scipyintegrate

SciPy integrate, or scipy.integrate, is a subpackage of SciPy focused on numerical integration and the solution of ordinary differential equations. It provides a collection of routines for definite integrals in one or more dimensions, as well as tools for solving initial-value problems and stiff systems. The module draws on established Fortran routines for reliability and offers Pythonic interfaces.

For one-dimensional integrals, the primary function is quad(func, a, b, ...), which returns the integral value and an estimate of the absolute error. Quad relies on the QUADPACK library. Additional one-dimensional options include fixed_quad for fixed Gauss-Legendre quadrature, quadrature for higher-precision adaptive rules, and romberg for Romberg extrapolation. For vector-valued integrands, quad_vec evaluates the integral of each component and returns an array of results.

Multi-dimensional integration is supported via dblquad and tplquad for nested integrals, and nquad for higher-dimensional problems. These functions accept limits that may be constants or functions of other variables and return a value with an error estimate. In addition, scipy.integrate provides simps for Simpson’s rule as a quick, low-cost approximation.

The module also covers numerical integration of ordinary differential equations. The most common tools are solve_ivp, which provides several integration methods such as RK45, RK23, DOP853, Radau, and BDF, and is the recommended interface for solving initial-value problems. ODEINT is an older interface available for backward compatibility and uses the LSODA solver from the ODEPACK library.

Overall, scipy.integrate offers a versatile suite of functions for precise, reliable numerical integration and differential equation solving within the SciPy ecosystem.