Integration and ODEs (:mod:`scipy.integrate`)

.. currentmodule:: scipy.integrate

Integrating functions, given function object

.. autosummary:: 
    :toctree:generated/
    quad          -- General purpose integration
    quad_vec      -- General purpose integration of vector-valued functions
    dblquad       -- General purpose double integration
    tplquad       -- General purpose triple integration
    nquad         -- General purpose N-D integration
    fixed_quad    -- Integrate func(x) using Gaussian quadrature of order n
    quadrature    -- Integrate with given tolerance using Gaussian quadrature
    romberg       -- Integrate func using Romberg integration
    quad_explain  -- Print information for use of quad
    newton_cotes  -- Weights and error coefficient for Newton-Cotes integration
    IntegrationWarning -- Warning on issues during integration
    AccuracyWarning  -- Warning on issues during quadrature integration

Integrating functions, given fixed samples

.. autosummary:: 
    :toctree:generated/
    trapezoid            -- Use trapezoidal rule to compute integral.
    cumulative_trapezoid -- Use trapezoidal rule to cumulatively compute integral.
    simpson              -- Use Simpson's rule to compute integral from samples.
    romb                 -- Use Romberg Integration to compute integral from
                         -- (2**k + 1) evenly-spaced samples.

.. seealso:: 
    :mod:`scipy.special` for orthogonal polynomials (special) for Gaussian
    quadrature roots and weights for other weighting factors and regions.

Solving initial value problems for ODE systems

The solvers are implemented as individual classes, which can be used directly (low-level usage) or through a convenience function.

.. autosummary:: 
    :toctree:generated/
    solve_ivp     -- Convenient function for ODE integration.
    RK23          -- Explicit Runge-Kutta solver of order 3(2).
    RK45          -- Explicit Runge-Kutta solver of order 5(4).
    DOP853        -- Explicit Runge-Kutta solver of order 8.
    Radau         -- Implicit Runge-Kutta solver of order 5.
    BDF           -- Implicit multi-step variable order (1 to 5) solver.
    LSODA         -- LSODA solver from ODEPACK Fortran package.
    OdeSolver     -- Base class for ODE solvers.
    DenseOutput   -- Local interpolant for computing a dense output.
    OdeSolution   -- Class which represents a continuous ODE solution.

Old API

These are the routines developed earlier for SciPy. They wrap older solvers implemented in Fortran (mostly ODEPACK). While the interface to them is not particularly convenient and certain features are missing compared to the new API, the solvers themselves are of good quality and work fast as compiled Fortran code. In some cases, it might be worth using this old API.

.. autosummary:: 
    :toctree:generated/
    odeint        -- General integration of ordinary differential equations.
    ode           -- Integrate ODE using VODE and ZVODE routines.
    complex_ode   -- Convert a complex-valued ODE to real-valued and integrate.

Solving boundary value problems for ODE systems

.. autosummary:: 
    :toctree:generated/
    solve_bvp     -- Solve a boundary value problem for a system of ODEs.

Examples

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

scipy.integrate._quadrature.simpson scipy.integrate._bvp.solve_bvp scipy.integrate._quadpack_py.dblquad scipy.integrate._odepack_py.odeint scipy.integrate._quadrature.quadrature scipy.integrate._quadrature.romb scipy.integrate._quadrature.romberg scipy.integrate._quadrature.fixed_quad scipy.integrate._quadrature.cumulative_trapezoid scipy.integrate._ivp.ivp.solve_ivp scipy.integrate._quad_vec.quad_vec scipy.integrate._quadpack_py.quad scipy.integrate._quadrature.newton_cotes scipy.integrate._quadpack_py.nquad scipy.integrate._ode.ode scipy.integrate._quadpack_py.tplquad

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