rref — Compute reduced row echelon form and pivot columns.
rref(A) returns the reduced row echelon form of a real or complex numeric matrix. [R, pivots] = rref(A) also returns the one-based pivot column indices. An optional tolerance controls which entries are considered zero during pivot selection.
Syntax
R = rref(A)
R = rref(A, tol)
[R, pivots] = rref(A)
[R, pivots] = rref(A, tol)Inputs
| Name | Type | Required | Default | Description |
|---|---|---|---|---|
A | Any | Yes | — | Input matrix. |
tol | NumericScalar | No | — | Pivot tolerance. |
Returns
| Name | Type | Description |
|---|---|---|
R | NumericArray | Reduced row echelon form of A. |
pivots | NumericArray | One-based pivot column indices. |
Returned values from rref depend on how many outputs the caller requests.
Errors
| Identifier | When | Message |
|---|---|---|
RunMat:rref:InvalidArgument | Optional tolerance argument is malformed or outside accepted bounds. | rref: invalid argument |
RunMat:rref:InvalidInput | Input shape/type cannot be processed for row reduction. | rref: invalid input |
RunMat:rref:TooManyOutputs | More than two output arguments are requested. | rref: too many output arguments |
RunMat:rref:Internal | Runtime fails while computing row reduction or executing gather/upload paths. | rref: internal runtime failure |
How rref works
- Inputs must behave like 2-D matrices. Trailing singleton dimensions are accepted; other higher-rank inputs raise
RunMat:rref:InvalidInput. - The default tolerance is
max(size(A)) * eps(norm(A, inf)), matching MATLAB row-reduction conventions. rref(A, tol)uses a caller-provided finite, non-negative scalar pivot tolerance.- The MATLAB-compatible matrix and tolerance classes are single and double. RunMat mode additionally accepts integer matrices and integer tolerances only through explicit exact-binary64 checks; strict MATLAB mode rejects those extensions.
- Complex inputs use complex Gauss-Jordan elimination with pivot selection by complex magnitude.
- The second output is a row vector of one-based pivot column indices.
- Non-finite values are accepted. The default tolerance still follows
max(size(A)) * eps(norm(A, inf));InfandNaNtherefore follow the runtime's IEEE floating-point tolerance and pivot arithmetic.
Does RunMat run rref on the GPU?
rref is an eager solve-style operation. It terminates fusion plans, gathers GPU-resident inputs for numerical row reduction, and returns ordinary MATLAB-compatible values.
GPU memory and residency
When the input is a real gpuArray, RunMat gathers it, computes the row-reduced matrix on the host, and re-uploads the reduced matrix through that handle's owning provider. If owner-preserving re-upload fails, the builtin raises RunMat:rref:Internal rather than silently changing residency. Pivot-column metadata is returned as a host vector.
Examples
Finding dependent columns
A = [1 2 3; 2 4 6; 1 1 1];
[R, pivots] = rref(A)Expected output:
R = [1 0 -1; 0 1 2; 0 0 0]
pivots = [1 2]Using a custom tolerance
A = diag([1, 1e-12]);
R = rref(A, 1e-6)Expected output:
R = [1 0; 0 0]Row-reducing an augmented system
A = [2 -1 3; 4 -2 6; 1 0 1];
b = [5; 10; 2];
RAug = rref([A b])Using rref with coding agents
Open a RunMat example with live inputs, then ask the agent to explain how rref changes the result.
Run a small rref example, explain the result, then change one input and compare the output.
FAQ
How does rref choose pivots?⌄
RunMat scans columns left to right, picks the largest available pivot by magnitude below the current pivot row, normalizes the pivot row, and eliminates the pivot column above and below the pivot.
Is the pivot vector one-based?⌄
Yes. The second output uses MATLAB's one-based column indices.
Does rref use the same tolerance as rank?⌄
No. rank uses an SVD tolerance. rref uses the MATLAB row-reduction tolerance based on the matrix infinity norm.
Related Linalg functions
Structure
bandwidth · isdiag · ishermitian · issymmetric · istril · istriu · symrcm
Ops
cross · ctranspose · dot · mldivide · mpower · mrdivide · mtimes · pagemtimes · pagetranspose · trace · transpose
Open-source implementation
Unlike proprietary runtimes, every RunMat function is open-source. Read exactly how rref is executed, line by line, in Rust.
- View the source for rref in Rust on GitHub
- Learn how the RunMat runtime works
- Found a bug? Open an issue with a minimal reproduction.
About RunMat
RunMat is an open-source runtime that executes MATLAB-syntax code blazing on any GPU. It is licensed under the Apache 2.0 license.
- RunMat automatically optimizes your math for GPU execution on Apple, Nvidia, and AMD hardware. No code changes needed. Simulations that took hours now take minutes.
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