quotient_ring

std.algebra.quotient_ring · Level L3

The quotient ring R/I by a two-sided ideal I: the classes a + I, labelled by their smallest element, with [a] + [b] = [a + b] and [a]·[b] = [ab], given as two tables indexed by elements; whether I is an ideal; whether it is proper (I ≠ R, so that 1 ≠ 0 in R/I); whether both operations are well defined, the same for every choice of representatives; and the number of classes.

[a] + [b] = [a + b], [a][b] = [ab]

Signature

quotient_ring(Ad: i64[n, n], Mu: i64[n, n], I: f64[n]) → f64[], f64[], f64[], f64[], f64[n, n], f64[n, n], f64[]

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 class labels of every sum and product, against those of the representatives in exact rational arithmetic, on all 40 test cases.
All 1,664 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

is_ideal left_cosets quotient_group

Called by

—

sha256:2953d15b359f46f51a99c3c8ba141acbfffd2ef23f19262529d9277e4d653488

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.quotient_ring · Ω:quotient_ring
Pins: sha256:d015307ea9e31e35cdf4c6d96076ceaf671e934bbb3026e55a28c7a9a3b8cb91this graph and the 12 it reaches through calls
Evidence: sha256:29095320cee93eab8bf845af696532a966bdd097ac4c6e4df6806560274895a0the 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.