sym_apply

std.linalg.sym_apply · Level L3

A function of a symmetric matrix, f(A) = V·f(Λ)·Vᵀ: f applied to the eigenvalues, the eigenvectors kept. A template over f, verified through f = square, where f(A) is A·A.

f(A) = V·diag(f(λ))·Vᵀ, A = V·diag(λ)·Vᵀ

Signature

sym_apply(A: f64[n, n]) → f64[n, 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.
Agrees with the reference witness f = square: A @ A to 80 digits (100-digit arithmetic), on all 40 test cases.
All 765 float64 results inside the running error bound; the closest uses 10% of it.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 39 of them inside the running error bound at u = 2⁻²⁴; the closest uses 2% of it. Uncertified, too ill-conditioned at float32 for a finite bound: 1.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 5%
bit-equal to the NumPy formula in float64: 5%
largest error, in units in the last place: 5.7e+3

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.

Note

Exact for every symmetric A in exact arithmetic, since A = VΛVᵀ. The eigenvectors' signs cancel in VΛVᵀ, but their error bound grows as eigenvalues approach each other, so the cases have eigenvalues 1, 2, …, n.

Identity

Calls

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Called by

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sha256:f1f682487cae9e55d6e49fbb2d1b8a180172c7a09f55c056d8bdbf7e2b8fafa1

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.sym_apply[f=std.elementwise.square]
Pins: sha256:0355b95ed8ec0bcf215a79653c3956c9f611b95701b69bb9e13b05e9815d62c9this graph and the 1 it reaches through calls
Evidence: sha256:65ec074a654e4603c74e1d136c6a19e276adf4eb82fd6a3822620127d0e3278ethe 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.