API

class idesolver.IDESolver(x: ~numpy.ndarray, y_0: float | ~numpy.float64 | complex | ~numpy.complex128 | ~numpy.ndarray | list, c: ~typing.Callable | None = None, d: ~typing.Callable | None = None, k: ~typing.Callable | None = None, f: ~typing.Callable | None = None, lower_bound: ~typing.Callable | None = None, upper_bound: ~typing.Callable | None = None, global_error_tolerance: float = 1e-06, max_iterations: int | None = None, ode_method: str = 'RK45', ode_atol: float = 1e-08, ode_rtol: float = 1e-08, int_atol: float = 1e-08, int_rtol: float = 1e-08, interpolation_kind: str = 'cubic', smoothing_factor: float = 0.5, store_intermediate_y: bool = False, global_error_function: ~typing.Callable = <function global_error>)[source]

A class that handles solving an integro-differential equation of the form

\[\begin{split}\frac{dy}{dx} & = c(y, x) + d(x) \int_{\alpha(x)}^{\beta(x)} k(x, s) \, F( y(s) ) \, ds, \\ & x \in [a, b], \quad y(a) = y_0.\end{split}\]
x

The positions where the solution is calculated (i.e., where \(y\) is evaluated).

Type:

numpy.ndarray

y

The solution \(y(x)\). None until IDESolver.solve() is finished.

Type:

numpy.ndarray

global_error

The final global error estimate. None until IDESolver.solve() is finished.

Type:

float

iteration

The current iteration. None until IDESolver.solve() starts.

Type:

int

y_intermediate

The intermediate solutions. Only exists if store_intermediate_y is True.

Parameters:

  • x (numpy.ndarray) – The array of \(x\) values to find the solution \(y(x)\) at. Generally something like numpy.linspace(a, b, num_pts).

  • y_0 (float or complex or numpy.ndarray) – The initial condition, \(y_0 = y(a)\) (can be multidimensional).

  • c – The function \(c(y, x)\). Defaults to \(c(y, x) = 0\).

  • d – The function \(d(x)\). Defaults to \(d(x) = 1\).

  • k – The kernel function \(k(x, s)\). Defaults to \(k(x, s) = 1\).

  • f – The function \(F(y)\). Defaults to \(f(y) = 0\).

  • lower_bound – The lower bound function \(\alpha(x)\). Defaults to the first element of x.

  • upper_bound – The upper bound function \(\beta(x)\). Defaults to the last element of x.

  • global_error_tolerance (float) – The algorithm will continue until the global errors goes below this or uses more than max_iterations iterations. If None, the algorithm continues until hitting max_iterations.

  • max_iterations (int) – The maximum number of iterations to use. If None, iteration will not stop unless the global_error_tolerance is satisfied. Defaults to None.

  • ode_method (str) – The ODE solution method to use. As the method option of scipy.integrate.solve_ivp(). Defaults to 'RK45', which is good for non-stiff systems.

  • ode_atol (float) – The absolute tolerance for the ODE solver. As the atol argument of scipy.integrate.solve_ivp().

  • ode_rtol (float) – The relative tolerance for the ODE solver. As the rtol argument of scipy.integrate.solve_ivp().

  • int_atol (float) – The absolute tolerance for the integration routine. As the epsabs argument of scipy.integrate.quad().

  • int_rtol (float) – The relative tolerance for the integration routine. As the epsrel argument of scipy.integrate.quad().

  • interpolation_kind (str) – The type of interpolation to use. As the kind argument of scipy.interpolate.interp1d. Defaults to 'cubic'.

  • smoothing_factor (float) – The smoothing factor used to combine the current guess with the new guess at each iteration. Defaults to 0.5.

  • store_intermediate_y (bool) – If True, the intermediate guesses for \(y(x)\) at each iteration will be stored in the attribute y_intermediate.

  • global_error_function – The function to use to calculate the global error. Defaults to global_error().

solve(callback: Callable | None = None) → ndarray[source]

Compute the solution to the IDE.

Will emit a warning message if the global error increases on an iteration. This does not necessarily mean that the algorithm is not converging, but may indicate that it’s having problems.

Will emit a warning message if the maximum number of iterations is used without reaching the global error tolerance.

Parameters:

callback – A function to call after each iteration. The function is passed the IDESolver instance, the current \(y\) guess, and the current global error.

Returns:

The solution to the IDE (i.e., \(y(x)\)).

Return type:

numpy.ndarray

idesolver.global_error(y1: ndarray, y2: ndarray) → float[source]

The default global error function.

The estimate is the square root of the sum of squared differences between y1 and y2.

Parameters:
Returns:

error – The global error estimate between y1 and y2.

Return type:

float

idesolver.complex_quad(integrand: ~typing.Callable, lower_bound: float, upper_bound: float, **kwargs) -> (<class 'complex'>, <class 'float'>, <class 'float'>, <class 'tuple'>, <class 'tuple'>)[source]

A thin wrapper over scipy.integrate.quad() that handles splitting the real and complex parts of the integral and recombining them. Keyword arguments are passed to both of the internal quad calls.