Overview
MINRES is a iterative linear solvers method for symmetric linear system solving. It works by minimizing residuals for symmetric systems without requiring positive definiteness.
ExtendedSource checked
MINRES
MINRES addresses symmetric linear system solving by minimizing residuals for symmetric systems without requiring positive definiteness.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Numerical Methods
- Difficulty
- Intermediate
- Family
- Iterative Linear Solvers
- Tier
- Extended
- Status
- Verified
- Review
- Source Checked
- Sources
- 3
- Formulas
- 0
- Relationships
- 6
- Code example
- Ready
- Visualization
- Ready
- Aliases
- Minimum Residual Method
Core Concepts
symmetric linear system solvingiterative linear solversresidual graphresidual minimizationnumerical 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 MINRES when solving symmetric linear system solving and you want to benefit from minimizing residuals for symmetric systems without requiring positive definiteness.
- Choose it when O(n^3) average/worst-case time and O(n^2) memory match your workload profile.
- Prefer it over GMRES or BiCGSTAB when the problem structure favors symmetric linear system solving.
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 symmetric linear system solving within Iterative Linear Solvers workloads (O(n^3) time, O(n^2) space).
- Works by minimizing residuals for symmetric systems without requiring positive definiteness.
- Maintains key invariants across symmetric linear system solving, iterative linear solvers, residual graph.
Use Cases
- Production and research workloads requiring symmetric linear system solving via minimizing residuals for symmetric systems without requiring positive definiteness.
- Iterative Linear Solvers pipelines where O(n^3) time and O(n^2) space fit system constraints.
- Engineering comparisons against GMRES and BiCGSTAB when tuning symmetric linear system solving.
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 MINRES.
- 2.Initialize the working buffers, basis vectors, or decomposition factors (symmetric linear system solving).
- 3.Execute the core numerical recurrence by minimizing residuals for symmetric systems without requiring positive definiteness.
- 4.Monitor residual error, stability bounds, or step convergence across iterations.
- 5.Return the computed symmetric linear system solving coefficients, factors, or transformed values.
Canonical Example
- Input
- Numbers, vectors, matrices, or functions configured for a representative symmetric linear system solving test case
- Output
- Computed numeric result satisfying MINRES invariants
- Explanation
- MINRES processes the input by minimizing residuals for symmetric systems without requiring positive definiteness, tracking symmetric linear system solving and iterative linear solvers to produce the verified symmetric linear system solving 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 minimizing residuals for symmetric systems without requiring positive definiteness on a small symmetric linear system solving example.
Step 1 of 5: Validate input dimensions, conditioning, and numerical tolerance parameters for MINRES.
numerical-methods-mechanismNotes
MINRES operates in O(n^3) time and O(n^2) space within the Iterative Linear Solvers family. Often compared with GMRES and BiCGSTAB.
Sources (3)
What each reference supports for this entry.
Templates for the Solution of Linear Systems
BookIterative solver reference for MINRES's sparse linear-system behavior and convergence tradeoffs.
Open sourceMatrix Computations
BookNumerical linear algebra reference for MINRES's matrix operations, Krylov behavior, or factorization mechanics.
Open sourcePaige-Saunders MINRES
PaperOriginal MINRES reference for solving sparse symmetric indefinite linear systems with Lanczos-based residual minimization.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Conceptual neighborGMRES
Numerical Methods · Core
MINRES and GMRES share a concrete iterative linear solvers mechanism for symmetric linear system solving.
Compare with thisRelationships
View full relationship pageConceptual neighborGMRES
MINRES and GMRES share a concrete iterative linear solvers mechanism for symmetric linear system solving.
Conceptual neighborBiCGSTAB
MINRES and BiCGSTAB share a concrete iterative linear solvers mechanism for symmetric linear system solving.
Conceptual neighborHouseholder QR
MINRES and Householder QR share a concrete iterative linear solvers mechanism for symmetric linear system solving.
Related Algorithms
Tags
Numerical Methods