HomeCalculatorsMathLinear AlgebraLeast Squares Calculator
Math calculator

Least Squares Calculator

Linear least squares for a small real matrix, square or rectangular. Full column rank uses QR; otherwise SVD. Same engine as Linear System. Not Statistics regression. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
Assurance
Core
Declared partition coverage
PASS · 19/19 declared partitions (solve, det, inverse, multiply, transpose, rank, eigen, lu, chol, qr, svd, lstsq, rref, condition, nullspace, columnspace, rowspace, leftnullspace, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
min ‖Ax − b‖₂ for a small real matrix (square or rectangular, m,n ∈ [2,4]).
Scope
Real matrices
Verification
Engine tested · Specification checked · v1.14.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.14.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-22.la-hardening-lu-svd
Verification revision
2026-09-22.la-hardening-lu-svd · 7/7 property · digest 7307c47e0dc1
Legacy regression
58/58 tests · Production surface contract 5/5
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
24/24 golden · 8/8 CVP boundary · 14/14 invalid · 7/7 property · 9/9 cross-interface · 2/2 cross-calculator · 5/5 CVP contract · Manifest
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
29 legacy golden · 6 legacy boundary · legacy regression suite · 24/24 oracle-backed golden · 14/14 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.14.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Objectivemin_x ‖Ax − b‖₂
Full column rank QRA = QR, Rx = Qᵀb
Rank-deficient / underdetermined SVDx = V Σ⁺ Uᵀ b
iA ∈ R^{m×n} with m,n ∈ [2,4]. Square order still works. Reuses QR/SVD factorizations on this same engine. Not a Statistics regression page. Float64.

How to use

1

Enter A and b

Choose 2×2, 3×3, 3×2 (overdetermined), or 2×3 (min-norm). Same fields as Linear System. Larger composition lives at /workspace/matrix.

2

Read x and residual₂

method is qr or svd_pinv. residual_2 is ‖Ax − b‖₂. Rank-deficient and underdetermined return the min-norm solution.

Example calculations

Common configurations with formula and result.

ϟ

Full rank square

2x + y = 5, x + y = 3

QR backsolve
x=2, y=1
ϟ

Overdetermined 3×2

[[1,1],[1,2],[1,3]] b=[1,2,2]

A = QR, Rx = Qᵀb
x=2/3, y=1/2
ϟ

Rank 1

[[1,2],[2,4]] b=[1,0]

x = V Σ⁺ Uᵀ b
x=0.04, y=0.08

Least Squares calculator specification

Version 1.14.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
x minimizes ‖Ax − b‖₂. Full column rank (including overdetermined m×n, m≥n) uses Householder QR; rank-deficient or underdetermined uses the SVD pseudoinverse. This page is mode=lstsq on math.linear_algebra. Not a second engine. Not y=a+bx regression.
What it calculates
min ‖Ax − b‖₂ for a small real matrix (square or rectangular, m,n ∈ [2,4]).
Inputs
  • order or rows+cols
  • a11…
  • b1…
Outputs
  • x
  • y
  • z
  • residual_2
  • rank_numeric
  • method
  • status
  • rows
  • cols
Formula
min ‖Ax − b‖₂
Assumptions
  • Real matrices
  • m,n ∈ [2,4]
  • Planar ℝ² figure when 2×2; other shapes omit it
  • Float64
  • Not a CAS
  • Not Statistics regression
Units
  • dimensionless
Boundary conditions
  • unknown mode (including jordan-form) → INVALID_MODE
  • rows or cols outside 2–4 → INVALID_INPUT
Example
[[1,1],[1,2],[1,3]] b=[1,2,2] → x=2/3, y=1/2
Validation cases

3 published on this page · 58/58 tests · Production surface contract 5/5 · View evidence

  • order=2 a11=2 a12=1 a21=1 a22=1 b1=5 b2=3 → x=2 y=1 residual_2≈0
  • rows=3 cols=2 a11=1 a12=1 a21=1 a22=2 a31=1 a32=3 b1=1 b2=2 b3=2 → x=2/3 y=1/2 method=qr
  • mode=jordan-form order=2 a11=1 a12=0 a21=0 a22=1 → INVALID_MODE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.14.0

Other calculators in this family: Cholesky Factorization Calculator, Column Space Calculator, Condition Number Calculator, Left Nullspace Calculator, Linear System Calculator, LU Factorization Calculator, Matrix Determinant Calculator, Matrix Eigenvalues Calculator . Explore all Linear Algebra.

→ Cholesky Factorization Calculator Lower Cholesky factorization of a small real symmetric positive-definite matrix. Same engine as Linear System. Not LU. Not a CAS. Runs locally. › → Column Space Calculator Column space (image) of a small real square matrix from RREF pivot columns. Same engine as Linear System. Trivial image is {0}. Left-nullspace is a sibling. Runs locally. › → Condition Number Calculator κ₁(A) = ‖A‖₁ ‖A⁻¹‖₁ for a small real square matrix. Same engine as Linear System. Singular matrices return infinite. Not a 2-norm product. Runs locally. › → Left Nullspace Calculator Left-nullspace (cokernel) of a small real square matrix as ker(Aᵀ). Same engine as Linear System. Trivial left kernel is {0}. Not Jordan form. Runs locally. › → Linear System Calculator Solve small linear systems Ax = b, plus det, inverse, rank, LU, Cholesky, QR, SVD, least squares, RREF, condition number, nullspace, column space, rowspace, left-nullspace, and eigenvalues. Same engine as the matrix workspace. Not a CAS. Runs locally. › → LU Factorization Calculator Doolittle LU factorization with partial pivoting for a small real matrix. Same engine as Linear System. Not a CAS. Runs locally. › → Matrix Determinant Calculator Determinant of a small real matrix. Same engine as Linear System and the matrix workspace. Not a CAS. Runs locally. › → Matrix Eigenvalues Calculator Eigenvalues and eigenvectors of a small real matrix via the characteristic polynomial. Same engine as Linear System. Not a CAS. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a second linear-algebra engine?

No. It is mode=lstsq on math.linear_algebra, the same engine as /calc/math/linear-system and /workspace/matrix.

Is this Statistics linear regression?

No. This is Ax ≈ b for a small matrix. A slope-intercept fit of a point cloud is a Statistics product, not this page.

Does rectangular mean a new Capability?

No. rows and cols grow this same math.linear_algebra contract. Solve / det / inverse / LU / Cholesky stay square. QR and SVD are sibling discovery URLs on this same engine and also accept rectangular A.