bundles / scipy 1.17.1 / scipy / integrate / _tanhsinh / tanhsinh

function

`scipy.integrate._tanhsinh:tanhsinh`

source: /scipy/integrate/_tanhsinh.py :30

Signature

   def     tanhsinh (    f    ,    a    ,    b    ,  * ,     args    =  ()   ,    log    =  False   ,    maxlevel    =  None   ,    minlevel    =  2   ,    atol    =  None   ,    rtol    =  None   ,    preserve_shape    =  False   ,    callback    =  None     )

Summary

Evaluate a convergent integral numerically using tanh-sinh quadrature.

Extended Summary

In practice, tanh-sinh quadrature achieves quadratic convergence for many integrands: the number of accurate digits scales roughly linearly with the number of function evaluations ^[1].

Either or both of the limits of integration may be infinite, and singularities at the endpoints are acceptable. Divergent integrals and integrands with non-finite derivatives or singularities within an interval are out of scope, but the latter may be evaluated be calling tanhsinh on each sub-interval separately.

Parameters

f : callable

The function to be integrated. The signature must be

f(xi: ndarray, *argsi) -> ndarray

where each element of xi is a finite real number and argsi is a tuple, which may contain an arbitrary number of arrays that are broadcastable with xi. f must be an elementwise function: see documentation of parameter preserve_shape for details. It must not mutate the array xi or the arrays in argsi. If f returns a value with complex dtype when evaluated at either endpoint, subsequent arguments x will have complex dtype (but zero imaginary part).

a, b : float array_like

Real lower and upper limits of integration. Must be broadcastable with one another and with arrays in args. Elements may be infinite.

args : tuple of array_like, optional

Additional positional array arguments to be passed to f. Arrays must be broadcastable with one another and the arrays of a and b. If the callable for which the root is desired requires arguments that are not broadcastable with x, wrap that callable with f such that f accepts only x and broadcastable *args.

log : bool, default: False

Setting to True indicates that f returns the log of the integrand and that atol and rtol are expressed as the logs of the absolute and relative errors. In this case, the result object will contain the log of the integral and error. This is useful for integrands for which numerical underflow or overflow would lead to inaccuracies. When log=True, the integrand (the exponential of f) must be real, but it may be negative, in which case the log of the integrand is a complex number with an imaginary part that is an odd multiple of π.

maxlevel : int, default: 10

The maximum refinement level of the algorithm.

At the zeroth level, f is called once, performing 16 function evaluations. At each subsequent level, f is called once more, approximately doubling the number of function evaluations that have been performed. Accordingly, for many integrands, each successive level will double the number of accurate digits in the result (up to the limits of floating point precision).

The algorithm will terminate after completing level maxlevel or after another termination condition is satisfied, whichever comes first.

minlevel : int, default: 2

The level at which to begin iteration (default: 2). This does not change the total number of function evaluations or the abscissae at which the function is evaluated; it changes only the number of times f is called. If minlevel=k, then the integrand is evaluated at all abscissae from levels 0 through k in a single call. Note that if minlevel exceeds maxlevel, the provided minlevel is ignored, and minlevel is set equal to maxlevel.

atol, rtol : float, optional

Absolute termination tolerance (default: 0) and relative termination tolerance (default: eps**0.75, where eps is the precision of the result dtype), respectively. Must be non-negative and finite if log is False, and must be expressed as the log of a non-negative and finite number if log is True. Iteration will stop when res.error < atol or res.error < res.integral * rtol.

preserve_shape : bool, default: False

In the following, "arguments of f" refers to the array xi and any arrays within argsi. Let shape be the broadcasted shape of a, b, and all elements of args (which is conceptually distinct from xi and argsi passed into f).

When preserve_shape=False (default), f must accept arguments of any broadcastable shapes.
When preserve_shape=True, f must accept arguments of shape shape or shape + (n,), where (n,) is the number of abscissae at which the function is being evaluated.

In either case, for each scalar element xi[j] within xi, the array returned by f must include the scalar f(xi[j]) at the same index. Consequently, the shape of the output is always the shape of the input xi.

See Examples.

callback : callable, optional

An optional user-supplied function to be called before the first iteration and after each iteration. Called as callback(res), where res is a _RichResult similar to that returned by tanhsinh (but containing the current iterate's values of all variables). If callback raises a StopIteration, the algorithm will terminate immediately and tanhsinh will return a result object. callback must not mutate res or its attributes.

Returns

res : _RichResult

An object similar to an instance of scipy.optimize.OptimizeResult with the following attributes. (The descriptions are written as though the values will be scalars; however, if f returns an array, the outputs will be arrays of the same shape.)

success: success
status: status
integral: integral
error: error
maxlevel: maxlevel
nfev: nfev

Notes

Implements the algorithm as described in ^[1] with minor adaptations for finite-precision arithmetic, including some described by ^[2] and ^[3]. The tanh-sinh scheme was originally introduced in ^[4].

Two error estimation schemes are described in ^[1] Section 5: one attempts to detect and exploit quadratic convergence; the other simply compares the integral estimates at successive levels. While neither is theoretically rigorous or conservative, both work well in practice. Our error estimate uses the minimum of these two schemes with a lower bound of eps * res.integral.

Due to floating-point error in the abscissae, the function may be evaluated at the endpoints of the interval during iterations, but the values returned by the function at the endpoints will be ignored.

Array API Standard Support

tanhsinh has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

====================  ====================  ====================
Library               CPU                   GPU
====================  ====================  ====================
NumPy                 ✅                     n/a                 
CuPy                  n/a                   ✅                   
PyTorch               ✅                     ✅                   
JAX                   ⛔                     ⛔                   
Dask                  ⛔                     n/a                 
====================  ====================  ====================

See dev-arrayapi for more information.

Examples

Evaluate the Gaussian integral:

import numpy as np
from scipy.integrate import tanhsinh
def f(x):
    return np.exp(-x**2)
res = tanhsinh(f, -np.inf, np.inf)

✓

res.integral  # true value is np.sqrt(np.pi), 1.7724538509055159
res.error  # actual error is 0

✗

The value of the Gaussian function (bell curve) is nearly zero for arguments sufficiently far from zero, so the value of the integral over a finite interval is nearly the same.

tanhsinh(f, -20, 20).integral

✗

However, with unfavorable integration limits, the integration scheme may not be able to find the important region.

tanhsinh(f, -np.inf, 1000).integral

✗

In such cases, or when there are singularities within the interval, break the integral into parts with endpoints at the important points.

tanhsinh(f, -np.inf, 0).integral + tanhsinh(f, 0, 1000).integral

✗

For integration involving very large or very small magnitudes, use log-integration. (For illustrative purposes, the following example shows a case in which both regular and log-integration work, but for more extreme limits of integration, log-integration would avoid the underflow experienced when evaluating the integral normally.)

res = tanhsinh(f, 20, 30, rtol=1e-10)

✓

res.integral, res.error

✗

def log_f(x):
    return -x**2
res = tanhsinh(log_f, 20, 30, log=True, rtol=np.log(1e-10))

✓

np.exp(res.integral), np.exp(res.error)

✗

The limits of integration and elements of `args` may be broadcastable arrays, and integration is performed elementwise.

from scipy import stats
dist = stats.gausshyper(13.8, 3.12, 2.51, 5.18)
a, b = dist.support()
x = np.linspace(a, b, 100)
res = tanhsinh(dist.pdf, a, x)
ref = dist.cdf(x)
np.allclose(res.integral, ref)

✓

By default, `preserve_shape` is False, and therefore the callable `f` may be called with arrays of any broadcastable shapes. For example:

shapes = []
def f(x, c):
   shape = np.broadcast_shapes(x.shape, c.shape)
   shapes.append(shape)
   return np.sin(c*x)
c = [1, 10, 30, 100]
res = tanhsinh(f, 0, 1, args=(c,), minlevel=1)
shapes

✓

To understand where these shapes are coming from - and to better understand how `tanhsinh` computes accurate results - note that higher values of ``c`` correspond with higher frequency sinusoids. The higher frequency sinusoids make the integrand more complicated, so more function evaluations are required to achieve the target accuracy:

res.nfev

✓

The initial ``shape``, ``(4,)``, corresponds with evaluating the integrand at a single abscissa and all four frequencies; this is used for input validation and to determine the size and dtype of the arrays that store results. The next shape corresponds with evaluating the integrand at an initial grid of abscissae and all four frequencies. Successive calls to the function double the total number of abscissae at which the function has been evaluated. However, in later function evaluations, the integrand is evaluated at fewer frequencies because the corresponding integral has already converged to the required tolerance. This saves function evaluations to improve performance, but it requires the function to accept arguments of any shape. "Vector-valued" integrands, such as those written for use with `scipy.integrate.quad_vec`, are unlikely to satisfy this requirement. For example, consider

def f(x):
   return [x, np.sin(10*x), np.cos(30*x), x*np.sin(100*x)**2]

✓

This integrand is not compatible with `tanhsinh` as written; for instance, the shape of the output will not be the same as the shape of ``x``. Such a function *could* be converted to a compatible form with the introduction of additional parameters, but this would be inconvenient. In such cases, a simpler solution would be to use `preserve_shape`.

shapes = []
def f(x):
    shapes.append(x.shape)
    x0, x1, x2, x3 = x
    return [x0, np.sin(10*x1), np.cos(30*x2), x3*np.sin(100*x3)]
a = np.zeros(4)
res = tanhsinh(f, a, 1, preserve_shape=True)
shapes

✓

Here, the broadcasted shape of `a` and `b` is ``(4,)``. With ``preserve_shape=True``, the function may be called with argument ``x`` of shape ``(4,)`` or ``(4, n)``, and this is what we observe.

Aliases

scipy.integrate.tanhsinh