scipy.integrate — Numerical integration

The scipy.integrate submodule provides numerical integration routines for real-valued, single-variable functions. The algorithms are not ported from CPython’s scipy.integrate but derived from the reference implementation at https://www.genivia.com/qthsh.html.

Numerical integration works best with float64 math enabled. With float32 math the routines still work, with reduced precision. The required error tolerance can be specified via the eps keyword argument; the default is the compile-time etolerance value (1e-14 for fp64, 1e-8 for fp32). Complex numbers are not supported.

Functions

scipy.integrate.quad(f: Callable[[float], float], a: float, b: float, *, order: int = 5, eps: float = etolerance) → tuple[float, float]

Integrate f from a to b using an Adaptive Gauss-Kronrod (G10, K21) quadrature. This is the recommended general-purpose integrator.

Parameters:

  • f – callable taking a single float and returning a float.

  • a – lower integration limit.

  • b – upper integration limit.

  • order – order of integration (default 5).

  • eps – error tolerance (default etolerance).

Returns:

a 2-tuple (result, error) of floats giving the value of the integral and an error estimate.

scipy.integrate.romberg(f: Callable[[float], float], a: float, b: float, *, steps: int = 100, eps: float = etolerance) → float

Integrate f from a to b using Romberg’s method, a Newton-Cotes formula that evaluates the integrand at equally spaced points. Best suited to integrands with continuous derivatives. Note that scipy.integrate.romberg is deprecated in CPython since SciPy 1.12.0; prefer quad() for new code.

Parameters:

  • f – callable taking a single float and returning a float.

  • a – lower integration limit.

  • b – upper integration limit.

  • steps – number of steps (default 100).

  • eps – error tolerance (default etolerance).

Returns:

the value of the integral as a float.

scipy.integrate.simpson(f: Callable[[float], float], a: float, b: float, *, steps: int = 100, eps: float = etolerance) → float

Integrate f from a to b using adaptive Simpson’s rule. Unlike CPython’s scipy.integrate.simpson this function takes a callable and chooses the sample spacing internally rather than taking an array of pre-computed function values.

Parameters:

  • f – callable taking a single float and returning a float.

  • a – lower integration limit.

  • b – upper integration limit.

  • steps – number of steps (default 100).

  • eps – error tolerance (default etolerance).

Returns:

the value of the integral as a float.

scipy.integrate.tanhsinh(f: Callable[[float], float], a: float, b: float, *, levels: int = 6, eps: float = etolerance) → tuple[float, float]

Integrate f from a to b using the Tanh-Sinh, Sinh-Sinh, and Exp-Sinh (double-exponential) quadrature family. This is the routine to use when the integrand has singularities or infinite derivatives at the endpoints, and it is the only routine in this submodule that accepts infinite integration limits (e.g. -np.inf, np.inf).

Parameters:

  • f – callable taking a single float and returning a float.

  • a – lower integration limit; may be -np.inf.

  • b – upper integration limit; may be np.inf.

  • levels – number of refinement levels (default 6).

  • eps – error tolerance (default etolerance).

Returns:

a 2-tuple (result, error) of floats giving the value of the integral and an error estimate.