ring_poly_step

std.algebra.ring_poly_step · Level L3

One pass of polynomial multiplication over a finite ring: add the term p_i·q, shifted i places, into the accumulated coefficients r, with the ring's own tables. The body ring_poly_mul scans.

r_j ← r_j + p_i·q_{j−i}

Signature

ring_poly_step(r: f64[L], i: f64[], p: i64[a], q: i64[c], Ad: i64[n, n], Mu: i64[n, n], zero: f64[]) → f64[L]

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 r_j + p_i·q_{j−i} by table lookups, zero outside q in exact rational arithmetic, on all 40 test cases.
All 157 float64 results are exact: the error is zero.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them exact, with zero error.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 100%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 0

Identity

Calls

—

Called by

ring_poly_mul

sha256:bcf102d0b9850b326ccc0ea0c255066ae21c01fb65900cda7eec483834812279

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

Control handle

Symbol: Ω:std.algebra.ring_poly_step · Ω:ring_poly_step
Pins: sha256:d868052fb66b1a286e973ff6b618619084927499c15d6ac640477746869c0b19this graph alone
Evidence: sha256:1b5728e90cea3066fc0794d26f192f0820d38527c2cf5f2187764ded83e1e3a1the 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.