en>Episcophagus: /* OR */ "the desired value" => 1, you can not use OR to set a bit that is 1 to 0, only 0 to 1.

2014-10-31T11:24:14Z

OR: "the desired value" => 1, you can not use OR to set a bit that is 1 to 0, only 0 to 1.

en>Dsimic: Undid revision 596905565 by 89.120.104.138 (talk) Sorry, this edit made the example incorrect – it is different, but that way shows beter the sign bit shifted in on the left

2014-02-25T05:53:55Z

Undid revision 596905565 by 89.120.104.138 (talk) Sorry, this edit made the example incorrect – it is different, but that way shows beter the sign bit shifted in on the left

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en>Dsimic: /* Mathematical equivalents */ Formatting improvement

2014-01-30T02:52:23Z

Mathematical equivalents: Formatting improvement

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In [[linear algebra]], a '''Hilbert matrix''', introduced by {{harvs|txt|last=Hilbert|year=1894|authorlink=David Hilbert}}, is a [[square matrix]] with entries being the [[unit fraction]]s

:<math> H_{ij} = \frac{1}{i+j-1}. </math>

For example, this is the 5 × 5 Hilbert matrix:

:<math>H = \begin{bmatrix}
1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\[4pt]
\frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} \\[4pt]
\frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} \\[4pt]
\frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} \\[4pt]
\frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9} \end{bmatrix}.</math>

The Hilbert matrix can be regarded as derived from the integral

:<math> H_{ij} = \int_{0}^{1} x^{i+j} \, dx, </math>

that is, as a [[Gramian matrix]] for powers of ''x''. It arises in the [[least squares]] approximation of arbitrary functions by [[polynomial]]s.

The Hilbert matrices are canonical examples of [[ill-conditioned]] matrices, making them notoriously difficult to use in numerical computation. For example, the 2-norm [[condition number]] of the matrix above is about 4.8 · 10<sup>5</sup>.

==Historical note==

{{harvtxt|Hilbert|1894}} introduced the Hilbert matrix to study the following question in [[approximation theory]]: "Assume that {{nowrap|''I'' {{=}} [''a'', ''b'']}} is a real interval. Is it then possible to find a non-zero polynomial ''P'' with integral coefficients, such that the integral

:<math>\int_{a}^b P(x)^2 dx</math>

is smaller than any given bound ''ε'' > 0, taken arbitrarily small?" To answer this question, Hilbert derives an exact formula for the [[determinant]] of the Hilbert matrices and investigates their asymptotics. He concludes that the answer to his question is positive if the length {{nowrap|''b'' − ''a''}} of the interval is smaller than 4.

==Properties==

The Hilbert matrix is [[symmetric matrix|symmetric]] and [[positive-definite matrix|positive definite]]. The Hilbert matrix is also [[totally positive]] (meaning the determinant of every [[submatrix]] is positive).

The Hilbert matrix is an example of a [[Hankel matrix]].

The determinant can be expressed in [[closed-form expression|closed form]], as a special case of the [[Cauchy determinant]]. The determinant of the ''n'' × ''n'' Hilbert matrix is

:<math>\det(H)={{c_n^{\;4}}\over {c_{2n}}}</math>

where

:<math>c_n = \prod_{i=1}^{n-1} i^{n-i}=\prod_{i=1}^{n-1} i!.\,</math>

Hilbert already mentioned the curious fact that the determinant of the Hilbert matrix is the reciprocal of an integer (see sequence {{OEIS2C|A005249}} in the [[OEIS]]) which also follows from the identity
: <math>{1 \over \det (H)}={{c_{2n}}\over {c_n^{\;4}}}=n!\cdot \prod_{i=1}^{2n-1} {i \choose [i/2]}.
</math>

Using [[Stirling's approximation]] of the [[factorial]] one can establish the following asymptotic result:

:<math>\det(H)=a_n\, n^{-1/4}(2\pi)^n \,4^{-n^2}</math>

where ''a''<sub>''n''</sub> converges to the constant <math>e^{1/4} 2^{1/12} A^{ - 3} \approx 0.6450 </math> as <math>n\rightarrow\infty</math>, where A is the [[Glaisher-Kinkelin constant]].

The [[matrix inverse|inverse]] of the Hilbert matrix can be expressed in closed form using [[binomial coefficient]]s; its entries are

:<math>(H^{-1})_{ij}=(-1)^{i+j}(i+j-1){n+i-1 \choose n-j}{n+j-1 \choose n-i}{i+j-2 \choose i-1}^2</math>

where ''n'' is the order of the matrix. It follows that the entries of the inverse matrix are all integer.

The condition number of the ''n''-by-''n'' Hilbert matrix grows as <math>O((1+\sqrt{2})^{4n}/\sqrt{n})</math>.

==References==

*{{Citation | last1=Hilbert | first1=David | author1-link=David Hilbert | title=Ein Beitrag zur Theorie des Legendre'schen Polynoms | publisher=Springer Netherlands | doi=10.1007/BF02418278 | jfm=25.0817.02 | year=1894 | journal=[[Acta Mathematica]] | issn=0001-5962 | volume=18 | pages=155–159}}. Reprinted in {{cite book|first=David|last= Hilbert|title=Collected papers|volume= II|chapter= article 21}}
* {{cite journal|doi=10.1007/PL00005392|last=Beckermann|first=Bernhard|title=The condition number of real Vandermonde, Krylov and positive definite Hankel matrices|journal= Numerische Mathematik|volume=85|issue=4|pages= 553–577|year= 2000}}
* {{cite journal|doi=10.2307/2975779|last=Choi|first= M.-D.|title= Tricks or Treats with the Hilbert Matrix|journal=American Mathematical Monthly|volume=90|issue=5|pages=301–312|year= 1983|jstor=2975779}}
* {{cite journal|last=Todd|first= John|title=The Condition Number of the Finite Segment of the Hilbert Matrix|journal=National Bureau of Standards, Applied Mathematics Series|volume=39|pages= 109–116|year=1954}}
* {{Cite book|last=Wilf|first=H. S.|title=Finite Sections of Some Classical Inequalities|location= Heidelberg|publisher= Springer|year= 1970|isbn=3-540-04809-X}}

{{Numerical linear algebra}}

[[Category:Numerical linear algebra]]
[[Category:Approximation theory]]
[[Category:Matrices]]
[[Category:Determinants]]

Bitwise operation - Revision history

en>Episcophagus: /* OR */ "the desired value" => 1, you can not use OR to set a bit that is 1 to 0, only 0 to 1.

en>Dsimic: Undid revision 596905565 by 89.120.104.138 (talk) Sorry, this edit made the example incorrect – it is different, but that way shows beter the sign bit shifted in on the left

en>Dsimic: /* Mathematical equivalents */ Formatting improvement