HomeCalculatorsMathAlgebra & EquationsQR Factorization Calculator
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QR Factorization Calculator

Householder QR factorization of a small real matrix. Same engine as Linear System. Not a CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 18/18 tests · Production surface contract 3/3 · v1.2.0
Input interpretation
Enter values to calculate.
Result
—
Model
A = QR for order 2–3 (engine up to 4).
Scope
Real square matrices
Verification
Engine tested · 18/18 tests · Production surface contract 3/3 · Specification checked · v1.2.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
7 golden · 2 boundary · 9 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: STALE · Last attested schema matched 1.2.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status STALE (1 capability; 163 remain CURRENT) @ 2026-09-19T00:00:17.039Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

FactorizationA = QR
OrthogonalityQᵀQ = I
iHouseholder QR. Not Gram–Schmidt as the product algorithm. Not SVD. Not a CAS. Order ≤ 4. Float64.

How to use

1

Enter a square matrix

2×2 or 3×3. Larger composition lives at /workspace/matrix.

2

Read Q and R

reconstruction_residual is ‖A − QR‖_F. orthogonality_residual is ‖QᵀQ − I‖_F.

Example calculations

Common configurations with formula and result.

ϟ

Upper triangular

[[1,1],[0,1]]

Q ≈ I, R ≈ A
residual ~ 0
ϟ

Classic 3×3

Golub/Van Loan example

A = QR
residual ~ 0

QR Factorization calculator specification

Version 1.2.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A = QR with Q orthogonal and R upper triangular. This page is a discovery surface on the linear-algebra engine (mode=qr). Least squares later reuses this factorization.
What it calculates
A = QR for order 2–3 (engine up to 4).
Inputs
  • order
  • a11…
Outputs
  • Q
  • R
  • reconstruction_residual
  • orthogonality_residual
  • condition_number
Formula
A = QR (Householder)
Assumptions
  • Real square matrices
  • Order ≤ 4
  • Float64
  • Not a CAS
  • Not least squares / SVD
Units
  • dimensionless
Boundary conditions
  • unknown mode (including svd) → INVALID_MODE
Example
[[1,1],[0,1]] → A=QR
Validation cases

1 published on this page · 18/18 tests · Production surface contract 3/3 · View evidence

  • order=2 a11=1 a12=1 a21=0 a22=1 → reconstruction_residual≈0
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.2.0

Other calculators in this family: Linear Equation Calculator, Linear System Calculator, LU Factorization Calculator, Matrix Determinant Calculator, Matrix Eigenvalues Calculator, Matrix Inverse Calculator, Matrix Rank Calculator, Polynomial Roots Calculator . Explore all Algebra & Equations.

→ Linear Equation Calculator Solve ax + b = rhs for x. Identity and inconsistent cases when a = 0. Not the quadratic formula. Runs locally. › → Linear System Calculator Solve small linear systems Ax = b, plus det, inverse, rank, LU, QR, 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. › → Matrix Inverse Calculator Inverse of a small real matrix via Gauss–Jordan. Same engine as Linear System. Singular matrices fail. Not a CAS. Runs locally. › → Matrix Rank Calculator Rank of a small real matrix via Gaussian elimination. Same engine as Linear System. Not a CAS. Runs locally. › → Polynomial Roots Calculator Find all roots of a real polynomial of degree 1–8. Closed form for linear and quadratic; Durand–Kerner for cubics and higher. Not the quadratic formula page. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a second QR solver?

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

Is this least squares?

Not yet. QR is the primitive least squares will call. Do not treat this page as a regression calculator.