What does the rank function do in MATLAB / RunMat?

r = rank(A) returns the numerical rank of a real or complex matrix A. The rank equals the number of singular values greater than a tolerance derived from the matrix size and the largest singular value. RunMat mirrors MATLAB’s logic exactly so that results agree bit-for-bit with the reference implementation.

How does the rank function behave in MATLAB / RunMat?

• Inputs must behave like 2-D matrices. Trailing singleton dimensions are accepted; other higher ranks result in "rank: inputs must be 2-D matrices or vectors".
• The default tolerance is tol = max(size(A)) * eps(max(s)), where s are the singular values from an SVD of A. You can override this by supplying a second argument: rank(A, tol).
• When you provide an explicit tolerance it must be a finite, non-negative scalar. Non-scalars, NaN, Inf, or negative values raise MATLAB-compatible errors.
• Logical and integer inputs are promoted to double precision before taking the SVD.
• rank([]) returns 0. Rank is always reported as a double scalar (e.g., 2.0).
• Complex inputs use a complex SVD so that conjugate transposes and magnitudes follow MATLAB’s conventions.

rank Function GPU Execution Behaviour

When a GPU acceleration provider is active, RunMat first offers the computation through the reserved rank provider hook. Backends that implement it can stay fully on-device and return a gpuTensor scalar. Providers without that hook—including today’s WGPU backend—gather the matrix to host memory, reuse the shared SVD logic, and then re-upload the scalar rank so downstream kernels continue on the GPU without user intervention. Auto-offload treats the builtin as an eager sink, so any fused producers flush before rank executes and residency bookkeeping remains consistent.

Examples of using the rank function in MATLAB / RunMat

Determining the rank of a full matrix

A = [1 2; 3 4];
rk = rank(A);

Expected output:

rk = 2

Detecting rank deficiency in a singular matrix

B = [1 2; 2 4];
rk = rank(B);

Expected output:

rk = 1

Applying a custom tolerance to suppress tiny singular values

C = diag([1, 1e-12]);
rk_default = rank(C);          % counts both singular values (rank 2)
rk_custom  = rank(C, 1e-9);    % treats the small value as zero (rank 1)

Computing the rank of a tall matrix

A = [1 0; 0 0; 0 1];
rk = rank(A);

Expected output:

rk = 2

Evaluating the rank of a complex matrix

Z = [1+1i 0; 0 2-3i];
rk = rank(Z);

Expected output:

rk = 2

Checking the rank of an empty matrix

E = [];
rk = rank(E);

Expected output:

rk = 0

Using rank with gpuArray data

G = gpuArray([1 2 3; 3 6 9; 0 1 0]);
rk = rank(G);      % Computation stays on the GPU when the provider supports it
rk_host = gather(rk);

Expected output:

rk_host = 2

GPU residency in RunMat (Do I need gpuArray?)

RunMat’s planner automatically keeps matrices on the GPU when a provider implements the rank hook. If the hook is missing, the builtin transparently gathers the matrix, computes the SVD on the CPU, and uploads the scalar result so later GPU work remains resident. You can still seed residency manually with gpuArray for MATLAB compatibility, but it is rarely required.

FAQ

How is the default tolerance chosen?

RunMat computes the default tolerance exactly as MATLAB: max(size(A)) * eps(max(s)), where s are the singular values of A. This scales the cutoff with matrix size and magnitude.

What does rank([]) return?

The rank of the empty matrix is 0. This matches MATLAB’s convention that an empty product has neutral value.

Does rank return an integer or a double?

rank returns a double-precision scalar, mirroring MATLAB’s numeric tower. The value is always an integer-valued double.

How does rank behave for vectors or scalars?

Scalars are treated as 1×1 matrices. rank([0]) returns 0, while rank([5]) returns 1. Row or column vectors behave as matrices with one dimension equal to 1.

Can rank detect symbolic rank or exact arithmetic?

No. Like MATLAB, RunMat’s rank relies on floating-point SVD and is subject to the chosen tolerance. For symbolic or exact arithmetic you would use a computer algebra system.

Will rank participate in fusion or auto-offload?

No. rank is a residency sink that eagerly computes an SVD. Fusion groups terminate before the call, and the planner treats the builtin as a scalar reduction.

Is the tolerance argument optional?

Yes. rank(A) uses the default tolerance and mirrors MATLAB. Supplying rank(A, tol) overrides the cutoff. Non-scalar or negative tolerances raise MATLAB-compatible errors.

What happens if the matrix contains NaNs or Infs?

Singular values involving NaN propagate and typically produce a rank of 0. Infinite values yield infinite singular values and therefore produce a rank equal to the number of infinite entries above tolerance, matching MATLAB’s behaviour.

Does rank allocate large temporary buffers?

Only enough memory for the SVD factors. For host execution this is handled by nalgebra (and LAPACK when enabled). GPU providers are free to reuse buffers or stream the computation.

See Also

pinv, svd, inv, det, null, gpuArray, gather

Source & Feedback

• Implementation: crates/runmat-runtime/src/builtins/math/linalg/solve/rank.rs
• Found a behavioural difference? Open an issue with a minimal reproduction.