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TT-cross approximation for multidimensional arrays
Abstract As is well known, a rank- r matrix can be recovered from a cross of r linearly independent columns and rows, and an arbitrary matrix can be interpolated on the cross entries. Other entriesExpand
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Breaking the Curse of Dimensionality, Or How to Use SVD in Many Dimensions
TLDR
A hierarchical data structure, called the Tree-Tucker format, is presented as an alternative to the canonical decomposition. Expand
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A Theory of Pseudoskeleton Approximations
Abstract Let an m × n matrix A be approximated by a rank- r matrix with an accuracy e. We prove that it is possible to choose r columns and r rows of A forming a so-called pseudoskeleton componentExpand
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How to find a good submatrix
Pseudoskeleton approximation and some other problems require the knowledge of sufficiently well-conditioned submatrix in a large-scale matrix. The quality of a submatrix can be measured by modulus ofExpand
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Optimal and Superoptimal Circulant Preconditioners
  • E. Tyrtyshnikov
  • Mathematics, Computer Science
  • SIAM J. Matrix Anal. Appl.
  • 1 April 1992
TLDR
Fast algorithms for finding superoptimal preconditioners, which inherit nonsingularity and positive-definiteness from A. Expand
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A brief introduction to numerical analysis
Lecture 1: metric space some useful definitions nested balls normed space popular vector norms matrix norms equivalent norms operator norms. Lecture 2: scalar product length of a vector isometricExpand
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Spectra of multilevel toeplitz matrices: Advanced theory via simple matrix relationships
We consider the eigenvalue and singular-value distributions for m-level Toeplitz matrices generated by a complex-valued periodic function ƒ of m real variables. We show that familiar formulations forExpand
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Some Remarks on the Elman Estimate for GMRES
TLDR
We propose some asymptotically tighter bound in terms of the ratio of the smallest eigenvalue of the hermitian part and the norm of some nonsymmetric matrix, based on a GMRES error estimate. Expand
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Incomplete Cross Approximation in the Mosaic-Skeleton Method
TLDR
The mosaic-skeleton method was bred in a simple observation that rather large blocks in very large matrices coming from integral formulations can be approximated accurately by a sum of just few rank-one matrices (skeletons). Expand
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