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Solving Sparse Linear Systems with Sparse Backward Error

https://doi.org/10.1137/0610013
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What this work says

Outgoing claims: the papers this work cites, builds on, or argues with.

inferredcites →https://doi.org/10.1007/bf01386090openalex-snapshot
Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides
inferredcites →https://doi.org/10.1090/s0025-5718-1980-0572859-4openalex-snapshot
Iterative Refinement Implies Numerical Stability for Gaussian Elimination
inferredcites →https://doi.org/10.1093/acprof:oso/9780198508380.001.0001openalex-snapshot
Direct Methods for Sparse Matrices
inferredcites →https://doi.org/10.1093/imamat/10.1.118openalex-snapshot
On the Automatic Scaling of Matrices for Gaussian Elimination
inferredcites →https://doi.org/10.1137/0716029openalex-snapshot
An Estimate for the Condition Number of a Matrix
inferredcites →https://doi.org/10.1137/0905023openalex-snapshot
Condition Estimates
inferredcites →https://doi.org/10.1137/0905062openalex-snapshot
Underflow and the Reliability of Numerical Software
inferredcites →https://doi.org/10.1137/0907080openalex-snapshot
Condition Number Estimators in a Sparse Matrix Software
inferredcites →https://doi.org/10.1137/1.9781611971811openalex-snapshot
LINPACK Users' Guide
inferredcites →https://doi.org/10.1137/1029112openalex-snapshot
A Survey of Condition Number Estimation for Triangular Matrices
inferredcites →https://doi.org/10.1145/321075.321076openalex-snapshot
Error Analysis of Direct Methods of Matrix Inversion
inferredcites →https://doi.org/10.1145/322139.322148openalex-snapshot
Scaling for Numerical Stability in Gaussian Elimination
inferredcites →https://doi.org/10.1145/62038.62043openalex-snapshot
Sparse matrix test problems
inferredcites →https://doi.org/10.1287/mnsc.3.3.255openalex-snapshot
The Elimination form of the Inverse and its Application to Linear Programming
inferredcites →https://doi.org/10.2307/2006212openalex-snapshot
LINPACK User's Guide.
inferredcites →https://doi.org/10.7146/dpb.v9i123.6541openalex-snapshot
Direct Methods for Sparse Matrices

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Incoming claims: where this work appears in other papers’ reference lists and arguments.

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Fully Parallel and Pipelined Sparse Direct Solver for Large Symmetric Indefinite Finite Element Problems
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Fully Parallel and Pipelined Sparse Direct Solver for Large Symmetric Indefinite Finite Element Problems
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Iterative refinement enhances the stability ofQR factorization methods for solving linear equations
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Perturbation theory and backward error forAX−XB=C
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Experiments with data perturbations to study condition numbers and numerical stability
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Stopping criteria for iterative methods:¶applications to PDE's
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Error Estimation in Preconditioned Conjugate Gradients
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Estimation of the Sensitivity of linear and nonlinear algebraic problems
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Componentwise perturbation theory for linear systems with multiple right-hand sides
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Gaussian elimination: when is scaling beneficial?
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Condition numbers and their condition numbers
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Accuracy and effectiveness of preconditioned conjugate gradient algorithms for large and ill-conditioned problems
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Extending LINPACK by verification routines for linear systems
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Copyright Page
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Preface
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Introduction
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Sparse matrices: storage schemes and simple operations
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Gaussian elimination for sparse matrices: an introduction
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Local pivotal strategies for sparse matrices
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Gaussian elimination using trees
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Graphs for symmetric and unsymmetric matrices
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The solve phase
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Other sparsity-oriented issues
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Matrix and vector norms
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Pictures of sparse matrices
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Solutions to selected exercises
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Large-scale 3-D EM modelling with a Block Low-Rank multifrontal direct solver
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Iterative refinement for linear systems and LAPACK
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Sparse Numerical Linear Algebra: Direct Methods and Preconditioning
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Mixed-precision iterative refinement using tensor cores on GPUs to accelerate solution of linear systems
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An adaptive methodology for multi-GPU programming in OpenCL
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Floating point fault tolerance with backward error assertions
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LAPACK: a portable linear algebra library for supercomputers
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A statistical approach to condition estimation
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LAPACK-based condition and error estimators for kalman filter design
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A Parallel Tiled Solver for Dense Symmetric Indefinite Systems on Multicore Architectures
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Harnessing GPU Tensor Cores for Fast FP16 Arithmetic to Speed up Mixed-Precision Iterative Refinement Solvers
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LAPACK: A portable linear algebra library for high-performance computers
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Parallel algorithm research at CERFACS
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Iterative Refinement of Accelerations in Real-Time Vehicle Dynamics
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Unsymmetric Ordering Using A Constrained Markowitz Scheme
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Towards Stable Mixed Pivoting Strategies for the Sequential and Parallel Solution of Sparse Symmetric Indefinite Systems
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Large Growth Factors in Gaussian Elimination with Pivoting
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The Strong Stability of Algorithms for Solving Symmetric Linear Systems
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Bounding the Error in Gaussian Eimination for Tridiagonal Systems
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The Componentwise Distance to the Nearest Singular Matrix
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Stopping Criteria for Iterative Solvers
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Backward Error and Condition of Structured Linear Systems
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Improved Error Bounds for Underdetermined System Solvers
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On Accurate Computations of the Perron Root
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Mixed, Componentwise, and Structured Condition Numbers
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A Componentwise Perturbation Analysis of the $QR$ Decomposition
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The Accuracy of Solutions to Triangular Systems
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Experience with a Matrix Norm Estimator
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Small-Sample Statistical Condition Estimates for General Matrix Functions
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Block Lanczos Techniques for Accelerating the Block Cimmino Method
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Matrix symmetrization and sparse direct solvers
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Improving Multifrontal Methods by Means of Block Low-Rank Representations
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Toward the Finite-Time Blowup of the 3D Axisymmetric Euler Equations: A Numerical Investigation
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On the Symmetric Componentwise Relative Backward Error for Linear Systems of Equations
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Accelerating the Solution of Linear Systems by Iterative Refinement in Three Precisions
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Formation of Finite-Time Singularities in the 3D Axisymmetric Euler Equations: A Numerics Guided Study
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Algebraic Error Analysis for Mixed-Precision Multigrid Solvers
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A Perturbation Analysis of the Generalized Sylvester Equation $( AR - LB,DR - LE ) = ( C,F )$
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On the Stability of Cholesky Factorization for Symmetric Quasidefinite Systems
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Computing the Smallest Eigenvalue of an M-Matrix
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A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra
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Statistical Condition Estimation for Linear Systems
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On the Solution of Equality Constrained Quadratic Programming Problems Arising in Optimization
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A numerical evaluation of HSL packages for the direct solution of large sparse, symmetric linear systems of equations
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Parallel sparse LU factorization on second-class message passing platforms
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An overview of SuperLU
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A numerical evaluation of sparse direct solvers for the solution of large sparse symmetric linear systems of equations
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Stability of block algorithms with fast level-3 BLAS
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The design of MA48
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Accelerating Linear System Solutions Using Randomization Techniques
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A combined unifrontal/multifrontal method for unsymmetric sparse matrices
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ReFloat: Low-Cost Floating-Point Processing in ReRAM for Accelerating Iterative Linear Solvers
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Combining Sparse Approximate Factorizations with Mixed-precision Iterative Refinement
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Algorithm 1050: SPEX Cholesky, LDL, and Backslash for Exactly Solving Sparse Linear Systems
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Analysis and comparison of two general sparse solvers for distributed memory computers
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SuperLU_DIST
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MA57---a code for the solution of sparse symmetric definite and indefinite systems
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A column pre-ordering strategy for the unsymmetric-pattern multifrontal method
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A NOTE ON THE STRONG COMPONENTWISE STABILITY OF ALGORITHMS FOR SOLVING SYMMETRIC LINEAR SYSTEMS
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Using Random Butterfly Transformations in Parallel Schur Complement-Based Preconditioning
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RESOLUÇÃO DE SISTEMAS DE EQUAÇÕES LINEARES DE GRANDE PORTE EM CLUSTERS MULTI-GPU UTILIZANDO O MÉTODO DO GRADIENTE CONJUGADO EM OPENCLTM
inferred← citeshttps://doi.org/10.20868/upm.thesis.48853openalex-snapshot
Contributions to Topological Formulations for the Dynamic Simulation of Vehicles
inferred← citeshttps://doi.org/10.21236/ada254994openalex-snapshot
Perturbation Theory for the Solution of Systems of Linear Equations
inferred← citeshttps://doi.org/10.2172/751785openalex-snapshot
SuperLU users' guide
inferred← citeshttps://doi.org/10.31223/x50627openalex-snapshot
A discontinuous Galerkin method for sequences of earthquakes and aseismic slip on multiple faults using unstructured curvilinear grids
inferred← citeshttps://doi.org/10.3390/inventions8010038openalex-snapshot
Reduction in the Computational Complexity of Calculating Losses on Eddy Currents in a Hydrogen Fuel Cell Using the Finite Element Analysis
inferred← citeshttps://doi.org/10.3970/cmes.2012.084.297openalex-snapshot
Hybrid Parallelism of Multifrontal Linear Solution Algorithm with Out Of Core Capability for Finite Element Analysis
inferred← citeshttps://doi.org/10.48550/arxiv.1512.04246openalex-snapshot
Numerical stability of iterative refinement with a relaxation for linear systems
inferred← citeshttps://doi.org/10.48550/arxiv.1608.00451openalex-snapshot
Numerical tolerance for spectral decompositions of random matrices
inferred← citeshttps://doi.org/10.5555/110382.110584openalex-snapshot
Parallel algorithm research at CERFACS
Editorial review for independent work. ICSAC — the Institute for Complexity Science and Advanced Computing — runs a structured editorial panel for researchers publishing outside journals, and publishes on CiteStamp rails. Submission is free and goes through ORCID: icsacinstitute.org/submit.