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Fast estimates of Hankel matrix condition numbers and numeric sparse interpolation

Erich L. Kaltofen, Wen-shin Lee, Zhengfeng Yang
<span title="">2011</span> <i title="ACM Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/r7ocu6k63rechaqyi4yeoiounm" style="color: black;">Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation - SNC &#39;11</a> </i> &nbsp;
We harness the Gohberg-Semencul formula for the inverse of a Hankel matrix to compute estimates for the structured condition numbers of all arising Hankel matrices in quadratic arithmetic time overall,  ...  Finally, we demonstrate by experiments that our condition number estimates lead to a viable termination criterion for polynomials with about 20 non-zero terms and of degree about 100, even in the presence  ...  Note added on Jan. 18, 2012: Changed dimensions of first singular leading principle Hankel matrix in Introduction on p. 131 to (t + 1) × (t + 1) from t × t.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/2331684.2331704">doi:10.1145/2331684.2331704</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/snc/KaltofenLY11.html">dblp:conf/snc/KaltofenLY11</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hrw7hwqqevbkreiiizy6tcyzxa">fatcat:hrw7hwqqevbkreiiizy6tcyzxa</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20140412030634/http://www4.ncsu.edu:80/~kaltofen/bibliography/11/KLY11.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/fd/a1/fda130693da3aebef744c68974f443d3a725c536.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/2331684.2331704"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> acm.org </button> </a>

Page 6359 of Mathematical Reviews Vol. , Issue 99i [page]

<span title="">1999</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The effect of using an approximate Schur complement is analyzed and condition number estimates for any type of ap- proximation of the Schur complement are derived.  ...  The authors consider Lanczos and polynomial algorithms with random start for estimating an eigenvector corresponding to the largest eigenvalue of a large positive-definite matrix.  ... 
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Page 1239 of Mathematical Reviews Vol. 58, Issue 2 [page]

<span title="">1979</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Author’s summary: “We consider the problem of solving, on elec- tronic digital computers, large systems of linear equations with a Numerical Analysis 53 #8188 65 sparse matrix.  ...  For a given order of reduced systems there is an optimal number of nonasymptotic coefficients that yields minimum error in the solu- tion.” Brisley, W. Two related estimation J. Austral. Math. Soc.  ... 
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k-Space Deep Learning for Accelerated MRI [article]

Yoseob Han, Leonard Sunwoo, Jong Chul Ye
<span title="2019-07-03">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The annihilating filter-based low-rank Hankel matrix approach (ALOHA) is one of the state-of-the-art compressed sensing approaches that directly interpolates the missing k-space data using low-rank Hankel  ...  The success of ALOHA is due to the concise signal representation in the k-space domain thanks to the duality between structured low-rankness in the k-space domain and the image domain sparsity.  ...  Note that the rank structure of the estimated Hankel matrix is hardwired as the number of filter channels, i.e. Q.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1805.03779v3">arXiv:1805.03779v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/a3pjg5lsgnezvejbsf6fbysi3u">fatcat:a3pjg5lsgnezvejbsf6fbysi3u</a> </span>
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Sparse polynomial interpolation: sparse recovery, super resolution, or Prony? [article]

Cédric Josz, Bernard Mourrain
<span title="2018-11-23">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We extend their result to the multivariate case, i.e., we show that exact recovery is guaranteed provided that a geometric spacing condition on the supports holds and the number of evaluations are sufficiently  ...  Finally we also describe the algebraic Prony method for sparse interpolation, which also recovers the exact decomposition but from less point evaluations and with no geometric spacing condition.  ...  Acknowledgement The work of the first two authors was funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement 666981 TAMING  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1708.06187v2">arXiv:1708.06187v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ulq5qbcaq5a33j3ysp2zekao3y">fatcat:ulq5qbcaq5a33j3ysp2zekao3y</a> </span>
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Sparse polynomial interpolation: sparse recovery, super-resolution, or Prony?

Cédric Josz, Jean Bernard Lasserre, Bernard Mourrain
<span title="2019-03-15">2019</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ld5cu4nusveezeot3mom6y4uku" style="color: black;">Advances in Computational Mathematics</a> </i> &nbsp;
We extend their result to the multivariate case, i.e., we show that exact recovery is guaranteed provided that a geometric spacing condition on the supports holds and the number of evaluations are sufficiently  ...  Finally we also describe the algebraic Prony method for sparse interpolation, which also recovers the exact decomposition but from less point evaluations and with no geometric spacing condition.  ...  Acknowledgement The work of the first two authors was funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement 666981 TAMING  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10444-019-09672-2">doi:10.1007/s10444-019-09672-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xxjngldxqneudhl2hwpssdotwq">fatcat:xxjngldxqneudhl2hwpssdotwq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200323075931/https://hal.archives-ouvertes.fr/hal-01575325/document" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/a4/6b/a46ba82d0237ae282b5f361320dce0752fd41818.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10444-019-09672-2"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Hybrid absorbing boundary conditions based on fast nonuniform grid integration for nonconvex scatterers

Amir Boag, Uri Shemer, Raphael Kastner
<span title="">2004</span> <i title="Wiley"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/e4432kfn7vhifc3aru3e7hyw5i" style="color: black;">Microwave and optical technology letters (Print)</a> </i> &nbsp;
Novel hybrid (local-global) absorbing boundary conditions (ABCs) are proposed for the truncation of computational domains in finite-method analysis of open-region problems.  ...  The pertinent boundary integration is accelerated by the nonuniform grid-interpolation scheme, resulting in a fast algorithm for treating electrically large objects with largely concave outer boundaries  ...  The resultant matrix is semi-sparse, with the interboundary interactions forming a fully populated block, the computational domain generating a sparse matrix, and the boundary-internal element interactions  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/mop.20389">doi:10.1002/mop.20389</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/yix45k7zhbcdxhqlpwnupvydeu">fatcat:yix45k7zhbcdxhqlpwnupvydeu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170809020607/http://www.eng.tau.ac.il/~kast/shemer-boag-44.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/c8/90/c890fb05ec61f4f9893b12f1f259a6afdf51d6b2.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/mop.20389"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> wiley.com </button> </a>

Symbolic and Numeric Methods for Exploiting Structure in Constructing Resultant Matrices

Ioannis Z Emiris, Victor Y Pan
<span title="">2002</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ezljl2d3lzga5efenbxdvvfcpa" style="color: black;">Journal of symbolic computation</a> </i> &nbsp;
First, a fast and numerically stable method for determining the rank of rectangular matrices, which works exclusively over oating point arithmetic.  ...  Second, exact polynomial arithmetic algorithms that improve upon the complexity of polynomial multiplication under our model of sparseness, o ering bounds linear in the number of variables and the number  ...  Acknowledgments We thank Erich Kaltofen for pointing out the references on Tellegen's theorem and Henryk Wo niakowski for furnishing us with the reprints of KW92, KW94].  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1006/jsco.2002.0520">doi:10.1006/jsco.2002.0520</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/q33ctvukenah7lv37y6pl5jqky">fatcat:q33ctvukenah7lv37y6pl5jqky</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190222182753/http://pdfs.semanticscholar.org/3b25/fd92136e70dd8174ab653e3744db50c80138.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/3b/25/3b25fd92136e70dd8174ab653e3744db50c80138.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1006/jsco.2002.0520"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Sparse + Low Rank Decomposition of Annihilating Filter-based Hankel Matrix for Impulse Noise Removal [article]

Kyong Hwan Jin, Jong Chul Ye
<span title="2015-10-19">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
By extending this idea, here we propose a novel impulse noise removal algorithm using sparse + low rank decomposition of an annihilating filter-based Hankel matrix.  ...  associated rank-deficient Hankel matrices in the image domain to estimate the missing pixels.  ...  denotes the estimated rank of the Hankel matrix.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1510.05559v1">arXiv:1510.05559v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vtwcqvo72jbxrkqy5wtfdyx4am">fatcat:vtwcqvo72jbxrkqy5wtfdyx4am</a> </span>
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A Fast Semidirect Least Squares Algorithm for Hierarchically Block Separable Matrices

Kenneth L. Ho, Leslie Greengard
<span title="">2014</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jkb4mlvirjanvdf6ozhoncfkyq" style="color: black;">SIAM Journal on Matrix Analysis and Applications</a> </i> &nbsp;
In essence, our scheme consists of a direct component, comprised of matrix compression and factorization, followed by an iterative component to enforce certain equality constraints.  ...  We illustrate the performance of the method on both over- and underdetermined systems in a variety of settings, with an emphasis on radial basis function approximation and efficient updating and downdating  ...  We would like to thank the anonymous referees for their careful reading and insightful remarks, which have improved the paper tremendously.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/120902677">doi:10.1137/120902677</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/n5rnrdu2avh4xhqdx4zm46u7ka">fatcat:n5rnrdu2avh4xhqdx4zm46u7ka</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170706031736/http://klho.github.io/pubs/ho-2014-siam-j-matrix-anal-appl.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/2c/96/2c9647e3bf14d6b07ca2c362ded42adc4031789b.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/120902677"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Two-Dimensional Augmented State–Space Approach with Applications to Sparse Representation of Radar Signatures

Wu, Xu
<span title="2019-10-24">2019</span> <i title="MDPI AG"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/taedaf6aozg7vitz5dpgkojane" style="color: black;">Sensors</a> </i> &nbsp;
Effectiveness of the proposed approach is verified by the numerical and measured inverse synthetic aperture radar (ISAR) data.  ...  In this work, we focus on sparse representation of two-dimensional (2-D) radar signatures for man-made targets.  ...  Thus, the number of non-repeated poles K 1 should satisfy the following condition.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3390/s19214631">doi:10.3390/s19214631</a> <a target="_blank" rel="external noopener" href="https://www.ncbi.nlm.nih.gov/pubmed/31653113">pmid:31653113</a> <a target="_blank" rel="external noopener" href="https://pubmed.ncbi.nlm.nih.gov/PMC6864621/">pmcid:PMC6864621</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zabtere5fjarva6l6hhqx2ewca">fatcat:zabtere5fjarva6l6hhqx2ewca</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200215091321/https://res.mdpi.com/d_attachment/sensors/sensors-19-04631/article_deploy/sensors-19-04631-v2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/0e/40/0e407c08893992de9ba16bc5191d7a5c04515613.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3390/s19214631"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> mdpi.com </button> </a> <a target="_blank" rel="external noopener" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6864621" title="pubmed link"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> pubmed.gov </button> </a>

Reconstruction of sparse Legendre and Gegenbauer expansions

Daniel Potts, Manfred Tasche
<span title="2015-12-29">2015</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/zhnaqo2cv5epniuk4xnpmuqb74" style="color: black;">BIT Numerical Mathematics</a> </i> &nbsp;
We present a new deterministic approximate algorithm for the reconstruction of sparse Legendre expansions from a small number of given samples.  ...  Furthermore we show that the suggested method can be extended to the reconstruction of sparse Gegenbauer expansions of low positive order.  ...  . , M 0 ) and x 1, j ∈ [−1, 1] ( j = 1, . . . , M 1 ) as eigenvalues of the matrix F Algorithm 3.1 (Sparse Legendre interpolation based on SVD) Input: L, K , N ∈ N (N 1, 3 ≤ L ≤ K ≤ N ), L is upper bound  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10543-015-0598-1">doi:10.1007/s10543-015-0598-1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fohuntfr2rbmflhpweq3vtw34a">fatcat:fohuntfr2rbmflhpweq3vtw34a</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170705102941/https://www-user.tu-chemnitz.de/~potts/paper/legendreexp.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/f5/95/f5951b03278da58b5d7a6278d0bd3613bf09b51e.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10543-015-0598-1"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

On low-complexity approximation of matrices

Alle-Jan van der Veen, Patrick Dewilde
<span title="">1994</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wsx3rzhpingfvewcn5nwhfkq3e" style="color: black;">Linear Algebra and its Applications</a> </i> &nbsp;
The operation of multiplication of a vector by a matrix can be represented by a computational scheme (or model) that acts sequentially on the entries of the vector.  ...  The number of intermediate quantities ("states") that are needed in the computations is a measure of the complexity of the model. If that complexity is low, then not only 1.  ...  The sum and product of matrices with sparse state structure have again a sparse state structure with number of states at each point not larger than the sum of the number of states of its component systems  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0024-3795(94)90383-2">doi:10.1016/0024-3795(94)90383-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/357qwhbo5vhcvm2ayzelgxmqmu">fatcat:357qwhbo5vhcvm2ayzelgxmqmu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20171001163156/http://publisher-connector.core.ac.uk/resourcesync/data/elsevier/pdf/a6c/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS8wMDI0Mzc5NTk0OTAzODMy.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/d5/72/d5721d2cf087eb789b74db558d71540d50473dd2.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0024-3795(94)90383-2"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Prony's method in several variables [article]

Tomas Sauer
<span title="2016-08-14">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The paper gives an extension of Prony's method to the multivariate case which is based on the relationship between polynomial interpolation, normal forms modulo ideals and H--bases.  ...  A detailed study of these question, the choice of an optimal matrix X and quantitative estimates are not in the scope of this paper, however.  ...  In addition, the algorithms are quite fast due to their numerical nature.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1602.02352v2">arXiv:1602.02352v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/j73mhbbgcbc7rbo4bfinzcp6pe">fatcat:j73mhbbgcbc7rbo4bfinzcp6pe</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200830071454/https://arxiv.org/pdf/1602.02352v2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/d0/99/d099b92aa9cf2504311eb8bfcb40209faf2b46e3.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1602.02352v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Chapter 10: Algebraic Algorithms [article]

Ioannis Z. Emiris, Victor Y. Pan, Elias P. Tsigaridas
<span title="2013-11-15">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Diaz-Herrera, editors, covers Algebraic Algorithms, both symbolic and numerical, for matrix computations and root-finding for polynomials and systems of polynomials equations.  ...  We cover part of these large subjects and include basic bibliography for further study.  ...  Condition number of a matrix is a scalar κ which grows large as the matrix approaches a singular matrix; then numeric inversion becomes an ill-conditioned problem. κ OUTPUT ERROR NORM ≈ INPUT ERROR NORM  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1311.3731v1">arXiv:1311.3731v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/whtgwztbmbgqbl44s4e663oulu">fatcat:whtgwztbmbgqbl44s4e663oulu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200827175250/https://arxiv.org/pdf/1311.3731v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/ad/ed/aded1608a613137cd55ea9763e7d5293201acd42.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1311.3731v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>
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