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Parameters ---------- f : callable Vector-valued function f(x) to integrate. a : float Initial point. b : float Final point. epsabs : float, optional Absolute tolerance. epsrel : float, optional Relative tolerance. norm : {'max', '2'}, optional Vector norm to use for error estimation. cache_size : int, optional Number of bytes to use for memoization. limit : float or int, optional An upper bound on the number of subintervals used in the adaptive algorithm. workers : int or map-like callable, optional If `workers` is an integer, part of the computation is done in parallel subdivided to this many tasks (using :class:`python:multiprocessing.pool.Pool`). Supply `-1` to use all cores available to the Process. Alternatively, supply a map-like callable, such as :meth:`python:multiprocessing.pool.Pool.map` for evaluating the population in parallel. This evaluation is carried out as ``workers(func, iterable)``. points : list, optional List of additional breakpoints. quadrature : {'gk21', 'gk15', 'trapezoid'}, optional Quadrature rule to use on subintervals. Options: 'gk21' (Gauss-Kronrod 21-point rule), 'gk15' (Gauss-Kronrod 15-point rule), 'trapezoid' (composite trapezoid rule). Default: 'gk21' for finite intervals and 'gk15' for (semi-)infinite full_output : bool, optional Return an additional ``info`` dictionary. args : tuple, optional Extra arguments to pass to function, if any. .. versionadded:: 1.8.0 Returns ------- res : {float, array-like} Estimate for the result err : float Error estimate for the result in the given norm info : dict Returned only when ``full_output=True``. Info dictionary. Is an object with the attributes: success : bool Whether integration reached target precision. status : int Indicator for convergence, success (0), failure (1), and failure due to rounding error (2). neval : int Number of function evaluations. intervals : ndarray, shape (num_intervals, 2) Start and end points of subdivision intervals. integrals : ndarray, shape (num_intervals, ...) Integral for each interval. Note that at most ``cache_size`` values are recorded, and the array may contains *nan* for missing items. errors : ndarray, shape (num_intervals,) Estimated integration error for each interval. Notes ----- The algorithm mainly follows the implementation of QUADPACK's DQAG* algorithms, implementing global error control and adaptive subdivision. The algorithm here has some differences to the QUADPACK approach: Instead of subdividing one interval at a time, the algorithm subdivides N intervals with largest errors at once. This enables (partial) parallelization of the integration. The logic of subdividing "next largest" intervals first is then not implemented, and we rely on the above extension to avoid concentrating on "small" intervals only. The Wynn epsilon table extrapolation is not used (QUADPACK uses it for infinite intervals). This is because the algorithm here is supposed to work on vector-valued functions, in an user-specified norm, and the extension of the epsilon algorithm to this case does not appear to be widely agreed. For max-norm, using elementwise Wynn epsilon could be possible, but we do not do this here with the hope that the epsilon extrapolation is mainly useful in special cases. References ---------- [1] R. Piessens, E. de Doncker, QUADPACK (1983). Examples -------- We can compute integrations of a vector-valued function: >>> from scipy.integrate import quad_vec >>> import numpy as np >>> import matplotlib.pyplot as plt >>> alpha = np.linspace(0.0, 2.0, num=30) >>> f = lambda x: x**alpha >>> x0, x1 = 0, 2 >>> y, err = quad_vec(f, x0, x1) >>> plt.plot(alpha, y) >>> plt.xlabel(r"$\alpha$") >>> plt.ylabel(r"$\int_{0}^{2} x^\alpha dx$") >>> plt.show() When using the argument `workers`, one should ensure that the main module is import-safe, for instance by rewriting the example above as: .. code-block:: python from scipy.integrate import quad_vec import numpy as np import matplotlib.pyplot as plt alpha = np.linspace(0.0, 2.0, num=30) x0, x1 = 0, 2 def f(x): return x**alpha if __name__ == "__main__": y, err = quad_vec(f, x0, x1, workers=2) Ngk15) epsabsepsrelnorm cache_sizelimitworkerspoints quadrature full_output)r0r1c3@K|]}j|ywrr7r`xpf2s r rbzquad_vec.. $CbRXXb\$Crprr&c3@K|]}j|ywrrtrus r rbzquad_vec..rxryr%)rc3@K|]}j|ywrrtrus r rbzquad_vec..s+JRBHHRL+Jryzinvalid integration bounds a=z, b=)Nmax2)Ngk21ritrapz trapezoidzunknown quadrature rzz`quadrature='trapz'` is deprecated in favour of `quadrature='trapezoid' and will raise an error from SciPy 1.16.0 onwards.) stacklevelzTarget precision reached.zTarget precision not reached.z* ( q5W|c!e';; ==3$,! 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