svd — Compute singular value decompositions with full, economy, and singular-value-only output forms.
svd(A) factors real or complex matrices into U * S * V', where S contains non-negative singular values in descending order. It supports MATLAB-compatible full, economy, and value-only call forms.
Syntax
S = svd(A)
S = svd(A, option)
S = svd(A, option1, option2)
[U, S] = svd(A)
[U, S] = svd(A, option)
[U, S] = svd(A, option1, option2)All supported svd forms
S = svd(A)
S = svd(A, option)
S = svd(A, option1, option2)
[U, S] = svd(A)
[U, S] = svd(A, option)
[U, S] = svd(A, option1, option2)
[U, S, V] = svd(A)
[U, S, V] = svd(A, option)
[U, S, V] = svd(A, option1, option2)Inputs
| Name | Type | Required | Default | Description |
|---|---|---|---|---|
A | NumericArray | Yes | — | Input matrix to decompose. |
option | Any | Yes | — | Option token (`0`, `econ`, `full`, `vector`, or `matrix`). |
option1 | Any | Yes | — | First option token (`0`, `econ`, `full`, `vector`, or `matrix`). |
option2 | Any | Yes | — | Second option token (`0`, `econ`, `full`, `vector`, or `matrix`). |
Returns
| Name | Type | Description |
|---|---|---|
S | NumericArray | Singular values (vector by default). |
U | NumericArray | Left singular vectors. |
S | NumericArray | Diagonal singular-value matrix or vector based on options. |
V | NumericArray | Right singular vectors. |
Returned values from svd depend on how many outputs the caller requests.
Errors
| Identifier | When | Message |
|---|---|---|
RunMat:svd:InvalidArgument | Option arguments or requested output count are invalid. | svd currently supports at most three outputs |
RunMat:svd:InvalidInput | Input is unsupported or not a 2-D numeric/logical matrix. | svd: expected numeric or logical values |
RunMat:svd:Internal | Runtime cannot materialize SVD outputs. | svd: internal runtime failure |
How svd works
- Single output
s = svd(A)returns the singular values as a column vector sorted in descending order. - Three outputs
[U,S,V] = svd(A)return the full-sized factors withUsquarem×m,Sshapedm×n, andVsquaren×n(m = size(A,1),n = size(A,2)). - Economy form
[U,S,V] = svd(A,'econ')(orsvd(A,0)) reduces the shapes to the rank-defining dimension so thatUandVdrop the redundant orthogonal columns. - Vector form
[U,s,V] = svd(A,'vector')supplies the singular values as a vector instead of a diagonal matrix. You can combine'vector'with'econ'. - The documented matrix classes are single and double. With RunMat extensions enabled, real logical and native integer matrices are also accepted; integer values must be exactly representable as double or the call rejects before factorisation.
- Typed-integer zero can select economy size only when RunMat extensions are enabled. The selector is parsed structurally without a floating conversion.
- Complex inputs yield unitary
UandV(conjugate-transpose preserves orthogonality) with real, non-negative singular values. - Empty matrices, row/column vectors, and scalars are all supported and follow MATLAB’s shape conventions.
Does RunMat run svd on the GPU?
RunMat reserves a dedicated svd provider hook; once a backend implements it, the factors can stay on the device as gpuTensor handles without round-tripping through host memory.
Today no provider ships that hook, so gpuArray inputs are gathered to the host, the CPU SVD executes, and the factors are returned as host tensors. You can re-establish residency with gpuArray(s) if you need to continue on the GPU.
Because SVD is a residency sink, the fusion planner treats it as a barrier—preceding GPU tensors are gathered and subsequent ops run on the host unless you manually promote them again.
Examples
Getting the singular values of a matrix
A = [1 2 3; 4 5 6; 7 8 9];
s = svd(A)Full SVD and reconstruction of a square matrix
A = [3 1; 0 2];
[U,S,V] = svd(A);
A_recon = U * S * V'Economy-size SVD for a tall matrix
A = randn(6, 3);
[U,S,V] = svd(A, 'econ');
size(U) % 6 x 3
size(S) % 3 x 3
size(V) % 3 x 3Economy-size SVD for a wide matrix
A = randn(3, 6);
[U,S,V] = svd(A, 'econ');
size(U) % 3 x 3
size(S) % 3 x 6
size(V) % 6 x 6Requesting vector form of the singular values
A = [10 0; 0 1];
[U,s,V] = svd(A, 'vector')Computing the SVD of a complex matrix
A = [1+2i, 2-1i; 0, 3i];
[U,S,V] = svd(A)Running svd on a gpuArray (automatic host fallback today)
G = gpuArray(randn(128, 64));
s = svd(G); % Values are gathered to host transparentlyUsing svd with coding agents
Open a RunMat example with live inputs, then ask the agent to explain how svd changes the result.
Run a small svd example, explain the result, then change one input and compare the output.
FAQ
How are the singular values ordered?⌄
They are returned in non-increasing order. MATLAB’s sign conventions are followed: values are non-negative and appear on the diagonal of S (or inside the vector form).
What is the difference between full and economy forms?⌄
Full SVD returns square U and V (m×m and n×n). Economy SVD trims them to m×min(m,n) and n×min(m,n). The S factor keeps the same column dimension as the input; it is min(m,n)×min(m,n) when m ≥ n and min(m,n)×n when m < n. Use economy when you do not need the redundant orthogonal columns.
What does the "vector" option change?⌄
It affects the second output. With "vector", S is returned as a column vector of singular values, matching svd(A) in the single-output form. Without it, S is a diagonal matrix.
Can I mix 'econ' and 'vector'?⌄
Yes. Any order of the options is accepted (svd(A,'vector','econ') and svd(A,'econ','vector') both work), and the returned dimensions mirror MATLAB’s behaviour.
What happens with scalars or empty matrices?⌄
svd of a scalar returns its absolute value. Empty matrices return empty factors with consistent dimensions so that downstream code can continue to operate without special cases.
Does RunMat require BLAS/LAPACK for svd?⌄
No. The builtin is always available. When BLAS/LAPACK is enabled, the host implementation leverages those libraries through nalgebra for performance; otherwise a pure-Rust algorithm is used under the hood.
Will the results stay on the GPU?⌄
Not yet. Presently the builtin gathers GPU operands to the host, runs the CPU factorisation, and returns host tensors. The GPU spec already reserves a hook so providers can keep everything device-resident once GPU kernels land.
Can svd accept an integer matrix?⌄
The compatibility surface documents single and double matrices. With RunMat extensions enabled, all eight real integer classes are accepted when every value is exactly representable as double. Wider values that would round at the binary64 boundary are rejected before factorisation; strict compatibility mode rejects typed-integer matrices.
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 svd is executed, line by line, in Rust.
- View the source for svd in Rust on GitHub
- Learn how the RunMat runtime works
- Found a bug? Open an issue with a minimal reproduction.
About RunMat
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