code: def romberg(
      function,
      a,
      b,
      args=(),
      tol=1.48e-8,
      rtol=1.48e-8,
      show=False,
      divmax=10,
      vec_func=False,
  ):
      """
      Romberg integration of a callable function or method.
  
      Returns the integral of `function` (a function of one variable)
      over the interval (`a`, `b`).
  
      If `show` is 1, the triangular array of the intermediate results
      will be printed. If `vec_func` is True (default is False), then
      `function` is assumed to support vector arguments.
  
      Parameters
      ----------
      function : callable
          Function to be integrated.
      a : float
          Lower limit of integration.
      b : float
          Upper limit of integration.
  
      Returns
      -------
      results  : float
          Result of the integration.
  
      Other Parameters
      ----------------
      args : tuple, optional
          Extra arguments to pass to function. Each element of `args` will
          be passed as a single argument to `func`. Default is to pass no
          extra arguments.
      tol, rtol : float, optional
          The desired absolute and relative tolerances. Defaults are 1.48e-8.
      show : bool, optional
          Whether to print the results. Default is False.
      divmax : int, optional
          Maximum order of extrapolation. Default is 10.
      vec_func : bool, optional
          Whether `func` handles arrays as arguments (i.e., whether it is a
          "vector" function). Default is False.
  
      See Also
      --------
      fixed_quad : Fixed-order Gaussian quadrature.
      quad : Adaptive quadrature using QUADPACK.
      dblquad : Double integrals.
      tplquad : Triple integrals.
      romb : Integrators for sampled data.
      simpson : Integrators for sampled data.
      cumulative_trapezoid : Cumulative integration for sampled data.
      ode : ODE integrator.
      odeint : ODE integrator.
  
      References
      ----------
      .. [1] 'Romberg's method' https://en.wikipedia.org/wiki/Romberg%27s_method
  
      Examples
      --------
      Integrate a gaussian from 0 to 1 and compare to the error function.
  
      >>> from scipy import integrate
      >>> from scipy.special import erf
      >>> gaussian = lambda x: 1/np.sqrt(np.pi) * np.exp(-x**2)
      >>> result = integrate.romberg(gaussian, 0, 1, show=True)
      Romberg integration of <function vfunc at ...> from [0, 1]
  
      ::
  
         Steps  StepSize  Results
             1  1.000000  0.385872
             2  0.500000  0.412631  0.421551
             4  0.250000  0.419184  0.421368  0.421356
             8  0.125000  0.420810  0.421352  0.421350  0.421350
            16  0.062500  0.421215  0.421350  0.421350  0.421350  0.421350
            32  0.031250  0.421317  0.421350  0.421350  0.421350  0.421350  0.421350
  
      The final result is 0.421350396475 after 33 function evaluations.
  
      >>> print("%g %g" % (2*result, erf(1)))
      0.842701 0.842701
  
      """
      if np.isinf(a) or np.isinf(b):
          raise ValueError(
              "Romberg integration only available " "for finite limits."
          )
      vfunc = vectorize1(function, args, vec_func=vec_func)
      n = 1
      interval = [a, b]
      intrange = b - a
      ordsum = _difftrap(vfunc, interval, n)
      result = intrange * ordsum
      resmat = [[result]]
      err = np.inf
      last_row = resmat[0]
      for i in range(1, divmax + 1):
          n *= 2
          ordsum += _difftrap(vfunc, interval, n)
          row = [intrange * ordsum / n]
          for k in range(i):
              row.append(_romberg_diff(last_row[k], row[k], k + 1))
          result = row[i]
          lastresult = last_row[i - 1]
          if show:
              resmat.append(row)
          err = abs(result - lastresult)
          if err < tol or err < rtol * abs(result):
              break
          last_row = row
      else:
          warnings.warn(
              "divmax (%d) exceeded. Latest difference = %e"
              % (divmax, err),
              AccuracyWarning,
          )
  
      if show:
          _printresmat(vfunc, interval, resmat)
      return result
  
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 18 for calling a global function with example usage: romberg(function, a, b, args, tol, rtol, show, divmax, vec_func) and returns: result