Modified Bessel function of the first kind, order 0.
LAX-backend implementation of
i0(). Original docstring below.
Usually denoted \(I_0\). This function does broadcast, but will not “up-cast” int dtype arguments unless accompanied by at least one float or complex dtype argument (see Raises below).
x (array_like, dtype float or complex) – Argument of the Bessel function.
out – The modified Bessel function evaluated at each of the elements of x.
- Return type
ndarray, shape = x.shape, dtype = x.dtype
TypeError – array cannot be safely cast to required type: If argument consists exclusively of int dtypes.
The scipy implementation is recommended over this function: it is a proper ufunc written in C, and more than an order of magnitude faster.
We use the algorithm published by Clenshaw 1 and referenced by Abramowitz and Stegun 2, for which the function domain is partitioned into the two intervals [0,8] and (8,inf), and Chebyshev polynomial expansions are employed in each interval. Relative error on the domain [0,30] using IEEE arithmetic is documented 3 as having a peak of 5.8e-16 with an rms of 1.4e-16 (n = 30000).
C. W. Clenshaw, “Chebyshev series for mathematical functions”, in National Physical Laboratory Mathematical Tables, vol. 5, London: Her Majesty’s Stationery Office, 1962.
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 10th printing, New York: Dover, 1964, pp. 379. http://www.math.sfu.ca/~cbm/aands/page_379.htm
>>> np.i0(0.) array(1.0) # may vary >>> np.i0([0., 1. + 2j]) array([ 1.00000000+0.j , 0.18785373+0.64616944j]) # may vary