API

class idesolver.IDESolver(x: numpy.ndarray, y_0: Union[float, numpy.float64, complex, numpy.complex128, numpy.ndarray, list], c: Optional[Callable] = None, d: Optional[Callable] = None, k: Optional[Callable] = None, f: Optional[Callable] = None, lower_bound: Optional[Callable] = None, upper_bound: Optional[Callable] = None, global_error_tolerance: float = 1e-06, max_iterations: Optional[int] = 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: 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: Optional[Callable] = None) → numpy.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: numpy.ndarray, y2: numpy.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: 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.