*Last updated: 2023-03-16.*
# tf_quant_finance.math.integration.gauss_kronrod
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Evaluates definite integral using adaptive Gauss-Kronrod quadrature.
```python
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%E2%80%93Kronrod_quadrature_formula
#### Example
```python
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.