matrix_rank

std.linalg.matrix_rank · Level L4

The numerical rank of a matrix with at least as many rows as columns (transpose a wide one first): how many singular values exceed rtol times the largest. NumPy's np.linalg.matrix_rank with tol = rtol·σ₁. For a rank-deficient matrix stored in floating point the exact rank is usually full, with tiny singular values; the tolerance is what makes the question answerable.

rank = #{i : σᵢ > rtol·σ₁}

Signature

matrix_rank(A: f64[n+k, n], rtol: 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.
Agrees with the reference np.linalg.matrix_rank(A, tol=rtol * s.max()) to 80 digits (100-digit 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 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

Note

Each singular value against the cut is a decision. The verification uses matrices with singular values k, k − 1, …, 1 and then zeros, so every decision is far from the cut; near it, the rank is not certain.

Identity

Calls

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Called by

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sha256:9a6a44e04f0f90d22cb5c7bd4fcc26fbf2069c0386e86bd0245541c448421d14

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Control handle

Symbol: Ω:std.linalg.matrix_rank · Ω:matrix_rank
Pins: sha256:86f9860c60352008d4d55d074c293f6489c5851d4d9d8659b257cd3ebcf26fdcthis graph alone
Evidence: sha256:02efff09a4d7aedfd57cfb679f36072eeba7410a5b2bf16142161a378d408805the 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.