jax.numpy.sinc

jax.numpy.sinc(x, /)

Calculate the normalized sinc function.

JAX implementation of numpy.sinc().

The normalized sinc function is given by

\[\mathrm{sinc}(x) = \frac{\sin({\pi x})}{\pi x}\]

where sinc(0) returns the limit value of 1. The sinc function is smooth and infinitely differentiable.

Parameters:

x (ArrayLike) – input array; will be promoted to an inexact type.

Returns:

An array of the same shape as x containing the result.

Return type:

Array

Examples

>>> x = jnp.array([-1, -0.5, 0, 0.5, 1])
>>> with jnp.printoptions(precision=3, suppress=True):
...   jnp.sinc(x)
Array([-0.   ,  0.637,  1.   ,  0.637, -0.   ], dtype=float32)

Compare this to the naive approach to computing the function, which is undefined at zero:

>>> with jnp.printoptions(precision=3, suppress=True):
...   jnp.sin(jnp.pi * x) / (jnp.pi * x)
Array([-0.   ,  0.637,    nan,  0.637, -0.   ], dtype=float32)

JAX defines a custom gradient rule for sinc to allow accurate evaluation of the gradient at zero even for higher-order derivatives:

>>> f = jnp.sinc
>>> for i in range(1, 6):
...   f = jax.grad(f)
...   print(f"(d/dx)^{i} f(0.0) = {f(0.0):.2f}")
...
(d/dx)^1 f(0.0) = 0.00
(d/dx)^2 f(0.0) = -3.29
(d/dx)^3 f(0.0) = 0.00
(d/dx)^4 f(0.0) = 19.48
(d/dx)^5 f(0.0) = 0.00