HomeCalculatorsMathLinear AlgebraQR Factorization Calculator
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QR Factorization Calculator

Householder QR factorization of a small real matrix, square or rectangular. Same engine as Linear System. Not a CAS. 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
A = QR for a small real matrix (square or rectangular, m,n ∈ [2,4]).
Scope
Real matrices
Verification
Engine tested · Source checked · v1.14.16 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.14.16 · CVP protocol 1.0.0-proposed · Evidence 2026-10-30.cholesky-numeric-contract-v2
Verification revision
2026-10-30.cholesky-numeric-contract-v2 · 10/10 property · digest 2f8af10fde34
Legacy regression
58/58 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 expected_values · O2 expected_values · O2 expected_values · O2 expected_values · 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 24/24 (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. Raw HTTP REST/JSON-RPC transport rejection 5/5; raw-number source and conflicting pins are transport-only checks, not engine/SSR input-source parity.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
478/478 golden · 8/8 CVP boundary · 78/78 invalid · 10/10 property · 2/2 metamorphic · 3/3 round-trip · 67/67 cross-interface · 2/2 cross-calculator · 6/6 CVP contract · Manifest
Sources
Sources
Evidence
29 legacy golden · 6 legacy boundary · legacy regression suite · 478/478 oracle-backed golden · 78/78 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.14.16 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.

FactorizationA = QR
OrthogonalityQᵀQ = I
iHouseholder QR. Not Gram–Schmidt as the product algorithm. Not a CAS. A ∈ R^{m×n} with m,n ∈ [2,4]. Float64. SVD is a sibling discovery URL.

How to use

1

Enter a matrix

2×2, 3×3, 3×2, or 2×3. 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

Overdetermined 3×2

[[1,1],[1,2],[1,3]]

A = QR
Q is 3×3, R is 3×2

QR Factorization calculator specification

Version 1.14.16 · Engine tested

Calculation status

Review policy · Evidence

Definition
A = QR with Q orthogonal and R upper triangular (or upper trapezoidal when A is rectangular). This page is a discovery surface on the linear-algebra engine (mode=qr). Least squares reuses this factorization.
What it calculates
A = QR for a small real matrix (square or rectangular, m,n ∈ [2,4]).
Inputs
  • order or rows+cols
  • a11…
Outputs
  • Q
  • R
  • reconstruction_residual
  • orthogonality_residual
  • rows
  • cols
  • condition_number
Formula
A = QR (Householder)
Assumptions
  • Real matrices
  • m,n ∈ [2,4]
  • Full Q (m×m) and R (m×n)
  • Planar ℝ² figure when 2×2; other shapes omit it
  • Float64
  • Not a CAS
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]] → A=QR
Validation cases

2 published on this page · 58/58 tests · Production surface contract 6/6 · View evidence

  • order=2 a11=1 a12=1 a21=0 a22=1 → reconstruction_residual≈0
  • rows=3 cols=2 a11=1 a12=1 a21=1 a22=2 a31=1 a32=3 → reconstruction_residual≈0
Sources
Calculation version
1.14.16

Background

Interpretation and common distinctions.

What does this qr factorization calculator compute?

A = QR with Q orthogonal and R upper triangular (or upper trapezoidal when A is rectangular).

The published formula or method is A = QR (Householder). The following references support the stated mathematics or method:

Which inputs and limits should you check?

Rank, inverse, subspace, and least-squares decisions depend on the stated tolerance and conditioning. Inspect residuals and warnings before interpreting a nearly singular matrix.

The worked example uses [[1,1],[0,1]] with Q ≈ I, R ≈ A. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.linear_algebra v1.14.16 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Input replay and numerical range

All entries share math.linear_algebra v1.14.16. Enter every cell as a finite decimal, including scientific notation. Empty cells and nonzero decimals below the binary64 range are rejected. QR, SVD, and least squares accept dimensions from 2 to 4; other operations require square matrices. Matching aliases may be supplied, while conflicting aliases or nested and flat cells are rejected.

A singular system may be consistent with infinitely many solutions or inconsistent; read the status and warnings. Tolerance-based rank and subspace decisions are not exact symbolic certificates. Results outside the numerical range refuse with NUMERIC_PRECISION_LIMIT. An unavailable auxiliary determinant is omitted with a warning, rather than reported as zero or null.

Calculations run locally in the browser. Shared query links contain inputs, are sent to the server when opened, and may appear in server or third-party logs. Anyone receiving a shared link can read its inputs. JSON and CSV exports include the original inputs, complete structured result, warnings, and calculation version.

Other calculators in this family: Cholesky Factorization Calculator, Column Space Calculator, Condition Number Calculator, Least Squares Calculator, Left Nullspace Calculator, Linear System Calculator, LU Factorization Calculator, Matrix Determinant 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. › → 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. › → 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. ›

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?

No. Least squares is /calc/math/least-squares on this same engine. This page only factors A = QR.

Does rectangular mean a new Capability?

No. rows and cols grow this same math.linear_algebra contract. Solve / det / inverse / LU / Cholesky stay square.