laplacian_2d

std.seq.laplacian_2d · Level L2

The five-point Laplacian (uᵢ₋₁,ⱼ + uᵢ₊₁,ⱼ + uᵢ,ⱼ₋₁ + uᵢ,ⱼ₊₁ − 4uᵢⱼ)/h² at the interior points of a grid of values with spacing h. It equals the exact Laplacian for every polynomial of degree at most 3, so a candidate solution of Poisson's equation can be checked on a grid.

Δₕu = (u_W + u_E + u_N + u_S − 4u)/h²

Signature

laplacian_2d(U: f64[m+2, n+2], h: f64[]) → f64[m, n]

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 (U[i−1,j] + U[i+1,j] + U[i,j−1] + U[i,j+1] − 4U[i,j])/h² in exact rational arithmetic, on all 40 test cases.
All 545 float64 results inside the running error bound; the closest uses 92% 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 79% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 68%
bit-equal to the NumPy formula in float64: 93%
largest error, in units in the last place: 28

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

—

Called by

—

sha256:e7b5534fa84db8946220eae0a33b804389c37d23f73aced3b521e2817e4d24d0

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.laplacian_2d · Ω:laplacian_2d
Pins: sha256:7cf87eb00639df03c86e78b3e383f25dd0270f8b619643b2ea39c2e653fe6cf6this graph alone
Evidence: sha256:2bb094aec520e06446c3d239ccb79ab330e72ab59758b64293957eedeacf4d7cthe 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.