poly_interp

std.seq.poly_interp · Level L2

The polynomial of degree n − 1 through the n points (xⱼ, yⱼ), x distinct, evaluated at each query point xq: Newton's divided differences, then Horner's rule for Newton's form. The same polynomial as Lagrange's formula, the reference.

p(xq) = Σⱼ yⱼ·Πₗ≠ⱼ (xq − xₗ)/(xⱼ − xₗ)

Signature

poly_interp(xq: f64[q], x: f64[n], y: f64[n]) → f64[q]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference Lagrange: Σⱼ y[j]·Πₗ≠ⱼ (xq − x[l])/(x[j] − x[l]) in exact rational arithmetic, on all 40 test cases.
All 183 float64 results inside the running error bound; the closest uses 29% of it.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them inside the running error bound at u = 2⁻²⁴; the closest uses 22% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 58%
bit-equal to the NumPy formula in float64: 49%
largest error, in units in the last place: 918

Large ulp counts appear where a result is tiny next to the numbers it is computed from (after cancellation, for example), so one unit in the last place is tiny too; the absolute error is still inside the bound. Results within their own error of zero are not counted.

Identity

Calls

divided_differences newton_eval

Called by

—

sha256:d1fb2ca4ef45e51ef1ebf2c98ac6b63ea9ef4fbe3652b82f7a6520e00d6d4809

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.seq.poly_interp · Ω:poly_interp
Pins: sha256:c1b1eb944c01996334d933bffd60d4d279ec2c7682bb6cca7d79b0bd50007689this graph and the 4 it reaches through calls
Evidence: sha256:e2289f2bb7a135e2800cd2ae15098a18f3b8f15581fa436d58ef1c69b8f50090the hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.