tf_quant_finance.math.integration.gauss_kronrod

Last updated: 2023-03-16.

tf_quant_finance.math.integration.gauss_kronrod#

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Evaluates definite integral using adaptive Gauss-Kronrod quadrature.

tf_quant_finance.math.integration.gauss_kronrod(
    func, lower, upper, tolerance, num_points=21, max_depth=20, dtype=None,
    name=None
)

Integrates func using adaptive Gauss-Kronrod quadrature [1].

Applies change of variables to the function to obtain the [-1,1] integration interval. Takes the sum of values obtained from evaluating the new function at points given by the roots of the Legendre polynomial of degree (num_points-1)//2 and the roots of the Stieltjes polynomial of degree (num_points+1)//2, multiplied with corresponding precalculated coefficients. Repeats procedure if not accurate enough by halving the intervals and dividing these into the same number of subintervals.

References#

[1] https://en.wikipedia.org/wiki/Gauss–Kronrod_quadrature_formula

Example#

  f = lambda x: x*x
  a = tf.constant([0.0])
  b = tf.constant([3.0])
  tol = 1e-5
  num_points = 21
  max_depth = 10
  gauss_kronrod(f, a, b, tol, num_points, max_depth) # [9.0]

Args:#

  • func: Represents a function to be integrated. It must be a callable of a single Tensor parameter and return a Tensor of the same shape and dtype as its input. It will be called with a Tensor of shape lower.shape + [n,  num_points] (where n is defined by the algorithm and represents the number of subintervals) and of the same dtype as lower.

  • lower: Represents the lower limits of integration. func will be integrated between each pair of points defined by lower and upper. Must be a 1-dimensional tensor of shape [batch_dim].

  • upper: Same shape and dtype as lower representing the upper limits of intergation.

  • tolerance: Represents the tolerance for the estimated error of the integral estimation, at which to stop further dividing the intervals.

  • num_points: Number of points at which the function func will be evaluated. Implemented for 15,21,31. Default value: 21.

  • max_depth: Maximum number of times to divide intervals into two parts and recalculate Gauss-Kronrod on them. Default value: 20.

  • dtype: If supplied, the dtype for the lower and upper. Result will have the same dtype. Default value: None which maps to dtype of lower.

  • name: The name to give to the ops created by this function. Default value: None which maps to ‘gauss_kronrod’.

Returns:#

Tensor of shape [batch_dim], containing value of the definite integral.

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