On the Stability of Some Hierarchical Rank Structured Matrix Algorithms

DOI10.1137/15M1026195zbMath1348.65064OpenAlexW2523571399MaRDI QIDQ2827062

Jian-Lin Xia, Yuanzhe Xi

Publication date: 12 October 2016

Published in: SIAM Journal on Matrix Analysis and Applications (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1137/15m1026195

zbMATH Keywords

error propagationbackward stabilityhierarchical rank structureHSS algorithmsstructured backward stabilityULV factorization

Mathematics Subject Classification ID

Related Items (11)

SuperDC: Superfast Divide-And-Conquer Eigenvalue Decomposition With Improved Stability for Rank-Structured Matrices ⋮ A Hierarchical Low Rank Schur Complement Preconditioner for Indefinite Linear Systems ⋮ A Fast Contour-Integral Eigensolver for Non-Hermitian Matrices ⋮ Fast structured Jacobi-Jacobi transforms ⋮ Structured condition numbers for linear systems with parameterized quasiseparable coefficient matrices ⋮ Structured generalized eigenvalue condition numbers for parameterized quasiseparable matrices ⋮ A stable matrix version of the fast multipole method: stabilization strategies and examples ⋮ Parallel Randomized and Matrix-Free Direct Solvers for Large Structured Dense Linear Systems ⋮ A Power Schur Complement Low-Rank Correction Preconditioner for General Sparse Linear Systems ⋮ Numerical algorithms for high-performance computational science ⋮ Robust and Accurate Stopping Criteria for Adaptive Randomized Sampling in Matrix-Free Hierarchically Semiseparable Construction

Uses Software

• mctoolbox

Cites Work

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• Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions
• Stable solutions of linear systems involving long chain of matrix multiplications
• A direct solver with \(O(N)\) complexity for integral equations on one-dimensional domains
• A fast implicit QR eigenvalue algorithm for companion matrices
• A sparse matrix arithmetic based on \({\mathfrak H}\)-matrices. I: Introduction to \({\mathfrak H}\)-matrices
• On a new class of structured matrices
• Introduction to hierarchical matrices with applications.
• A sparse \({\mathcal H}\)-matrix arithmetic. II: Application to multi-dimensional problems
• An Algebraic Multilevel Preconditioner with Low-Rank Corrections for Sparse Symmetric Matrices
• Superfast and Stable Structured Solvers for Toeplitz Least Squares via Randomized Sampling
• On the Complexity of Some Hierarchical Structured Matrix Algorithms
• A Fast Randomized Eigensolver with Structured LDL Factorization Update
• Fast Sparse Selected Inversion
• Robust Approximate Cholesky Factorization of Rank-Structured Symmetric Positive Definite Matrices
• Fast algorithms for hierarchically semiseparable matrices
• A Fast Randomized Algorithm for Computing a Hierarchically Semiseparable Representation of a Matrix
• Construction of Data-Sparse $\mathcal{H}^2$-Matrices by Hierarchical Compression
• Hierarchische Matrizen
• Accuracy and Stability of Numerical Algorithms
• Efficient Algorithms for Computing a Strong Rank-Revealing QR Factorization
• A Superfast Structured Solver for Toeplitz Linear Systems via Randomized Sampling
• A Fast Direct Solver for Structured Linear Systems by Recursive Skeletonization
• A Fast $ULV$ Decomposition Solver for Hierarchically Semiseparable Representations
• Efficient Structured Multifrontal Factorization for General Large Sparse Matrices
• Stability of QR-based fast system solvers for a subclass of quasiseparable rank one matrices
• Nested Product Decomposition of Quasiseparable Matrices
• Some Fast Algorithms for Sequentially Semiseparable Representations
• A bibliography on semiseparable matrices
• A fast algorithm for particle simulations

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