Evaluate a splined radial basis

[sirmarcel]

Hello, as discussed on Slack, here's a feature request: It'd be nice to be able to evaluate a (splined) radial basis at arbitrary points for testing purposes. This would allow fine-grained testing of alternative implementations, as opposed to end-to-end testing via the SphericalExpansion calculators.

[Luthaf]

This would mainly require copying the code from here to a Python function:

rascaline/rascaline/src/math/splines.rs

Lines 228 to 284 in 5ac2ee9

	// notation in this function follows
	// https://en.wikipedia.org/wiki/Cubic_Hermite_spline
	let mut k = match self.points.binary_search_by(
	|v| v.position.partial_cmp(&x).expect("got NaN")
	) {
	Ok(k) => k,
	Err(k) => k - 1,
	};
	// If we are evaluating at exactly the last spline point, use the
	// previous point as a basis, and t will be 1 below.
	if k == self.points.len() - 1 {
	k -= 1;
	}
	let point_k = &self.points[k];
	let point_k_1 = &self.points[k + 1];
	let x_k = point_k.position;
	let x_k_1 = point_k_1.position;
	debug_assert!(x_k <= x && x <= x_k_1);
	let delta = x_k_1 - x_k;
	let t = (x - x_k) / delta;
	let t_2 = t * t;
	let t_3 = t_2 * t;
	// Hermit base polynomials
	let h00 = 2.0 * t_3 - 3.0 * t_2 + 1.0;
	let h10 = t_3 - 2.0 * t_2 + t;
	let h01 = -2.0 * t_3 + 3.0 * t_2;
	let h11 = t_3 - t_2;
	let p_k = &point_k.values;
	let p_k_1 = &point_k_1.values;
	let m_k = &point_k.derivatives;
	let m_k_1 = &point_k_1.derivatives;
	azip!((v in values, p_k in p_k, p_k_1 in p_k_1, m_k in m_k, m_k_1 in m_k_1) {
	*v = h00 * p_k + h10 * delta * m_k + h01 * p_k_1 + h11 * delta * m_k_1;
	});
	if let Some(gradients) = gradients {
	let d_h00_dt = 6.0 * (t_2 - t);
	let d_h10_dt = 3.0 * t_2 - 4.0 * t + 1.0;
	let d_h01_dt = -d_h00_dt;
	let d_h11_dt = 3.0 * t_2 - 2.0 * t;
	let dx_dt = 1.0 / delta;
	azip!((g in gradients, p_k in p_k, p_k_1 in p_k_1, m_k in m_k, m_k_1 in m_k_1) {
	*g = d_h00_dt * p_k * dx_dt + d_h10_dt * m_k + d_h01_dt * p_k_1 * dx_dt + d_h11_dt * m_k_1;
	});
	}
	}