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In [[numerical mathematics]], '''hierarchical matrices (H-matrices)'''
<ref name="HA99">{{cite journal|last=Hackbusch|first=Wolfgang|date=1999|title=A sparse matrix arithmetic based on H-matrices. Part I: Introduction to H-matrices|journal=Computing|volume=62|pages=89–108|doi=10.1007/s006070050015}}</ref>
<ref name="GRHA02">{{cite journal|last=Grasedyck|first=Lars|last2=Hackbusch|first2=Wolfgang|date=2003|title=Construction and arithmetics of H-matrices|journal=Computing|volume=70|pages=295–334|url=https://dx.doi.org/10.1007/s00607-003-0019-1|doi=10.1007/s00607-003-0019-1}}</ref>
<ref name="HA09">{{cite book|last=Hackbusch|first=Wolfgang|date=2015|title=Hierarchical matrices: Algorithms and Analysis|publisher=Springer|url=https://dx.doi.org/10.1007/978-3-662-47324-5}}</ref>
are used as data-sparse approximations of non-sparse matrices.
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<ref name="MB08">{{cite book|last=Bebendorf|first=Mario|date=2008|title=Hierarchical matrices: A means to efficiently solve elliptic boundary value problems|publisher=Springer}}</ref>
<ref name="HAKH00">{{cite journal|last=Hackbusch|first=Wolfgang|last2=Khoromskij|first2=Boris N.|date=2000|title=A sparse H-Matrix Arithmetic. Part II: Application to Multi-Dimensional Problems|journal=Computing|volume=64|pages=21–47}}</ref>
<ref name="MB00">{{cite journal|last=Bebendorf|first=Mario|title=Approximation of boundary element matrices|date=2000|journal=Num. Math.|volume=86|pages=565–589|doi=10.1007/pl00005410}}</ref>
<ref name="BERJ03">{{cite journal|last=Bebendorf|first=Mario|last2=Rjasanow|first2=Sergej|date=2003|title=Adaptive low-rank approximation of collocation matrices|journal=Computing|volume=70|pages=1–24|doi=10.1007/s00607-002-1469-6}}</ref>
<ref name="BOGR05">{{cite journal|last=Börm|first=Steffen|last2=Grasedyck|first2=Lars|date=2005|title=Hybrid cross approximation of integral operators|journal=Num. Math.|volume=101|pages=221–249|doi=10.1007/s00211-005-0618-1}}</ref>
<ref name="BOCH16">{{cite journal|last=Börm|first=Steffen|last2=Christophersen|first2=Sven|date=2016|title=Approximation of integral operators by Green quadrature and nested cross approximation|journal=Num. Math.|volume=133|pages=409–442|url=https://dx.doi.org/10.1007/s00211-015-0757-y|doi=10.1007/s00211-015-0757-y}}</ref>,
preconditioning the resulting systems of linear equations
,<ref name="FAMEPR16">{{cite journal|last=Faustmann|first=Markus|last2=Melenk|first2=J. Markus|last3=Praetorius|first3=Dirk|date=2016|title=Existence of H-matrix approximants to the inverses of BEM matrices: The simple-layer operator|journal=Math. Comp.|volume=85|pages=119–152|url=https://dx.doi.org/10.1090/mcom/2990|doi=10.1090/mcom/2990}}</ref>
or solving elliptic partial differential equations
<ref name="BEHA03">{{cite journal|last=Bebendorf|first=Mario|last2=Hackbusch|first2=Wolfgang|date=2003|title=Existence of H-matrix approximants to the inverse FE-matrix of elliptic operators with <math>L^\infty</math>-coefficients|journal=Num. Math.|volume=95|pages=1–28|doi=10.1007/s00211-002-0445-6}}</ref>
<ref name="BO10">{{cite journal|last=Börm|first=Steffen|date=2010|title=Approximation of solution operators of elliptic partial differential equations by H- and H<sup>2</sup>-matrices|journal=Num. Math.|volume=115|pages=165–193|url=https://dx.doi.org/10.1007/s00211-009-0278-7|doi=10.1007/s00211-009-0278-7}}</ref>
<ref name ="FAMEPR13">{{cite journal|last=Faustmann|first=Markus|last2=Melenk|first2=J. Markus|last3=Praetorius|first3=Dirk|date=2015|title=H-matrix approximability of the inverses of FEM matrices|journal=Num. Math.|volume=131|pages=615–642|url=https://dx.doi.org/10.1007/s00211-015-0706-9|doi=10.1007/s00211-015-0706-9}}</ref>,
a rank proportional to <math>\log(1/\epsilon)^\gamma</math> with a small constant <math>\gamma</math> is sufficient to ensure an
accuracy of <math>\epsilon</math>.
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<ref name="MB00"/>
<ref name="BERJ03"/>
<ref name="TY00">{{cite journal|last=Tyrtyshnikov|first=Eugene|date=2000|title=Incomplete cross approximation in the mosaic-skeleton method|journal=Computing|volume=64|pages=367–380|doi=10.1007/s006070070031}}</ref>
that use only the entries of the original matrix <math>G</math> to construct a [[low rank approximation|low-rank approximation]].
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[[Schur complement]]s of diagonal blocks and combining both using the matrix-matrix multiplication.
In a similar way, the [[LU decomposition]]
<ref name="BE07">{{cite journal|last=Bebendorf|first=Mario|date=2007|title=Why finite element discretizations can be factored by triangular hierarchical matrices|journal=SIAM J. Num. Anal.|volume=45|pages=1472–1494|doi=10.1137/060669747}}</ref>
<ref name="GRKRBO09">{{cite journal|last=Grasedyck|first=Lars|last2=Kriemann|first2=Ronald|last3=Le Borne|first3=Sabine|date=2009|title=Domain decomposition based H-LU preconditioning|journal=Num. Math.|volume=112|pages=565–600|doi=10.1007/s00211-009-0218-6}}</ref>
can be constructed using only recursion and multiplication.
Both operations also require <math>O(n k^2\,\log(n)^2)</math> operations.
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In the context of boundary integral operators, replacing the fixed rank <math>k</math> by block-dependent ranks
leads to approximations that preserve the rate of convergence of the underlying boundary element method
at a complexity of <math>O(n).</math><ref name="SA00">{{cite journal|last=Sauter|first=Stefan|date=2000|title=Variable order panel clustering|journal=Computing|volume=64|pages=223–261|url=https://link.springer.com/article/10.1007/s006070050045|doi=10.1007/s006070050045}}</ref>
<ref name="BOSA05">{{cite journal|last=Börm|first=Steffen|last2=Sauter|first2=Stefan|date=2005|title=BEM with linear complexity for the classical boundary integral operators|journal=Math. Comp.|volume=74|pages=1139–1177|url=http://www.ams.org/journals/mcom/2005-74-251/S0025-5718-04-01733-8|doi=10.1090/s0025-5718-04-01733-8}}</ref>
Arithmetic operations like multiplication, inversion, and Cholesky or LR factorization of H<sup>2</sup>-matrices
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