Fixed-node quadrature

Quadrature rules turn an integral into a weighted sum:

\int f(x)\,\omega(x)\,dx \approx \sum_i w_i f(x_i).

(1)

jaxstro treats the nodes and weights as setup constants. They may be generated on the host, but the differentiable part is the same JAX expression every time: evaluate the integrand at fixed nodes and sum weighted values. This is why gradients flow through f(x_i) and not through node construction.

Learning objectives¶

After this chapter, you should be able to match a quadrature rule to its weight function, state its polynomial exactness, and distinguish exactness evidence from convergence evidence on a non-polynomial integrand.

Concept check: count moments, not nodes alone¶

Predict the highest polynomial degree integrated exactly by a Gaussian rule. Compute its moments, then audit the first degree where the exactness guarantee no longer applies.

Gaussian Rules¶

The Gaussian rules are exact for polynomials through degree 2n - 1 when using n nodes:

gauss_legendre_nodes(n) integrates on [-1, 1] with unit weight.
gauss_laguerre_nodes(n) integrates on [0, \infty) with weight exp(-x).
gauss_hermite_nodes(n) computes expectations under a standard normal density, using the probabilists’ Hermite convention.

The nodes and weights are generated with NumPy’s polynomial routines using the classical Golub-Welsch construction, then frozen as JAX arrays. They are not model parameters.

Clenshaw-Curtis¶

clenshaw_curtis_nodes(n) places nodes at Chebyshev-Lobatto points:

x_i = \cos\left(\frac{i\pi}{n-1}\right).

(2)

The weights come from the standard cosine-series construction. Clenshaw-Curtis is often competitive with Gaussian quadrature for smooth functions and has useful endpoint behavior because it includes -1 and 1.

Cumulative Simpson¶

cumulative_simpson(y, x=None, dx=1, axis=-1) lives with the integration helpers because it integrates sampled values on a uniform grid. Its shape contract is explicit: it returns cumulative values only at Simpson panel endpoints. If the input has n samples along the integration axis, n must be odd and the output axis has length (n + 1) // 2.

That contract avoids pretending Simpson’s three-point panel gives trustworthy cumulative values at every individual sample. Use cumulative_trapz when every sample needs a cumulative value.

Differentiability¶

For fixed nodes, all quadrature outputs are linear combinations of integrand values. For fixed sampled grids, Simpson and cumulative Simpson are linear combinations of y. The validation suite checks finite-difference versus AD gradients through those values.

Node generation and shape choices are intentionally static. Adaptive quadrature and convergence-loop integration are deferred because their data-dependent control flow needs a separate AD policy.