quadax.quadcc

Global adaptive quadrature using Clenshaw-Curtis rule.

Integrate fun from interval[0] to interval[-1] using a h-adaptive scheme with error estimate. Breakpoints can be specified in interval where integration difficulty may occur.

A good general purpose integrator for most reasonably well behaved functions over finite or infinite intervals.

Parameters:

fun (callable) – Function to integrate, should have a signature of the form fun(x, *args) -> float, Array. Should be JAX transformable.
interval (array-like) – Lower and upper limits of integration with possible breakpoints. Use np.inf to denote infinite intervals.
args (tuple, optional) – Extra arguments passed to fun.
full_output (bool, optional) – If True, return the full state of the integrator. See below for more information.
epsabs (float, optional) – Absolute and relative error tolerance. Default is square root of machine precision. Algorithm tries to obtain an accuracy of abs(i-result) <= max(epsabs, epsrel*abs(i)) where i = integral of fun over interval, and result is the numerical approximation.
epsrel (float, optional) – Absolute and relative error tolerance. Default is square root of machine precision. Algorithm tries to obtain an accuracy of abs(i-result) <= max(epsabs, epsrel*abs(i)) where i = integral of fun over interval, and result is the numerical approximation.
max_ninter (int, optional) – An upper bound on the number of sub-intervals used in the adaptive algorithm.
order ({8, 16, 32, 64, 128, 256}) – Order of local integration rule.
norm (int, callable) – Norm to use for measuring error for vector valued integrands. No effect if the integrand is scalar valued. If an int, uses p-norm of the given order, otherwise should be callable.

Returns:

y (float, Array) – The integral of fun from a to b.
info (QuadratureInfo) – Named tuple with the following fields:
- err : (float) Estimate of the error in the approximation.
- neval : (int) Total number of function evaluations.
- status : (int) Flag indicating reason for termination. status of 0 means normal termination, any other value indicates a possible error. A human readable message can be obtained by print(quadax.STATUS[status])
- info : (dict or None) Other information returned by the algorithm. Only present if full_output is True. Contains the following:
  - ’ninter’ : (int) The number, K, of sub-intervals produced in the subdivision process.
  - ’a_arr’ : (ndarray) rank-1 array of length max_ninter, the first K elements of which are the left end points of the (remapped) sub-intervals in the partition of the integration range.
  - ’b_arr’ : (ndarray) rank-1 array of length max_ninter, the first K elements of which are the right end points of the (remapped) sub-intervals.
  - ’r_arr’ : (ndarray) rank-1 array of length max_ninter, the first K elements of which are the integral approximations on the sub-intervals.
  - ’e_arr’ : (ndarray) rank-1 array of length max_ninter, the first K elements of which are the moduli of the absolute error estimates on the sub-intervals.

Notes

Adaptive algorithms are inherently somewhat sequential, so perfect parallelism is generally not achievable. The local quadrature rule vmaps integrand evaluation at order points, so using higher order methods will generally be more efficient on GPU/TPU.