is_positive_definite

std.linalg.is_positive_definite · Level L4

Whether an integer matrix is symmetric, and whether it is positive definite, each as 1 or 0, by Sylvester's criterion: a symmetric matrix is positive definite exactly when every leading principal minor is positive, decided exactly by leading_minors. Then ⟨x, y⟩ = xᵀAy is an inner product on Rⁿ. Testing ⟨x, x⟩ > 0 on finitely many vectors would not prove it.

A = Aᵀ ∧ Δ_k > 0 ∀k

Signature

is_positive_definite(A: f64[n, n]) → f64[], 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 symmetry, and every leading minor positive in exact rational arithmetic, on all 40 test cases.
All 80 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

leading_minors

Called by

—

sha256:5c793f3a29b1ac8e55486b63e399b1e9ff7408e2a92170bebe9bf870b736cf94

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

Control handle

Symbol: Ω:std.linalg.is_positive_definite · Ω:is_positive_definite
Pins: sha256:0451498e98670529ba27968eb8fb34f69bc12cda8b29da7ea3455672654a931athis graph and the 2 it reaches through calls
Evidence: sha256:2d96e28a4fa739c13b2230b1d9da7be85b3ef4383dbcc45cc17af24c3cb7d650the 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.