Overview
Householder QR is a matrix decomposition method for qr decomposition. It works by reflecting columns to zero out subdiagonal entries.
CoreSource checked
Householder QR
Householder QR addresses qr decomposition by reflecting columns to zero out subdiagonal entries.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Numerical Methods
- Difficulty
- Intermediate
- Family
- Matrix Decomposition
- Tier
- Core
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 0
- Relationships
- 5
- Code example
- Ready
- Visualization
- Ready
Core Concepts
qr decompositionmatrix decompositionnumerical methods
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(n^2)
Average
O(n^3)
Worst
O(n^3)
Space
O(n^2)
When to Use
- Use Householder QR when solving qr decomposition and you want to benefit from reflecting columns to zero out subdiagonal entries.
- Choose it when O(n^3) average/worst-case time and O(n^2) memory match your workload profile.
- Prefer it over GMRES or MINRES when the problem structure favors qr decomposition.
When Not to Use
- Avoid when the input is ill-conditioned and the method lacks a stability safeguard.
- Avoid when exact symbolic results are required instead of numeric approximations.
What It Does
- Computes qr decomposition within Matrix Decomposition workloads (O(n^3) time, O(n^2) space).
- Works by reflecting columns to zero out subdiagonal entries.
- Maintains key invariants across qr decomposition, matrix decomposition, numerical methods.
Use Cases
- Production and research workloads requiring qr decomposition via reflecting columns to zero out subdiagonal entries.
- Matrix Decomposition pipelines where O(n^3) time and O(n^2) space fit system constraints.
- Engineering comparisons against GMRES and MINRES when tuning qr decomposition.
Inputs
- Numbers, vectors, matrices, or functions
- Tolerance or precision settings
Outputs
- Computed numeric result
- Residual, factorization, or approximation
Pseudocode
- 1.Validate input dimensions, conditioning, and numerical tolerance parameters for Householder QR.
- 2.Initialize the working buffers, basis vectors, or decomposition factors (qr decomposition).
- 3.Execute the core numerical recurrence by reflecting columns to zero out subdiagonal entries.
- 4.Monitor residual error, stability bounds, or step convergence across iterations.
- 5.Return the computed qr decomposition coefficients, factors, or transformed values.
Canonical Example
- Input
- Numbers, vectors, matrices, or functions configured for a representative qr decomposition test case
- Output
- Computed numeric result satisfying Householder QR invariants
- Explanation
- Householder QR processes the input by reflecting columns to zero out subdiagonal entries, tracking qr decomposition and matrix decomposition to produce the verified qr decomposition result.
Implementation Notes
- Document parameter meanings such as tolerance, precision, and iteration limit.
- Use residual or error checks rather than trusting iteration count alone.
- Prefer stable formulations when inputs are ill-conditioned.
Common Pitfalls
- Ignoring overflow, cancellation, or loss of precision.
- Applying the method outside its stability assumptions.
- Testing only well-conditioned inputs.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow reflecting columns to zero out subdiagonal entries on a small qr decomposition example.
Step 1 of 5: Validate input dimensions, conditioning, and numerical tolerance parameters for Householder QR.
numerical-methods-mechanismNotes
Householder QR operates in O(n^3) time and O(n^2) space within the Matrix Decomposition family. Often compared with GMRES and MINRES.
Sources (1)
What each reference supports for this entry.
Matrix Computations
BookNumerical linear algebra reference for Householder QR's matrix operations, Krylov behavior, or factorization mechanics.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
UsesGMRES
Numerical Methods · Core
Householder QR uses GMRES as a subroutine, representation, or supporting idea while reflecting columns to zero out subdiagonal entries.
Compare with thisRelationships
View full relationship pageUsesGMRES
Householder QR uses GMRES as a subroutine, representation, or supporting idea while reflecting columns to zero out subdiagonal entries.
UsesMINRES
Householder QR uses MINRES as a subroutine, representation, or supporting idea while reflecting columns to zero out subdiagonal entries.
UsesBiCGSTAB
Householder QR uses BiCGSTAB as a subroutine, representation, or supporting idea while reflecting columns to zero out subdiagonal entries.
Related Algorithms
Tags
Numerical MethodsLinear Algebra