Polynomial Evaluation via polyval

Why this matters

jnp.polyval(coeffs, x_pts) evaluates a polynomial at an array of points. Coefficients are given in descending power order — the same convention as jnp.polyfit. That is: coeffs[0] * x^(n-1) + coeffs[1] * x^(n-2) + ... + coeffs[-1].

polyval is the natural complement to polyfit:

• Prediction — evaluate a fitted polynomial at new x values.
• Function approximation — quickly compute polynomial models at many points.
• Visualization — generate smooth curve points for plotting.

The function is vectorized over x_pts — pass a 1-D array and get one output per input point.

Worked mini-example

import jax.numpy as jnp

coeffs = jnp.array([1.0, 0.0, 0.0])   # x^2 (descending: [1, 0, 0])
x_pts  = jnp.array([0.0, 1.0, 2.0, 3.0])
out = jnp.polyval(coeffs, x_pts)
# out = [0.0, 1.0, 4.0, 9.0]

Common pitfalls

• Ascending vs descending — jnp.polyval expects descending order. If you accidentally pass ascending-order coefficients (e.g., from jnp.Polynomial), the evaluation will be wrong.
• Scalar vs array x — works on both; for a single point, wrap in an array or expect a scalar output.
• No in-place mutations — JAX arrays are immutable; polyval returns a new array.

Problem

Implement polyval_at(coeffs, x_pts) that evaluates a polynomial at each point in x_pts.

• coeffs: 1-D jax array of shape (deg+1,) — coefficients in descending power order.
• x_pts: 1-D jax array — points to evaluate at.
• Returns: 1-D array same shape as x_pts — polynomial values.