code: def quadrature(
      func,
      a,
      b,
      args=(),
      tol=1.49e-8,
      rtol=1.49e-8,
      maxiter=50,
      vec_func=True,
      miniter=1,
  ):
      """
      Compute a definite integral using fixed-tolerance Gaussian quadrature.
  
      Integrate `func` from `a` to `b` using Gaussian quadrature
      with absolute tolerance `tol`.
  
      Parameters
      ----------
      func : function
          A Python function or method to integrate.
      a : float
          Lower limit of integration.
      b : float
          Upper limit of integration.
      args : tuple, optional
          Extra arguments to pass to function.
      tol, rtol : float, optional
          Iteration stops when error between last two iterates is less than
          `tol` OR the relative change is less than `rtol`.
      maxiter : int, optional
          Maximum order of Gaussian quadrature.
      vec_func : bool, optional
          True or False if func handles arrays as arguments (is
          a "vector" function). Default is True.
      miniter : int, optional
          Minimum order of Gaussian quadrature.
  
      Returns
      -------
      val : float
          Gaussian quadrature approximation (within tolerance) to integral.
      err : float
          Difference between last two estimates of the integral.
  
      See also
      --------
      romberg: adaptive Romberg quadrature
      fixed_quad: fixed-order Gaussian quadrature
      quad: adaptive quadrature using QUADPACK
      dblquad: double integrals
      tplquad: triple integrals
      romb: integrator for sampled data
      simpson: integrator for sampled data
      cumulative_trapezoid: cumulative integration for sampled data
      ode: ODE integrator
      odeint: ODE integrator
  
      Examples
      --------
      >>> from scipy import integrate
      >>> f = lambda x: x**8
      >>> integrate.quadrature(f, 0.0, 1.0)
      (0.11111111111111106, 4.163336342344337e-17)
      >>> print(1/9.0)  # analytical result
      0.1111111111111111
  
      >>> integrate.quadrature(np.cos, 0.0, np.pi/2)
      (0.9999999999999536, 3.9611425250996035e-11)
      >>> np.sin(np.pi/2)-np.sin(0)  # analytical result
      1.0
  
      """
      if not isinstance(args, tuple):
          args = (args,)
      vfunc = vectorize1(func, args, vec_func=vec_func)
      val = np.inf
      err = np.inf
      maxiter = max(miniter + 1, maxiter)
      for n in range(miniter, maxiter + 1):
          newval = fixed_quad(vfunc, a, b, (), n)[0]
          err = abs(newval - val)
          val = newval
  
          if err < tol or err < rtol * abs(val):
              break
      else:
          warnings.warn(
              "maxiter (%d) exceeded. Latest difference = %e"
              % (maxiter, err),
              AccuracyWarning,
          )
      return val, err
  
imports: from __future__ import annotations
  from typing import TYPE_CHECKING, Callable, Dict, Tuple, Any, cast
  import functools
  import numpy
  import math
  import types
  import warnings
  from numpy import trapz
  from scipy.special import roots_legendre
  from scipy.special import gammaln
label_desc: Use image node_id 7 for calling a global function with example usage: quadrature(func, a, b, args, tol, rtol, maxiter, vec_func, miniter) and returns: val,  err