in_span

std.linalg.in_span · Level L4

Whether an integer vector t lies in the span of the integer vectors V (rows), as 1 or 0: adding t does not raise the exact rank. Spans are finite linear combinations; this decides membership over the reals, which a bounded search for coefficients could not. y is in the image of a matrix A exactly when it is in the span of A's columns.

t ∈ span V ⇔ rank [V; t] = rank V

Signature

in_span(V: f64[k, n], t: f64[n]) → 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 exact ranks with and without t in exact rational arithmetic, on all 40 test cases.
All 40 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 38 of them exact, with zero error. Uncertified, too ill-conditioned at float32 for a finite bound: 2.

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

rank_exact

Called by

same_coset

sha256:5f21464878e99e23a8617ede51e09c07226eba7d3cd4a381d5152693a4232a27

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.in_span · Ω:in_span
Pins: sha256:a92ce4fccb9b9844e25d02841aa3451b8f10d0499e22e192c450300ae1b8e31fthis graph and the 2 it reaches through calls
Evidence: sha256:6b452790110e4a17707e64ec372f1bd8cbd02d03dc59decef3b27ed15420a81bthe 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.