poisson_fd_1d

std.seq.poisson_fd_1d · Level L3

The finite-difference solution of the boundary-value problem −u″ = f on a uniform grid with spacing h and Dirichlet values u_a, u_b at the two ends: the values at the n interior points, from (2uᵢ − uᵢ₋₁ − uᵢ₊₁)/h² = fᵢ. One solve of the tridiagonal system. It is exact for every quadratic u. The reference is the Thomas algorithm.

(2uᵢ − uᵢ₋₁ − uᵢ₊₁)/h² = fᵢ, u₀ = u_a, uₙ₊₁ = u_b

Signature

poisson_fd_1d(f: f64[n], h: f64[], ua: f64[], ub: f64[]) → f64[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 the Thomas algorithm on tridiag(−1, 2, −1)·u = h²f + boundary terms in exact rational arithmetic, on all 40 test cases.
All 140 float64 results inside the running error bound; the closest uses 30% of it.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 32 of them inside the running error bound at u = 2⁻²⁴; the closest uses 14% of it. Uncertified, too ill-conditioned at float32 for a finite bound: 8.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 39%
bit-equal to the NumPy formula in float64: 51%
largest error, in units in the last place: 59

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

—

sha256:9811263894c0733772c77bf73cb91dc00567f5d86a649a2e8e13c795200bca82

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.poisson_fd_1d · Ω:poisson_fd_1d
Pins: sha256:a0f9d2e5f1c8219d9dd347688163c51702bc2007f8362e4231e359b454c9117ethis graph alone
Evidence: sha256:b455bea05b22a586e0080150fa816df3258239a467b3e95e6c30532be9c77eb3the 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.