induced_norms

std.linalg.induced_norms · Level L1

The operator norms of a matrix A : Rⁿ → Rᵐ induced by ℓ₁ and by ℓ∞: ‖A‖₁ = max over columns of Σ|a_ij|, and ‖A‖∞ = max over rows of Σ|a_ij|. They are the least C with ‖Ax‖ ≤ C‖x‖ for every x, so A is a bounded operator; a few vectors x satisfying the bound would not be a certificate.

‖A‖₁ = max_j Σ_i |a_ij|, ‖A‖∞ = max_i Σ_j |a_ij|

Signature

induced_norms(A: f64[m, 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 absolute column and row sums in exact rational arithmetic, on all 40 test cases.
All 80 float64 results inside the running error bound; the closest uses 47% 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 93% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 94%
bit-equal to the NumPy formula in float64: 94%
largest error, in units in the last place: 0.75

Identity

Calls

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Called by

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sha256:752656903cae97442b293f860b1fe147f27bad6eb84b54f1015948350cdbcaad

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.induced_norms · Ω:induced_norms
Pins: sha256:7e37107a92723712fb2f30cc16b45b2062486ba453ce317ce0584ceae76f78f4this graph alone
Evidence: sha256:2000609d9db61714432e2ef9ed4f98131eb4e302677df3f39bde1acf2a2aaa48the 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.