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CategoryMath: Linalg & Solve
GPUYes
BLAS/LAPACK

rcond — Estimate the reciprocal condition number of a square matrix (1 / cond(A)) with MATLAB-compatible behaviour.

rc = rcond(A) returns an estimate of the reciprocal of the condition number of a square matrix A in the 1-norm. Well-conditioned matrices yield values close to 1, whereas singular or numerically singular matrices yield values near zero. The estimate mirrors MATLAB: it is based on the ratio of the smallest to largest singular values.

How does the rcond function behave in MATLAB / RunMat?

  • Inputs must be square matrices (scalars count as 1 × 1). Non-square inputs trigger the MATLAB error "rcond: input must be a square matrix."
  • For the empty matrix (0 × 0) the reciprocal condition number is Inf.
  • rcond(0) returns 0. Non-zero scalars return 1.
  • Logical and integer arrays are promoted to double precision before analysis.
  • Complex matrices are supported; the singular values are computed in complex arithmetic.
  • The implementation runs an SVD and computes min(s) / max(s), honouring MATLAB's tolerance logic.

GPU behavior

When the input resides on a GPU, RunMat first invokes the active provider's rcond hook that this builtin registers. When that hook is unavailable the runtime attempts to reuse the provider's linsolve factorisations to recover the reciprocal condition number. If neither path is implemented the matrix is gathered to the host, the shared SVD-based estimator is executed, and the scalar result is uploaded back to the device so subsequent GPU work keeps residency.

GPU residency

Manually calling gpuArray is rarely necessary. When matrices already live on the device, RunMat invokes the provider hook if one exists, or automatically gathers, evaluates, and re-uploads the scalar result. This preserves MATLAB compatibility while keeping the runtime efficient and ergonomic.

Examples of using rcond in MATLAB / RunMat

Reciprocal condition of the identity matrix

A = eye(3);
rc = rcond(A)

Expected output:

rc = 1

Detecting a nearly singular matrix

A = diag([1, 1e-8]);
rc = rcond(A)

Expected output:

rc = 1e-8

Reciprocal condition of a singular matrix

A = [1 2; 2 4];
rc = rcond(A)

Expected output:

rc = 0

Reciprocal condition for a complex matrix

A = [1+2i 0; 3i 2-1i];
rc = rcond(A)

Expected output:

rc = 0.1887

Handling the empty matrix

rc = rcond([])

Expected output:

rc = Inf

Estimating conditioning on GPU data

G = gpuArray([2 0; 0 0.5]);
rc_gpu = rcond(G);
rc = gather(rc_gpu)

Expected output:

rc = 0.25

FAQ

What range of values should I expect from rcond?

rcond returns a value in [0, 1]. Values near 1 indicate a well-conditioned matrix. Values near 0 indicate singular or ill-conditioned matrices.

Why does rcond return Inf for the empty matrix?

The empty matrix is perfectly conditioned by convention, so the reciprocal condition number is Inf (1 / 0).

Does rcond warn me about singular systems automatically?

rcond itself does not issue warnings. Use the value to decide whether the matrix is singular to working precision, similar to MATLAB workflows (if rcond(A) < eps, ...).

Is the SVD tolerance the same as MATLAB?

Yes. The estimator matches MATLAB's definition by deriving the result from the singular values. The same tolerance logic is shared with pinv and rank.

Will calling rcond move my data off the GPU?

Only if the active provider lacks a dedicated implementation. In that case RunMat transparently downloads the matrix, computes the estimate on the host, and re-uploads the scalar.

See also

inv, pinv, rank, det, linsolve, gpuArray, gather

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