டோப்ளிட்சு அணி
நேரியல் இயற்கணிதத்தில் '''டோப்ளிட்சு அணி''' அல்லது '''மூலைவிட்ட-மாறிலி அணி''' (''Toeplitz matrix'' , ''diagonal-constant matrix'') என்பது இடமிருந்து வலமாக இறங்கும் மூலைவிட்டங்கள் ஒவ்வொன்றிலும்  மாறாத  உறுப்புகளைக் கொண்ட அணியாகும்.
 
 எடுத்துக்காட்டு:
 :&lt;math&gt;
 \begin{bmatrix}
 a &amp; b &amp; c &amp; d &amp; e \\
 f &amp; a &amp; b &amp; c &amp; d \\
 g &amp; f &amp; a &amp; b &amp; c \\
 h &amp; g &amp; f &amp; a &amp; b \\
 i &amp; h &amp; g &amp; f &amp; a 
 \end{bmatrix}.
 &lt;/math&gt;
 
 ''n''&amp;times;''n'' வரிசை கொண்ட டோப்ளிட்சு அணி ''A'' இன் அமைப்பு:
 :&lt;math&gt;
 A =
 \begin{bmatrix}
   a_{0} &amp; a_{-1} &amp; a_{-2} &amp; \ldots  &amp; \ldots &amp; a_{-(n-1)}  \\
   a_{1} &amp; a_0    &amp; a_{-1} &amp; \ddots  &amp;        &amp; \vdots \\
   a_{2} &amp; a_{1}  &amp; \ddots &amp; \ddots  &amp; \ddots &amp; \vdots \\ 
  \vdots &amp; \ddots &amp; \ddots &amp; \ddots  &amp; a_{-1} &amp; a_{-2}\\
  \vdots &amp;        &amp; \ddots &amp; a_{1}   &amp; a_{0}  &amp; a_{-1} \\
 a_{n-1} &amp; \ldots &amp; \ldots &amp; a_{2}   &amp; a_{1}  &amp; a_{0}
 \end{bmatrix}
 &lt;/math&gt;
 
 ''A'' இன் ''i'',''j'' ஆவது உறுப்பு ''A''&lt;sub&gt;''i'',''j''&lt;/sub&gt; எனில், 
 :&lt;math&gt;A_{i,j} = A_{i+1,j+1} = a_{i-j}.\ &lt;/math&gt;
 
 டோப்ளிட்சு அணி ஒரு [[சதுர அணி]]யாக இருக்கவேண்டிய அவசியமில்லை.
 
 ==குறிப்புகள்==
 {{reflist}}
 
 ==மேற்கோள்கள்==
 *{{citation | last1 = Bareiss | first1 = E.H. | year = 1969 | title = Numerical solution of linear equations with Toeplitz and vector Toeplitz matrices | url = | journal = Numerische Mathematik | volume = 13 | issue = | pages = 404–424 | doi = 10.1007/BF02163269 }}
 *{{citation | last1 = Bojanczyk | first1 = A.W. | last2 = Brent | first2 = R.P. | last3 = Hoog | first3 = F.R. De | last4 = Sweet | first4 = D.R. | year = 1995 | title = On the stability of the Bareiss and related Toeplitz factorization algorithms | url = | journal = [[SIAM Journal on Matrix Analysis and Applications]] | volume = 16 | issue = | pages = 40–57 | doi = 10.1137/S0895479891221563 }}
 *{{citation|first1=Albrecht| last1= Böttcher| first2=Sergei M. | last2= Grudsky|title=Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis|url=https://books.google.com/books?id=Dmr0BwAAQBAJ&amp;pg=PA1|year=2012|publisher=Birkhäuser|isbn=978-3-0348-8395-5|pages=1–}}
 *[[Richard P. Brent|Brent R.P.]] (1999), &quot;Stability of fast algorithms for structured linear systems&quot;, ''Fast Reliable Algorithms for Matrices with Structure'' (editors&amp;mdash;T. Kailath, A.H. Sayed), ch.4 ([[Society for Industrial and Applied Mathematics|SIAM]]).
 *{{citation|last1=Chan |first1=R. H.-F.| last2= Jin| first2= X.-Q. | year=2007 | title= An Introduction to Iterative Toeplitz Solvers | publisher= [[Society for Industrial and Applied Mathematics|SIAM]]}}.
 *{{citation | last1 = Chandrasekeran | first1 = S. | last2 = Gu | first2 = M. | last3 = Sun | first3 = X. | last4 = Xia | first4 = J. | last5 = Zhu | first5 = J. | year = 2007 | title = A superfast algorithm for Toeplitz systems of linear equations | url = | journal = SIAM Journal on Matrix Analysis and Applications | volume = 29 | issue = | pages = 1247–1266 | doi = 10.1137/040617200 }}
 *{{citation | last1 = Chen | first1 = W.W. | last2 = Hurvich | first2 = C.M. | last3 = Lu | first3 = Y. | year = 2006 | title = On the correlation matrix of the discrete Fourier transform and the fast solution of large Toeplitz systems for long-memory time series | url = | journal = Journal of the American Statistical Association | volume = 101 | issue = | pages = 812–822 | doi = 10.1198/016214505000001069 }}
 *Gene H. Golub|Golub G.H., [[Charles F. Van Loan|van Loan C.F.]] (1996), ''Matrix Computations''  (Johns Hopkins University Press) &amp;sect;4.7&amp;mdash;Toeplitz and Related Systems.
 *Gray R.M., ''[http://ee.stanford.edu/~gray/toeplitz.pdf Toeplitz and Circulant Matrices: A Review]'' ([http://www.nowpublishers.com Now Publishers]).
 *{{citation | last1 = Krishna | first1 = H. | last2=Wang | first2= Y.  | title = The Split Levinson Algorithm is weakly stable | journal = SIAM Journal on Numerical Analysis | volume = 30 | issue = 5 | pages = 1498–1508 | year = 1993 | url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&amp;id=SJNAAM000030000005001498000001&amp;idtype=cvips&amp;gifs=yes | doi = 10.1137/0730078}}.
 *{{citation| last1=Monahan | first1= J.F. | year=2011 | title=Numerical Methods of Statistics | publisher= Cambridge University Press }}. 
 *{{citation | title=Structured Matrices and Polynomials: unified superfast algorithms | first=Victor Y. | last=Pan | authorlink=Victor Pan | publisher=Birkhäuser | year=2001 | isbn=0817642404 }}
 *{{citation|last1=Press | first1=W.H. | last2= Teukolsky | first2= S.A. | last3= Vetterling | first3= W.T. | last4= Flannery | first4= B.P. | year=2007 | title= Numerical Recipes: The Art of Scientific Computing | edition= Third | publisher= Cambridge University Press | isbn= 978-0-521-88068-8}}.
 *{{citation | last1 = Stewart | first1 = M | year = 2003 | title = A superfast Toeplitz solver with improved numerical stability | url = | journal = [[SIAM Journal on Matrix Analysis and Applications]] | volume = 25 | issue = | pages = 669–693 | doi = 10.1137/S089547980241791X }}
 
 [[பகுப்பு:அணிகள்]]