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Algorithms for numerical tasks in finite precision simultaneously seek to minimize the number of floating point operations performed, and also the number of bits of precision required by each floating point operation.
The QR transformation a unitary analogue to the LR transformation—Part 1
John GF Francis · 1961
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The QR transformation—Part 2
John GF Francis · 1962
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Global convergene of tridiagonal qr algorithm with origin shifts
J.H. Wilkinson · 1968
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The shifted qr algorithm for hermitian matrices
TJ Dekker and JF Traub · 1971
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A computational method for eigenvalues and eigenvectors of a matrix with real eigenvalues
A. N. Beavers and E. D. Denman · 1973
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A new similarity transformation method for eigenvalues and eigenvectors
A.N. Beavers and E.D. Denman · 1974
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The matrix sign function and computations in systems
Eugene D. Denman and Alex N. Beavers · 1976
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A new proof of global convergence for the tridiagonal $ql$ algorithm
W. Hoffmann and B. N. Parlett · 1978
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Eigenvalues and condition numbers of random matrices
Alan Edelman · 1988
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Algorithms for the polar decomposition
Walter Gander · 1990
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Design of a parallel nonsymmetric eigenroutine toolbox, part i
Zhaojun Bai and James W. Demmel · 1993
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The matrix sign function
Charles S. Kenney and Alan J Laub · 1995
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An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems
Zhaojun Bai, James Demmel, and Ming Gu · 1997
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Numerical behaviour of higham’s scaled method for polar decomposition
Andrzej Kiełbasiński and Krystyna Zietak · 2003
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Fast linear algebra is stable
James Demmel, Ioana Dumitriu, and Olga Holtz · 2007
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Fast matrix multiplication is stable
James Demmel, Ioana Dumitriu, Olga Holtz, and Robert Kleinberg · 2007
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Stable and efficient spectral divide and conquer algorithms for the symmetric eigenvalue decomposition and the svd
Yuji Nakatsukasa and Nicholas J. Higham · 2013
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A stable scaling of newton-schulz for improving the sign function computation of a hermitian matrix
Jie Chen, Edmond Chow, and The Newton-Schulz · 2014
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Computing fundamental matrix decompositions accurately via the matrix sign function in two iterations: The power of Zolotarev’s functions
Yuji Nakatsukasa and Roland W. Freund · 2016
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A stable, polynomial-time algorithm for the eigenpair problem
Diego Armentano, Carlos Beltrán, Peter Bürgisser, Felipe Cucker, and Michael Shub · 2018
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Pseudospectral shattering, the sign function, and diagonalization in nearly matrix multiplication time
Jess Banks, Jorge Garza-Vargas, Archit Kulkarni, and Nikhil Srivastava · 2020
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A new scaling for Newton’s iteration for the polar decomposition and its backward stability
Ralph Byers and Hongguo Xu · 2008
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Yuji Nakatsukasa, Zhaojun Bai, and François Gygi · 2010
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Minimizing communication for eigenproblems and the singular value decomposition
Grey Ballard, James Demmel, and Ioana Dumitriu · 2011
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Backward stability of iterations for computing the polar decomposition
Yuji Nakatsukasa and Nicholas J. Higham · 2012
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Jess Banks, Archit Kulkarni, Satyaki Mukherjee, and Nikhil Srivastava · 2021
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Global convergence of hessenberg shifted qr ii: Numerical stability
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Global Convergence of Hessenberg Shifted QR I: Exact Arithmetic
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Note on ”a new scaling for newton’s iteration for the polar decomposition and its backward stability” by r. byers and h. xu*
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Invariant subspaces and pca in nearly matrix multiplication time, 2024
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