qty, arg_dim = arg_stack.shape
result = np.zeros(arg_qty)
# Reshape to vectorize calculations
x = arg_stack.reshape(arg_qty, 1, arg_dim)
sub_matrix = x_data - x
distances_squared = np.sum(sub_matrix**2, axis=2)
# Remove zero distances from further calculations
zero_distances = np.where(distances_squared == 0)
valid_indexes = np.ones(arg_qty, dtype=bool)
valid_indexes[zero_distances[0]] = False
weights = distances_squared[valid_indexes] ** (-1.5)
numerator_sum = np.sum(y_data * weights, axis=1)
denominator_sum = np.sum(weights, axis=1)
result[valid_indexes] = numerator_sum / denominator_sum
result[~valid_indexes] = y_data[zero_distances[1]]
return result if len(result) > 1 else result[0]
[docs]
def __neg__(self):
"""Negates the Function object. The result has the same effect as
multiplying the Function by -1.
Returns
-------
Function
The negated Function object.
"""
if isinstance(self.source, np.ndarray):
neg_source = np.column_stack((self.x_array, -self.y_array))
return Function(
neg_source,
self.__inputs__,
self.__outputs__,
self.__interpolation__,
self.__extrapolation__,
)
else:
return Function(
lambda x: -self.source(x),
self.__inputs__,
self.__outputs__,
self.__interpolation__,
self.__extrapolation__,
)
[docs]
def __ge__(self, other):
"""Greater than or equal to comparison operator. It can be used to
compare a Function object with a scalar or another Function object.
This has the same effect as comparing numpy arrays.
Note that it only works for Functions if at least one of them is
defined by a set of points so that the bounds of the domain can be
set.
If both are defined by a set of points, they must have the same
discretization.
Parameters
----------
other : scalar or Function
Returns
-------
numpy.ndarray of bool
The result of the comparison one by one.
"""
other_is_function = isinstance(other, Function)
if isinstance(self.source, np.ndarray):
if other_is_function:
try:
return self.y_array >= other.y_array
except AttributeError:
# Other is lambda based Function
return self.y_array >= other(self.x_array)
except ValueError as exc:
raise ValueError(
"Comparison not supported between instances of the "
"Function class with different domain discretization."
) from exc
else:
# Other is not a Function
try:
return self.y_array >= other
except TypeError as exc:
raise TypeError(
"Comparison not supported between instances of "
f"'Function' and '{type(other)}'."
) from exc
else:
# self is lambda based Function
if other_is_function:
try:
return self(other.x_array) >= other.y_array
except AttributeError as exc:
raise TypeError(
"Comparison not supported between two instances of "
"the Function class with callable sources."
) from exc
return None
[docs]
def __le__(self, other):
"""Less than or equal to comparison operator. It can be used to
compare a Function object with a scalar or another Function object.
This has the same effect as comparing numpy arrays.
Note that it only works for Functions if at least one of them is
defined by a set of points so that the bounds of the domain can be
set.
If both are defined by a set of points, they must have the same
discretization.
Parameters
----------
other : scalar or Function
Returns
-------
numpy.ndarray of bool
The result of the comparison one by one.
"""
other_is_function = isinstance(other, Function)
if isinstance(self.source, np.ndarray):
if other_is_function:
try:
return self.y_array <= other.y_array
except AttributeError:
# Other is lambda based Function
return self.y_array <= other(self.x_array)
except ValueError as exc:
raise ValueError(
"Operands should have the same discretization."
) from exc
else:
# Other is not a Function
try:
return self.y_array <= other
except TypeError as exc:
raise TypeError(
"Comparison not supported between instances of "
f"'Function' and '{type(other)}'."
) from exc
else:
# self is lambda based Function
if other_is_function:
try:
return self(other.x_array) <= other.y_array
except AttributeError as exc:
raise TypeError(
"Comparison not supported between two instances of "
"the Function class with callable sources."
) from exc
return None
[docs]
def __gt__(self, other):
"""Greater than comparison operator. It can be used to compare a
Function object with a scalar or another Function object. This has
the same effect as comparing numpy arrays.
Note that it only works for Functions if at least one of them is
defined by a set of points so that the bounds of the domain can be
set.
If both are defined by a set of points, they must have the same
discretization.
Parameters
----------
other : scalar or Function
Returns
-------
numpy.ndarray of bool
The result of the comparison one by one.
"""
return ~self.__le__(other)
[docs]
def __lt__(self, other):
"""Less than comparison operator. It can be used to compare a
Function object with a scalar or another Function object. This has
the same effect as comparing numpy arrays.
Note that it only works for Functions if at least one of them is
defined by a set of points so that the bounds of the domain can be
set.
If both are defined by a set of points, they must have the same
discretization.
Parameters
----------
other : scalar or Function
Returns
-------
numpy.ndarray of bool
The result of the comparison one by one.
"""
return ~self.__ge__(other)
# Define all possible algebraic operations
[docs]
def __add__(self, other):
"""Sums a Function object and 'other', returns a new Function
object which gives the result of the sum. Only implemented for
1D domains.
Parameters
----------
other : Function, int, float, callable
What self will be added to. If other and self are Function
objects which are based on a list of points, have the exact same
domain (are defined in the same grid points) and have the same
dimension, then a special implementation is used.
This implementation is faster, however behavior between grid
points is only interpolated, not calculated as it would be;
the resultant Function has the same interpolation as self.
Returns
-------
result : Function
A Function object which gives the result of self(x)+other(x).
"""
# If other is Function try...
try:
# Check if Function objects source is array or callable
# Check if Function objects have the same domain discretization
if (
isinstance(other.source, np.ndarray)
and isinstance(self.source, np.ndarray)
and self.__dom_dim__ == other.__dom_dim__
and np.array_equal(self.x_array, other.x_array)
):
# Operate on grid values
ys = self.y_array + other.y_array
xs = self.x_array
source = np.concatenate(([xs], [ys])).transpose()
# Retrieve inputs, outputs and interpolation
inputs = self.__inputs__[:]
outputs = self.__outputs__[0] + " + " + other.__outputs__[0]
outputs = "(" + outputs + ")"
interpolation = self.__interpolation__
extrapolation = self.__extrapolation__
# Create new Function object
return Function(source, inputs, outputs, interpolation, extrapolation)
else:
return Function(lambda x: (self.get_value(x) + other(x)))
# If other is Float except...
except AttributeError:
if isinstance(other, NUMERICAL_TYPES):
# Check if Function object source is array or callable
if isinstance(self.source, np.ndarray):
# Operate on grid values
ys = self.y_array + other
xs = self.x_array
source = np.concatenate(([xs], [ys])).transpose()
# Retrieve inputs, outputs and interpolation
inputs = self.__inputs__[:]
outputs = self.__outputs__[0] + " + " + str(other)
outputs = "(" + outputs + ")"
interpolation = self.__interpolation__
extrapolation = self.__extrapolation__
# Create new Function object
return Function(
source, inputs, outputs, interpolation, extrapolation
)
else:
return Function(lambda x: (self.get_value(x) + other))
# Or if it is just a callable
elif callable(other):
return Function(lambda x: (self.get_value(x) + other(x)))
[docs]
def __radd__(self, other):
"""Sums 'other' and a Function object and returns a new Function
object which gives the result of the sum. Only implemented for
1D domains.
Parameters
----------
other : int, float, callable
What self will be added to.
Returns
-------
result : Function
A Function object which gives the result of other(x)/+self(x).
"""
return self + other
[docs]
def __sub__(self, other):
"""Subtracts from a Function object and returns a new Function object
which gives the result of the subtraction. Only implemented for 1D
domains.
Parameters
----------
other : Function, int, float, callable
What self will be subtracted by. If other and self are Function
objects which are based on a list of points, have the exact same
domain (are defined in the same grid points) and have the same
dimension, then a special implementation is used.
This implementation is faster, however behavior between grid
points is only interpolated, not calculated as it would be;
the resultant Function has the same interpolation as self.
Returns
-------
result : Function
A Function object which gives the result of self(x)-other(x).
"""
try:
return self + (-other)
except TypeError:
return Function(lambda x: (self.get_value(x) - other(x)))
[docs]
def __rsub__(self, other):
"""Subtracts a Function object from 'other' and returns a new Function
object which gives the result of the subtraction. Only implemented for
1D domains.
Parameters
----------
other : int, float, callable
What self will subtract from.
Returns
-------
result : Function
A Function object which gives the result of other(x)-self(x).
"""
return other + (-self)
[docs]
def __mul__(self, other):
"""Multiplies a Function object and returns a new Function object
which gives the result of the multiplication. Only implemented for 1D
domains.
Parameters
----------
other : Function, int, float, callable
What self will be multiplied by. If other and self are Function
objects which are based on a list of points, have the exact same
domain (are defined in the same grid points) and have the same
dimension, then a special implementation is used.
This implementation is faster, however behavior between grid
points is only interpolated, not calculated as it would be;
the resultant Function has the same interpolation as self.
Returns
-------
result : Function
A Function object which gives the result of self(x)*other(x).
"""
# If other is Function try...
try:
# Check if Function objects source is array or callable
# Check if Function objects have the same domain discretization
if (
isinstance(other.source, np.ndarray)
and isinstance(self.source, np.ndarray)
and self.__dom_dim__ == other.__dom_dim__
and np.array_equal(self.x_array, other.x_array)
):
# Operate on grid values
ys = self.y_array * other.y_array
xs = self.x_array
source = np.concatenate(([xs], [ys])).transpose()
# Retrieve inputs, outputs and interpolation
inputs = self.__inputs__[:]
outputs = self.__outputs__[0] + "*" + other.__outputs__[0]
outputs = "(" + outputs + ")"
interpolation = self.__interpolation__
extrapolation = self.__extrapolation__
# Create new Function object
return Function(source, inputs, outputs, interpolation, extrapolation)
else:
return Function(lambda x: (self.get_value(x) * other(x)))
# If other is Float except...
except AttributeError:
if isinstance(other, NUMERICAL_TYPES):
# Check if Function object source is array or callable
if isinstance(self.source, np.ndarray):
# Operate on grid values
ys = self.y_array * other
xs = self.x_array
source = np.concatenate(([xs], [ys])).transpose()
# Retrieve inputs, outputs and interpolation
inputs = self.__inputs__[:]
outputs = self.__outputs__[0] + "*" + str(other)
outputs = "(" + outputs + ")"
interpolation = self.__interpolation__
extrapolation = self.__extrapolation__
# Create new Function object
return Function(
source, inputs, outputs, interpolation, extrapolation
)
else:
return Function(lambda x: (self.get_value(x) * other))
# Or if it is just a callable
elif callable(other):
return Function(lambda x: (self.get_value(x) * other(x)))
[docs]
def __rmul__(self, other):
"""Multiplies 'other' by a Function object and returns a new Function
object which gives the result of the multiplication. Only implemented for
1D domains.
Parameters
----------
other : int, float, callable
What self will be multiplied by.
Returns
-------
result : Function
A Function object which gives the result of other(x)*self(x).
"""
return self * other
[docs]
def __truediv__(self, other):
"""Divides a Function object and returns a new Function object
which gives the result of the division. Only implemented for 1D
domains.
Parameters
----------
other : Function, int, float, callable
What self will be divided by. If other and self are Function
objects which are based on a list of points, have the exact same
domain (are defined in the same grid points) and have the same
dimension, then a special implementation is used.
This implementation is faster, however behavior between grid
points is only interpolated, not calculated as it would be;
the resultant Function has the same interpolation as self.
Returns
-------
result : Function
A Function object which gives the result of self(x)/other(x).
"""
# If other is Function try...
try:
# Check if Function objects source is array or callable
# Check if Function objects have the same domain discretization
if (
isinstance(other.source, np.ndarray)
and isinstance(self.source, np.ndarray)
and self.__dom_dim__ == other.__dom_dim__
and np.array_equal(self.x_array, other.x_array)
):
# operate on grid values
with np.errstate(divide="ignore", invalid="ignore"):
ys = self.source[:, 1] / other.source[:, 1]
ys = np.nan_to_num(ys)
xs = self.source[:, 0]
source = np.concatenate(([xs], [ys])).transpose()
# retrieve inputs, outputs and interpolation
inputs = self.__inputs__[:]
outputs = self.__outputs__[0] + "/" + other.__outputs__[0]
outputs = "(" + outputs + ")"
interpolation = self.__interpolation__
extrapolation = self.__extrapolation__
# Create new Function object
return Function(source, inputs, outputs, interpolation, extrapolation)
else:
return Function(lambda x: (self.get_value_opt(x) / other(x)))
# If other is Float except...
except AttributeError:
if isinstance(other, NUMERICAL_TYPES):
# Check if Function object source is array or callable
if isinstance(self.source, np.ndarray):
# Operate on grid values
ys = self.y_array / other
xs = self.x_array
source = np.concatenate(([xs], [ys])).transpose()
# Retrieve inputs, outputs and interpolation
inputs = self.__inputs__[:]
outputs = self.__outputs__[0] + "/" + str(other)
outputs = "(" + outputs + ")"
interpolation = self.__interpolation__
extrapolation = self.__extrapolation__
# Create new Function object
return Function(
source, inputs, outputs, interpolation, extrapolation
)
else:
return Function(lambda x: (self.get_value_opt(x) / other))
# Or if it is just a callable
elif callable(other):
return Function(lambda x: (self.get_value_opt(x) / other(x)))
[docs]
def __rtruediv__(self, other):
"""Divides 'other' by a Function object and returns a new Function
object which gives the result of the division. Only implemented for
1D domains.
Parameters
----------
other : int, float, callable
What self will divide.
Returns
-------
result : Function
A Function object which gives the result of other(x)/self(x).
"""
# Check if Function object source is array and other is float
if isinstance(other, NUMERICAL_TYPES):
if isinstance(self.source, np.ndarray):
# Operate on grid values
ys = other / self.y_array
xs = self.x_array
source = np.concatenate(([xs], [ys])).transpose()
# Retrieve inputs, outputs and interpolation
inputs = self.__inputs__[:]
outputs = str(other) + "/" + self.__outputs__[0]
outputs = "(" + outputs + ")"
interpolation = self.__interpolation__
extrapolation = self.__extrapolation__
# Create new Function object
return Function(source, inputs, outputs, interpolation, extrapolation)
else:
return Function(lambda x: (other / self.get_value_opt(x)))
# Or if it is just a callable
elif callable(other):
return Function(lambda x: (other(x) / self.get_value_opt(x)))
[docs]
def __pow__(self, other):
"""Raises a Function object to the power of 'other' and
returns a new Function object which gives the result. Only
implemented for 1D domains.
Parameters
----------
other : Function, int, float, callable
What self will be raised to. If other and self are Function
objects which are based on a list of points, have the exact same
domain (are defined in the same grid points) and have the same
dimension, then a special implementation is used.
This implementation is faster, however behavior between grid
points is only interpolated, not calculated as it would be;
the resultant Function has the same interpolation as self.
Returns
-------
result : Function
A Function object which gives the result of self(x)**other(x).
"""
# If other is Function try...
try:
# Check if Function objects source is array or callable
# Check if Function objects have the same domain discretization
if (
isinstance(other.source, np.ndarray)
and isinstance(self.source, np.ndarray)
and self.__dom_dim__ == other.__dom_dim__
and np.any(self.x_array - other.x_array) is False
and np.array_equal(self.x_array, other.x_array)
):
# Operate on grid values
ys = self.y_array**other.y_array
xs = self.x_array
source = np.concatenate(([xs], [ys])).transpose()
# Retrieve inputs, outputs and interpolation
inputs = self.__inputs__[:]
outputs = self.__outputs__[0] + "**" + other.__outputs__[0]
outputs = "(" + outputs + ")"
interpolation = self.__interpolation__
extrapolation = self.__extrapolation__
# Create new Function object
return Function(source, inputs, outputs, interpolation, extrapolation)
else:
return Function(lambda x: (self.get_value_opt(x) ** other(x)))
# If other is Float except...
except AttributeError:
if isinstance(other, NUMERICAL_TYPES):
# Check if Function object source is array or callable
if isinstance(self.source, np.ndarray):
# Operate on grid values
ys = self.y_array**other
xs = self.x_array
source = np.concatenate(([xs], [ys])).transpose()
# Retrieve inputs, outputs and interpolation
inputs = self.__inputs__[:]
outputs = self.__outputs__[0] + "**" + str(other)
outputs = "(" + outputs + ")"
interpolation = self.__interpolation__
extrapolation = self.__extrapolation__
# Create new Function object
return Function(
source, inputs, outputs, interpolation, extrapolation
)
else:
return Function(lambda x: (self.get_value(x) ** other))
# Or if it is just a callable
elif callable(other):
return Function(lambda x: (self.get_value(x) ** other(x)))
[docs]
def __rpow__(self, other):
"""Raises 'other' to the power of a Function object and returns
a new Function object which gives the result. Only implemented
for 1D domains.
Parameters
----------
other : int, float, callable
The object that will be exponentiated by the function.
Returns
-------
result : Function
A Function object which gives the result of other(x)**self(x).
"""
# Check if Function object source is array and other is float
if isinstance(other, NUMERICAL_TYPES):
if isinstance(self.source, np.ndarray):
# Operate on grid values
ys = other**self.y_array
xs = self.x_array
source = np.concatenate(([xs], [ys])).transpose()
# Retrieve inputs, outputs and interpolation
inputs = self.__inputs__[:]
outputs = str(other) + "**" + self.__outputs__[0]
outputs = "(" + outputs + ")"
interpolation = self.__interpolation__
extrapolation = self.__extrapolation__
# Create new Function object
return Function(source, inputs, outputs, interpolation, extrapolation)
else:
return Function(lambda x: (other ** self.get_value(x)))
# Or if it is just a callable
elif callable(other):
return Function(lambda x: (other(x) ** self.get_value(x)))
[docs]
def __matmul__(self, other):
"""Operator @ as an alias for composition. Therefore, this
method is a shorthand for Function.compose(other).
Parameters
----------
other : Function
Function object to be composed with self.
Returns
-------
result : Function
A Function object which gives the result of self(other(x)).
See Also
--------
Function.compose
"""
return self.compose(other)
[docs]
def integral(self, a, b, numerical=False):
"""Evaluate a definite integral of a 1-D Function in the interval
from a to b.
Parameters
----------
a : float
Lower limit of integration.
b : float
Upper limit of integration.
numerical : bool
If True, forces the definite integral to be evaluated numerically.
The current numerical method used is scipy.integrate.quad.
If False, try to calculate using interpolation information.
Currently, only available for spline and linear interpolation. If
unavailable, calculate numerically anyways.
Returns
-------
ans : float
Evaluated integral.
"""
# Guarantee a < b
integration_sign = np.sign(b - a)
if integration_sign == -1:
a, b = b, a
# Different implementations depending on interpolation
if self.__interpolation__ == "spline" and numerical is False:
x_data = self.x_array
y_data = self.y_array
coeffs = self.__spline_coefficients__
ans = 0
# Check to see if interval starts before point data
if a < x_data[0]:
if self.__extrapolation__ == "constant":
ans += y_data[0] * (min(x_data[0], b) - a)
elif self.__extrapolation__ == "natural":
c = coeffs[:, 0]
sub_b = a - x_data[0]
sub_a = min(b, x_data[0]) - x_data[0]
ans += (
(c[3] * sub_a**4) / 4
+ (c[2] * sub_a**3 / 3)
+ (c[1] * sub_a**2 / 2)
+ c[0] * sub_a
)
ans -= (
(c[3] * sub_b**4) / 4
+ (c[2] * sub_b**3 / 3)
+ (c[1] * sub_b**2 / 2)
+ c[0] * sub_b
)
else:
# self.__extrapolation__ = 'zero'
pass
# Integrate in subintervals between xs of given data up to b
i = max(np.searchsorted(x_data, a, side="left") - 1, 0)
while i < len(x_data) - 1 and x_data[i] < b:
if x_data[i] <= a <= x_data[i + 1] and x_data[i] <= b <= x_data[i + 1]:
sub_a = a - x_data[i]
sub_b = b - x_data[i]
elif x_data[i] <= a <= x_data[i + 1]:
sub_a = a - x_data[i]
sub_b = x_data[i + 1] - x_data[i]
elif b <= x_data[i + 1]:
sub_a = 0
sub_b = b - x_data[i]
else:
sub_a = 0
sub_b = x_data[i + 1] - x_data[i]
c = coeffs[:, i]
ans += (
(c[3] * sub_b**4) / 4
+ (c[2] * sub_b**3 / 3)
+ (c[1] * sub_b**2 / 2)
+ c[0] * sub_b
)
ans -= (
(c[3] * sub_a**4) / 4
+ (c[2] * sub_a**3 / 3)
+ (c[1] * sub_a**2 / 2)
+ c[0] * sub_a
)
i += 1
# Check to see if interval ends after point data
if b > x_data[-1]:
if self.__extrapolation__ == "constant":
ans += y_data[-1] * (b - max(x_data[-1], a))
elif self.__extrapolation__ == "natural":
c = coeffs[:, -1]
sub_a = max(x_data[-1], a) - x_data[-2]
sub_b = b - x_data[-2]
ans -= (
(c[3] * sub_a**4) / 4
+ (c[2] * sub_a**3 / 3)
+ (c[1] * sub_a**2 / 2)
+ c[0] * sub_a
)
ans += (
(c[3] * sub_b**4) / 4
+ (c[2] * sub_b**3 / 3)
+ (c[1] * sub_b**2 / 2)
+ c[0] * sub_b
)
else:
# self.__extrapolation__ = 'zero'
pass
elif self.__interpolation__ == "linear" and numerical is False:
# Integrate from a to b using np.trapz
x_data = self.x_array
y_data = self.y_array
# Get data in interval
x_integration_data = x_data[(x_data >= a) & (x_data <= b)]
y_integration_data = y_data[(x_data >= a) & (x_data <= b)]
# Add integration limits to data
if self.__extrapolation__ == "zero":
if a >= x_data[0]:
x_integration_data = np.concatenate(([a], x_integration_data))
y_integration_data = np.concatenate(([self(a)], y_integration_data))
if b <= x_data[-1]:
x_integration_data = np.concatenate((x_integration_data, [b]))
y_integration_data = np.concatenate((y_integration_data, [self(b)]))
else:
x_integration_data = np.concatenate(([a], x_integration_data))
y_integration_data = np.concatenate(([self(a)], y_integration_data))
x_integration_data = np.concatenate((x_integration_data, [b]))
y_integration_data = np.concatenate((y_integration_data, [self(b)]))
# Integrate using np.trapz
ans = np.trapz(y_integration_data, x_integration_data)
else:
# Integrate numerically
ans, _ = integrate.quad(self, a, b, epsabs=1e-4, epsrel=1e-3, limit=1000)
return integration_sign * ans
[docs]
def differentiate(self, x, dx=1e-6, order=1):
"""Differentiate a Function object at a given point.
Parameters
----------
x : float
Point at which to differentiate.
dx : float
Step size to use for numerical differentiation.
order : int
Order of differentiation.
Returns
-------
ans : float
Evaluated derivative.
"""
if order == 1:
return (self.get_value(x + dx) - self.get_value(x - dx)) / (2 * dx)
elif order == 2:
return (
self.get_value(x + dx) - 2 * self.get_value(x) + self.get_value(x - dx)
) / dx**2
[docs]
def identity_function(self):
"""Returns a Function object that correspond to the identity mapping,
i.e. f(x) = x.
If the Function object is defined on an array, the identity Function
follows the same discretization, and has linear interpolation and
extrapolation.
If the Function is defined by a lambda, the identity Function is the
identity map 'lambda x: x'.
Returns
-------
result : Function
A Function object that corresponds to the identity mapping.
"""
# Check if Function object source is array
if isinstance(self.source, np.ndarray):
return Function(
np.column_stack((self.x_array, self.x_array)),
inputs=self.__inputs__,
outputs=f"identity of {self.__outputs__}",
interpolation="linear",
extrapolation="natural",
)
else:
return Function(
lambda x: x,
inputs=self.__inputs__,
outputs=f"identity of {self.__outputs__}",
)
[docs]
def derivative_function(self):
"""Returns a Function object which gives the derivative of the Function object.
Returns
-------
result : Function
A Function object which gives the derivative of self.
"""
# Check if Function object source is array
if isinstance(self.source, np.ndarray):
# Operate on grid values
ys = np.diff(self.y_array) / np.diff(self.x_array)
xs = self.source[:-1, 0] + np.diff(self.x_array) / 2
source = np.column_stack((xs, ys))
# Retrieve inputs, outputs and interpolation
inputs = self.__inputs__[:]
outputs = f"d({self.__outputs__[0]})/d({inputs[0]})"
else:
def source_function(x):
return self.differentiate(x)
source = source_function
inputs = self.__inputs__[:]
outputs = f"d({self.__outputs__[0]})/d({inputs[0]})"
# Create new Function object
return Function(
source, inputs, outputs, self.__interpolation__, self.__extrapolation__
)
[docs]
def integral_function(self, lower=None, upper=None, datapoints=100):
"""Returns a Function object representing the integral of the Function
object.
Parameters
----------
lower : scalar, optional
The lower limit of the interval in which the function is to be
evaluated at. If the Function is given by a dataset, the default
value is the start of the dataset.
upper : scalar, optional
The upper limit of the interval in which the function is to be
evaluated at. If the Function is given by a dataset, the default
value is the end of the dataset.
datapoints : int, optional
The number of points in which the integral will be evaluated for
plotting it, which draws lines between each evaluated point.
The default value is 100.
Returns
-------
result : Function
The integral of the Function object.
"""
if isinstance(self.source, np.ndarray):
lower = self.source[0, 0] if lower is None else lower
upper = self.source[-1, 0] if upper is None else upper
x_data = np.linspace(lower, upper, datapoints)
y_data = np.zeros(datapoints)
for i in range(datapoints):
y_data[i] = self.integral(lower, x_data[i])
return Function(
np.column_stack((x_data, y_data)),
inputs=self.__inputs__,
outputs=[o + " Integral" for o in self.__outputs__],
)
else:
lower = 0 if lower is None else lower
return Function(
lambda x: self.integral(lower, x),
inputs=self.__inputs__,
outputs=[o + " Integral" for o in self.__outputs__],
)
[docs]
def isbijective(self):
"""Checks whether the Function is bijective. Only applicable to
Functions whose source is a list of points, raises an error otherwise.
Returns
-------
result : bool
True if the Function is bijective, False otherwise.
"""
if isinstance(self.source, np.ndarray):
x_data_distinct = set(self.x_array)
y_data_distinct = set(self.y_array)
distinct_map = set(zip(x_data_distinct, y_data_distinct))
return len(distinct_map) == len(x_data_distinct) == len(y_data_distinct)
else:
raise TypeError(
"Only Functions whose source is a list of points can be "
"checked for bijectivity."
)
[docs]
def is_strictly_bijective(self):
"""Checks whether the Function is "strictly" bijective.
Only applicable to Functions whose source is a list of points,
raises an error otherwise.
Notes
-----
By "strictly" bijective, this implementation considers the
list-of-points-defined Function bijective between each consecutive pair
of points. Therefore, the Function may be flagged as not bijective even
if the mapping between the set of points which define the Function is
bijective.
Returns
-------
result : bool
True if the Function is "strictly" bijective, False otherwise.
Examples
--------
>>> f = Function([[0, 0], [1, 1], [2, 4]])
>>> f.isbijective()
True
>>> f.is_strictly_bijective()
True
>>> f = Function([[-1, 1], [0, 0], [1, 1], [2, 4]])
>>> f.isbijective()
False
>>> f.is_strictly_bijective()
False
A Function which is not "strictly" bijective, but is bijective, can be
constructed as x^2 defined at -1, 0 and 2.
>>> f = Function([[-1, 1], [0, 0], [2, 4]])
>>> f.isbijective()
True
>>> f.is_strictly_bijective()
False
"""
if isinstance(self.source, np.ndarray):
# Assuming domain is sorted, range must also be
y_data = self.y_array
# Both ascending and descending order means Function is bijective
y_data_diff = np.diff(y_data)
return np.all(y_data_diff >= 0) or np.all(y_data_diff <= 0)
else:
raise TypeError(
"Only Functions whose source is a list of points can be "
"checked for bijectivity."
)
[docs]
def inverse_function(self, approx_func=None, tol=1e-4):
"""
Returns the inverse of the Function. The inverse function of F is a
function that undoes the operation of F. The inverse of F exists if
and only if F is bijective. Makes the domain the range and the range
the domain.
If the Function is given by a list of points, its bijectivity is
checked and an error is raised if it is not bijective.
If the Function is given by a function, its bijection is not
checked and may lead to inaccuracies outside of its bijective region.
Parameters
----------
approx_func : callable, optional
A function that approximates the inverse of the Function. This
function is used to find the starting guesses for the inverse
root finding algorithm. This is better used when the inverse
in complex but has a simple approximation or when the root
finding algorithm performs poorly due to default start point.
The default is None in which case the starting point is zero.
tol : float, optional
The tolerance for the inverse root finding algorithm. The default
is 1e-4.
Returns
-------
result : Function
A Function whose domain and range have been inverted.
"""
if isinstance(self.source, np.ndarray):
if self.is_strictly_bijective():
# Swap the columns
source = np.flip(self.source, axis=1)
else:
raise ValueError(
"Function is not bijective, so it does not have an inverse."
)
else:
if approx_func is not None:
def source_function(x):
return self.find_input(x, start=approx_func(x), tol=tol)
else:
def source_function(x):
return self.find_input(x, start=0, tol=tol)
source = source_function
return Function(
source,
inputs=self.__outputs__,
outputs=self.__inputs__,
interpolation=self.__interpolation__,
extrapolation=self.__extrapolation__,
)
[docs]
def average(self, lower, upper):
"""
Returns the average of the function.
Parameters
----------
lower : float
Lower point of the region that the average will be calculated at.
upper : float
Upper point of the region that the average will be calculated at.
Returns
-------
result : float
The average of the function.
"""
return self.integral(lower, upper) / (upper - lower)
[docs]
def average_function(self, lower=None):
"""
Returns a Function object representing the average of the Function
object.
Parameters
----------
lower : float
Lower limit of the new domain. Only required if the Function's
source is a callable instead of a list of points.
Returns
-------
result : Function
The average of the Function object.
"""
if isinstance(self.source, np.ndarray):
if lower is None:
lower = self.source[0, 0]
upper = self.source[-1, 0]
x_data = np.linspace(lower, upper, 100)
y_data = np.zeros(100)
y_data[0] = self.source[:, 1][0]
for i in range(1, 100):
y_data[i] = self.average(lower, x_data[i])
return Function(
np.concatenate(([x_data], [y_data])).transpose(),
inputs=self.__inputs__,
outputs=[o + " Average" for o in self.__outputs__],
)
else:
lower = 0 if lower is None else lower
return Function(
lambda x: self.average(lower, x),
inputs=self.__inputs__,
outputs=[o + " Average" for o in self.__outputs__],
)
[docs]
def compose(self, func, extrapolate=False):
"""
Returns a Function object which is the result of inputting a function
into a function (i.e. f(g(x))). The domain will become the domain of
the input function and the range will become the range of the original
function.
Parameters
----------
func : Function
The function to be inputted into the function.
extrapolate : bool, optional
Whether or not to extrapolate the function if the input function's
range is outside of the original function's domain. The default is
False.
Returns
-------
result : Function
The result of inputting the function into the function.
"""
# Check if the input is a function
if not isinstance(func, Function):
raise TypeError("Input must be a Function object.")
if isinstance(self.source, np.ndarray) and isinstance(func.source, np.ndarray):
# Perform bounds check for composition
if not extrapolate:
if func.min < self.x_initial or func.max > self.x_final:
raise ValueError(
f"Input Function image {func.min, func.max} must be within "
f"the domain of the Function {self.x_initial, self.x_final}."
)
return Function(
np.concatenate(([func.x_array], [self(func.y_array)])).T,
inputs=func.__inputs__,
outputs=self.__outputs__,
interpolation=self.__interpolation__,
extrapolation=self.__extrapolation__,
)
else:
return Function(
lambda x: self(func(x)),
inputs=func.__inputs__,
outputs=self.__outputs__,
interpolation=self.__interpolation__,
extrapolation=self.__extrapolation__,
)
[docs]
def savetxt(
self,
filename,
lower=None,
upper=None,
samples=None,
fmt="%.6f",
delimiter=",",
newline="\n",
encoding=None,
):
"""Save a Function object to a text file. The first line is the header
with inputs and outputs. The following lines are the data. The text file
can have any extension, but it is recommended to use .csv or .txt.
Parameters
----------
filename : str
The name of the file to be saved, with the extension.
lower : float or int, optional
The lower bound of the range for which data is to be generated.
This is required if the source is a callable function.
upper : float or int, optional
The upper bound of the range for which data is to be generated.
This is required if the source is a callable function.
samples : int, optional
The number of sample points to generate within the specified range.
This is required if the source is a callable function.
fmt : str, optional
The format string for each line of the file, by default "%.6f".
delimiter : str, optional
The string used to separate values, by default ",".
newline : str, optional
The string used to separate lines in the file, by default "\n".
encoding : str, optional
The encoding to be used for the file, by default None (which means
using the system default encoding).
Raises
------
ValueError
Raised if `lower`, `upper`, and `samples` are not provided when
the source is a callable function. These parameters are necessary
to generate the data points for saving.
"""
# create the header
header_line = delimiter.join(self.__inputs__ + self.__outputs__)
# create the datapoints
if callable(self.source):
if lower is None or upper is None or samples is None:
raise ValueError(
"If the source is a callable, lower, upper and samples"
+ " must be provided."
)
# Generate the data points using the callable
x = np.linspace(lower, upper, samples)
data_points = np.column_stack((x, self.source(x)))
else:
# If the source is already an array, use it as is
data_points = self.source
if lower and upper and samples:
data_points = self.set_discrete(
lower, upper, samples, mutate_self=False
).source
# export to a file
with open(filename, "w", encoding=encoding) as file:
file.write(header_line + newline)
np.savetxt(file, data_points, fmt=fmt, delimiter=delimiter, newline=newline)
class PiecewiseFunction(Function):
"""Class for creating piecewise functions. These kind of functions are
defined by a dictionary of functions, where the keys are tuples that
represent the domain of the function. The domains must be disjoint.
"""
def __new__(
cls,
source,
inputs=None,
outputs=None,
interpolation="spline",
extrapolation=None,
datapoints=100,
):
"""
Creates a piecewise function from a dictionary of functions. The keys of
the dictionary must be tuples that represent the domain of the function.
The domains must be disjoint. The piecewise function will be evaluated
at datapoints points to create Function object.
Parameters
----------
source: dictionary
A dictionary of Function objects, where the keys are the domains.
inputs : list of strings
A list of strings that represent the inputs of the function.
outputs: list of strings
A list of strings that represent the outputs of the function.
interpolation: str
The type of interpolation to use. The default value is 'spline'.
extrapolation: str
The type of extrapolation to use. The default value is None.
datapoints: int
The number of points in which the piecewise function will be
evaluated to create a base function. The default value is 100.
"""
if inputs is None:
inputs = ["Scalar"]
if outputs is None:
outputs = ["Scalar"]
# Check if source is a dictionary
if not isinstance(source, dict):
raise TypeError("source must be a dictionary")
# Check if all keys are tuples
for key in source.keys():
if not isinstance(key, tuple):
raise TypeError("keys of source must be tuples")
# Check if all domains are disjoint
for key1 in source.keys():
for key2 in source.keys():
if key1 != key2:
if key1[0] < key2[1] and key1[1] > key2[0]:
raise ValueError("domains must be disjoint")
# Crate Function
def calc_output(func, inputs):
"""Receives a list of inputs value and a function, populates another
list with the results corresponding to the same results.
Parameters
----------
func : Function
The Function object to be
inputs : list, tuple, np.array
The array of points to applied the func to.
Examples
--------
>>> inputs = [0, 1, 2, 3, 4, 5]
>>> def func(x):
... return x*10
>>> calc_output(func, inputs)
[0, 10, 20, 30, 40, 50]
Notes
-----
In the future, consider using the built-in map function from python.
"""
output = np.zeros(len(inputs))
for j, value in enumerate(inputs):
output[j] = func.get_value(value)
return output
input_data = []
output_data = []
for key in sorted(source.keys()):
i = np.linspace(key[0], key[1], datapoints)
i = i[~np.in1d(i, input_data)]
input_data = np.concatenate((input_data, i))
f = Function(source[key])
output_data = np.concatenate((output_data, calc_output(f, i)))
return Function(
np.concatenate(([input_data], [output_data])).T,
inputs=inputs,
outputs=outputs,
interpolation=interpolation,
extrapolation=extrapolation,
)
def funcify_method(*args, **kwargs):
"""Decorator factory to wrap methods as Function objects and save them as
cached properties.
Parameters
----------
*args : list
Positional arguments to be passed to rocketpy.Function.
**kwargs : dict
Keyword arguments to be passed to rocketpy.Function.
Returns
-------
decorator : function
Decorator function to wrap callables as Function objects.
Examples
--------
There are 3 types of methods that this decorator supports:
1. Method which returns a valid rocketpy.Function source argument.
>>> from rocketpy.mathutils import funcify_method
>>> class Example():
... @funcify_method(inputs=['x'], outputs=['y'])
... def f(self):
... return lambda x: x**2
>>> example = Example()
>>> example.f
'Function from R1 to R1 : (x) → (y)'
Normal algebra can be performed afterwards:
>>> g = 2*example.f + 3
>>> g(2)
11
2. Method which returns a rocketpy.Function instance. An interesting use is
to reset input and output names after algebraic operations.
>>> class Example():
... @funcify_method(inputs=['x'], outputs=['x**3'])
... def cube(self):
... f = Function(lambda x: x**2)
... g = Function(lambda x: x**5)
... return g / f
>>> example = Example()
>>> example.cube
'Function from R1 to R1 : (x) → (x**3)'
3. Method which is itself a valid rocketpy.Function source argument.
>>> class Example():
... @funcify_method('x', 'f(x)')
... def f(self, x):
... return x**2
>>> example = Example()
>>> example.f
'Function from R1 to R1 : (x) → (f(x))'
In order to reset the cache, just delete de attribute from the instance:
>>> del example.f
Once it is requested again, it will be re-created as a new Function object:
>>> example.f
'Function from R1 to R1 : (x) → (f(x))'
"""
func = None
if len(args) == 1 and callable(args[0]):
func = args[0]
args = []
class funcify_method_decorator:
"""Decorator class to transform a cached property that is being defined
inside a class to a Function object. This improves readability of the
code since it will not require the user to directly invoke the Function
class.
"""
# pylint: disable=C0103,R0903
def __init__(self, func):
self.func = func
self.attrname = None
self.__doc__ = func.__doc__
def __set_name__(self, owner, name):
self.attrname = name
def __get__(self, instance, owner=None):
if instance is None:
return self
cache = instance.__dict__
try:
# If cache is ready, return it
val = cache[self.attrname]
except KeyError:
# If cache is not ready, create it
try:
# Handle methods which return Function instances
val = self.func(instance).reset(*args, **kwargs)
except AttributeError:
# Handle methods which return a valid source
source = self.func(instance)
val = Function(source, *args, **kwargs)
except TypeError:
# Handle methods which are the source themselves
def source_function(*_):
return self.func(instance, *_)
source = source_function
val = Function(source, *args, **kwargs)
# pylint: disable=W0201
val.__doc__ = self.__doc__
val.__cached__ = True
cache[self.attrname] = val
return val
if func:
return funcify_method_decorator(func)
else:
return funcify_method_decorator
def reset_funcified_methods(instance):
"""Resets all the funcified methods of the instance. It does so by
deleting the current Functions, which will make the interpreter redefine
them when they are called. This is useful when the instance has changed
and the methods need to be recalculated.
Parameters
----------
instance : object
The instance of the class whose funcified methods will be recalculated.
The class must have a mutable __dict__ attribute.
Return
------
None
"""
for key in list(instance.__dict__):
if hasattr(instance.__dict__[key], "__cached__"):
instance.__dict__.pop(key)
if __name__ == "__main__":
import doctest
results = doctest.testmod()
if results.failed < 1:
print(f"All the {results.attempted} tests passed!")
else:
print(f"{results.failed} out of {results.attempted} tests failed.")