quadratic_pde_residual

std.seq.quadratic_pde_residual · Level L3

The residual of a quadratic candidate u = u₀ + u₁x + u₂y + u₃x² + u₄xy + u₅y² in the linear constant-coefficient equation a·u_xx + b·u_xy + c·u_yy + d·u_x + e·u_y + f·u = g, where g is a quadratic in the same basis, as a coefficient vector: u solves the equation everywhere exactly when every coefficient is 0 (a zero residual at one point proves only that point). Also the discriminant b² − 4ac of the principal part (by this convention: negative elliptic, zero parabolic, positive hyperbolic).

a u_xx + b u_xy + c u_yy + d u_x + e u_y + f u − g; b² − 4ac

Signature

quadratic_pde_residual(u: f64[6], coef: f64[6], g: f64[6]) → f64[6], 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 the derivatives of the quadratic, collected by monomial in exact rational arithmetic, on all 40 test cases.
All 280 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

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Called by

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sha256:46f03d318aa4212dee07a3f2af1b516c18f0ac8136ca6dabf4a5b8b8eb5836ed

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.quadratic_pde_residual · Ω:quadratic_pde_residual
Pins: sha256:749e3cd7268b0579cd3f7687b9b9017c4325d65733187db45141832f4d4e00f8this graph alone
Evidence: sha256:b32151368c2431471968f60f8c3535002e97451ecc45f5368869357695723c50the 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.