References
- [1]
- S. M. Rump. Fast and Parallel Interval Arithmetic. BIT Numerical Mathematics 39, 534–554 (1999).
- [2]
- N. Revol and P. Théveny. Parallel Implementation of Interval Matrix Multiplication. Reliable Computing 19, 91 (2013).
- [3]
- S. Miyajima. Fast enclosure for solutions of Sylvester equations. Linear Algebra and its Applications 439, 856–878 (2013).
- [4]
- S. M. Rump and S. Oishi. Fast enclosure of matrix eigenvalues and singular values via rounding mode controlled computation. Linear Algebra and its Applications 324, 133–146 (2001).
- [5]
- S. Miyajima. Fast verified matrix multiplication. Journal of Computational and Applied Mathematics 233, 2994–3004 (2010).
- [6]
- T. Ogita and K. Aishima. Iterative refinement for singular value decomposition based on matrix multiplication. Journal of Computational and Applied Mathematics 369, 112512 (2020).
- [7]
- S. Miyajima. Numerical enclosure for each eigenvalue in generalized eigenvalue problem. Journal of Computational and Applied Mathematics 236, 2545–2552 (2012).
- [8]
- S. Miyajima. Verified computation of invariant subspaces. SIAM Journal on Matrix Analysis and Applications 35, 1205–1225 (2014).
- [9]
- Z. Bujanović, D. Kressner and C. Schröder. Iterative refinement of Schur decompositions. Numerical Algorithms 95, 247–267 (2024). Preprint: arXiv:2203.10879v1, 2022.
- [10]
- S. M. Rump. Verified bounds for singular values, in particular for the spectral norm of a matrix and its inverse. BIT Numerical Mathematics 51, 367–384 (2011).
- [11]
- S. Miyajima. Verified bounds for all the singular values of matrix. Japan Journal of Industrial and Applied Mathematics 31, 513–539 (2014).
- [12]
- S. M. Rump. Verification methods: Rigorous results using floating-point arithmetic. Acta Numerica 19, 287–449 (2010).
- [13]
- L. Qi. Some simple estimates for singular values of a matrix. Linear Algebra and its Applications 56, 105–119 (1984).
- [14]
- N. J. Higham. Accuracy and Stability of Numerical Algorithms: Second Edition (Philadelphia, PA: SIAM, 1996).
- [15]
- S. Oishi. Lower bounds for the smallest singular values of generalized asymptotic diagonal dominant matrices. Japan Journal of Industrial and Applied Mathematics 40, 1569–1585 (2023).
- [16]
- S. M. Rump and S. Oishi. A note on Oishi's lower bound for the smallest singular value of linearized Galerkin equations. Japan Journal of Industrial and Applied Mathematics 41, 1097–1104 (2024).
- [17]
- T. Ogita and K. Aishima. Iterative refinement for symmetric eigenvalue decomposition. Japan Journal of Industrial and Applied Mathematics 35, 1007–1035 (2018). Algorithm 1: RefSyEv.
- [18]
- N. J. Higham. Computing the Polar Decomposition—with Applications. SIAM Journal on Scientific and Statistical Computing 7, 1160–1174 (1986).
- [19]
- N. J. Higham. Functions of Matrices: Theory and Computation (Philadelphia, PA: SIAM, 2008). Chapter 8 covers the polar decomposition.
- [20]
- Y. Nakatsukasa, Z. Bai and F. Gygi. Optimizing Halley's Iteration for Computing the Matrix Polar Decomposition. SIAM Journal on Scientific Computing 32, 2700–2720 (2010). QDWH algorithm for polar decomposition.
- [21]
- J. H. Wilkinson. Rounding Errors in Algebraic Processes (Prentice-Hall, 1963). Classic text on rounding error analysis.
- [22]
- R. S. Martin, G. Peters and J. H. Wilkinson. Iterative Refinement of the Solution of a Positive Definite System of Equations. Numerische Mathematik 8, 203–216 (1971). Cholesky factor refinement.
- [23]
- Y. Yamamoto, Y. Nakatsukasa, Y. Yanagisawa and T. Fukaya. Roundoff error analysis of the CholeskyQR2 algorithm. Numerische Mathematik 131, 297–322 (2015). CholeskyQR2 for QR reorthogonalization.
- [24]
- T. Fukaya, R. Kannan, G. Ballard and J. Dongarra. LU-Cholesky QR Algorithms for Thin QR Decomposition. SIAM Journal on Scientific Computing 42, A1401–A1423 (2020).
- [25]
- E. Carson and N. J. Higham. A New Analysis of Iterative Refinement and its Application to Accurate Solution of Ill-Conditioned Sparse Linear Systems. SIAM Journal on Scientific Computing 39, A2834–A2856 (2017). Mixed precision iterative refinement.
- [26]
- S. M. Rump and T. Ogita. Error-free transformations and verified matrix decompositions. Numerical Algorithms (2024). Verified LU, Cholesky, QR, polar decompositions.