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<p><b>नया पृष्ठ</b></p><div>[[गणित]] में कौशी के नाम से नामकरण किया गया '''कौशी आव्यूह''' अथवा '''कौशी मैट्रिक्स''' एक m&times;n का [[आव्यूह]] है जहाँ ''a''<sub>''ij''</sub> निम्न प्रकार परिभाषित है<br />
<br />
:<math><br />
a_{ij}={\frac{1}{x_i-y_j}};\quad x_i-y_j\neq 0,\quad 1 \le i \le m,\quad 1 \le j \le n<br />
</math><br />
जहाँ <math>x_i</math> और <math>y_j</math> [[क्षेत्र (गणित)|क्षेत्र]] <math>\mathcal{F}</math> के अवयव हैं और <math>(x_i)</math> और <math>(y_j)</math> [[एकैकी फलन|एकैकी]] अनुक्रम हैं (इनमें पुनरावृत्त अवयव समाहित नहीं हैं अर्थात सभी अवयव ''भिन्न'' हैं).<br />
[[चित्र:LogoQM.tif|अंगूठाकार|कौशी]]<br />
<br />
[[हिल्बर्ट आव्यूह]] कौशी आव्यूह की विशेष स्थिति है, जहाँ<br />
:<math>x_i-y_j = i+j-1. \;</math><br />
कौशी आव्यूह का प्रत्येक [[उपाआव्यूह]] अपने आप में एक कौशी आव्यूह है।<br />
<br />
== कौशी सारणिक ==<br />
{{main|कौशी सारणिक}}<br />
कौशी आव्यूह का सारणिक प्राचलों <math>(x_i)</math> और <math>(y_j)</math> का स्पष्ट रूप से एक परिमेय फलन होगा। <br />
<!--The determinant of a Cauchy matrix is clearly a [[rational fraction]] in the parameters <math>(x_i)</math> and <math>(y_j)</math>. If the sequences were not injective, the determinant would vanish, and tends to infinity if some <math>x_i</math> tends to <math>y_j</math>. A subset of its zeros and poles are thus known. The fact is that there are no more zeros and poles:<br />
<br />
The determinant of a square Cauchy matrix '''A''' is known as a '''Cauchy determinant''' and can be given explicitly as<br />
:<math> \det \mathbf{A}={{\prod_{i=2}^n \prod_{j=1}^{i-1} (x_i-x_j)(y_j-y_i)}\over {\prod_{i=1}^n \prod_{j=1}^n (x_i-y_j)}}</math> &emsp;&emsp;&emsp;&emsp;(Schechter 1959, eqn 4).<br />
It is always nonzero, and thus all square Cauchy matrices are [[invertible matrix|invertible]]. The inverse '''A'''<sup>−1</sup> = '''B''' = [b<sub>ij</sub>] is given by <br />
:<math>b_{ij} = (x_j - y_i) A_j(y_i) B_i(x_j) \,</math> &emsp;&emsp;&emsp;&emsp;(Schechter 1959, Theorem 1)<br />
where ''A''<sub>i</sub>(x) and ''B''<sub>i</sub>(x) are the [[Lagrange polynomials]] for <math>(x_i)</math> and <math>(y_j)</math>, respectively. That is, <br />
:<math>A_i(x) = \frac{A(x)}{A^\prime(x_i)(x-x_i)} \quad\text{and}\quad B_i(x) = \frac{B(x)}{B^\prime(y_i)(x-y_i)}, </math><br />
with<br />
:<math>A(x) = \prod_{i=1}^n (x-x_i) \quad\text{and}\quad B(x) = \prod_{i=1}^n (x-y_i). </math><br />
--><br />
== व्यापकीकरण ==<br />
एक आव्यूह '''C''' '''कौशी स्दृश्य''' कहलाता है यदि इसे निम्न रूप में लिखा जा सके<br />
<br />
:<math>C_{ij}=\frac{r_i s_j}{x_i-y_j}.</math><br />
<br />
'''X'''=diag(x<sub>i</sub>), '''Y'''=diag(y<sub>i</sub>) परिभषित करने पर, दोनों कौशी और कौशी सदृश आव्यूह विस्तापन समीकरण सन्तुष्ट करते हैं<br />
<br />
:<math>\mathbf{XC}-\mathbf{CY}=rs^\mathrm{T}</math><br />
<br />
(जहां कौशी आव्यूह के लिए <math>r=s=(1,1,\ldots,1)</math>)। अतः कौशी स्दृश आव्यूह एक सामान्य विस्थापन आव्यूह है, <!--which can be exploited while working with the matrix. For example, there are known algorithms in literature for<br />
* approximate Cauchy matrix-vector multiplication with <math>O(n \log n)</math> [[Floating point| ops]] (e.g. the [[fast multipole method]]),<br />
* ([[pivoting|pivoted]]) [[LU factorization]] with <math>O(n^2)</math> ops (GKO algorithm), and thus linear system solving,<br />
* approximated or unstable algorithms for linear system solving in <math>O(n \log^2 n)</math>.<br />
Here <math>n</math> denotes the size of the matrix (one usually deals with square matrices, though all algorithms can be easily generalized to rectangular matrices).<br />
--><br />
== ये भी देखें==<br />
[[टोएपलित्ज़ आव्यूह]]<br />
== सन्दर्भ ==<br />
* {{cite journal |author=A. Gerasoulis |title=A fast algorithm for the multiplication of generalized Hilbert matrices with vectors |journal=Mathematics of Computation |year=1988 |volume=50 |issue=181 |pages=179–188 |url=http://www.ams.org/journals/mcom/1988-50-181/S0025-5718-1988-0917825-9/S0025-5718-1988-0917825-9.pdf |access-date=5 मई 2013 |archive-url=https://web.archive.org/web/20121024030124/http://www.ams.org/journals/mcom/1988-50-181/S0025-5718-1988-0917825-9/S0025-5718-1988-0917825-9.pdf |archive-date=24 अक्तूबर 2012 |url-status=live }}<br />
* {{cite journal |author=I. Gohberg, T. Kailath, V. Olshevsky |title=Fast Gaussian elimination with partial pivoting for matrices with displacement structure |journal=Mathematics of Computation |year=1995 |volume=64 |issue=212 |pages=1557–1576 |url=http://www.ams.org/journals/mcom/1995-64-212/S0025-5718-1995-1312096-X/S0025-5718-1995-1312096-X.pdf |access-date=5 मई 2013 |archive-url=https://web.archive.org/web/20121024030156/http://www.ams.org/journals/mcom/1995-64-212/S0025-5718-1995-1312096-X/S0025-5718-1995-1312096-X.pdf |archive-date=24 अक्तूबर 2012 |url-status=live }}<br />
* {{cite journal |author=P. G. Martinsson, M. Tygert, V. Rokhlin |title=An <math>O(N \log^2 N)</math> algorithm for the inversion of general Toeplitz matrices |journal=Computers &amp; Mathematics with Applications |year=2005 |volume=50 |pages=741–752 |url=http://amath.colorado.edu/faculty/martinss/Pubs/2004_toeplitz.pdf |access-date=5 मई 2013 |archive-url=https://web.archive.org/web/20110927002711/http://amath.colorado.edu/faculty/martinss/Pubs/2004_toeplitz.pdf |archive-date=27 सितंबर 2011 |url-status=live }}<br />
* {{cite journal |author=S. Schechter |title=On the inversion of certain matrices |journal=Mathematical Tables and Other Aids to Computation |year=1959 |volume=13 |issue=66 |pages=73–77 |url=http://www.ams.org/journals/mcom/1959-13-066/S0025-5718-1959-0105798-2/S0025-5718-1959-0105798-2.pdf |access-date=5 मई 2013 |archive-url=https://web.archive.org/web/20121024025943/http://www.ams.org/journals/mcom/1959-13-066/S0025-5718-1959-0105798-2/S0025-5718-1959-0105798-2.pdf |archive-date=24 अक्तूबर 2012 |url-status=live }}<br />
{{कौशी नामकरण}}<br />
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[[श्रेणी:गणित]]<br />
[[श्रेणी:सम्मिश्र विश्लेषण]]<br />
[[श्रेणी:कौशी नामकरण]]<br />
[[श्रेणी:आव्यूह सिद्धान्त]]<br />
[[श्रेणी:सारणिक]]<br />
[[श्रेणी:चित्र जोड़ें]]</div>imported>InternetArchiveBot