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(Created page with "** Nr 1. Cf. A priori and a posteriori bounds in matrix computations <pre> Markup: .BDIS {A15} p \eql {fnnme max} from {1 \...")
 
m
 
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\$\$ \tag{A15}
 
\$\$ \tag{A15}
 
p =  \displaystyle\max _ {1 <= i <= n}  
 
p =  \displaystyle\max _ {1 <= i <= n}  
−
{{\left \| {b _ i - \sum _ {j = 1} ^ n a _ {ij} {\hat x} _ j} \right \|}  
+
{{\left | {b _ i - \sum _ {j = 1} ^ n a _ {ij} {\hat x} _ j} \right |}  
−
\over {BN + AN \cdot \sum _ {j = 1} ^ n \left \| {{\hat x} _ j} \right \|}} ,
+
\over {BN + AN \cdot \sum _ {j = 1} ^ n \left | {{\hat x} _ j} \right |}} ,
 
\$\$
 
\$\$
 
</pre>
 
</pre>
  
−
TeX Code displayed:
+
TeX Code displayed [https://www.encyclopediaofmath.org/legacyimages/a/a110/a110010/a110010108.png from this a110010108.png ]
 
$$ \tag{A15}
 
$$ \tag{A15}
 
p =  \displaystyle\max _ {1 <= i <= n}  
 
p =  \displaystyle\max _ {1 <= i <= n}  
−
{{\left \| {b _ i - \sum _ {j = 1} ^ n a _ {ij} {\hat x} _ j} \right \|}  
+
{{\left | {b _ i - \sum _ {j = 1} ^ n a _ {ij} {\hat x} _ j} \right |}  
−
\over {BN + AN \cdot \sum _ {j = 1} ^ n \left \| {{\hat x} _ j} \right \|}} ,
+
\over {BN + AN \cdot \sum _ {j = 1} ^ n \left | {{\hat x} _ j} \right |}} ,
 
$$
 
$$
  
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TeX rendered (inline)
 
TeX rendered (inline)
 
$ H _ {\mathfrak A / \mathfrak A _ 1} (A,\, B) $
 
$ H _ {\mathfrak A / \mathfrak A _ 1} (A,\, B) $
−
 
+
[https://www.encyclopediaofmath.org/legacyimages/a/a010/a010200/a01020061.png from this a01020061.png]
 
----------------------------------------------------------
 
----------------------------------------------------------
  
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TeX rendered (inline)
 
TeX rendered (inline)
 
$  {\operatorname\Spec} \, \mathbf Z [ 1 / n ,\, \xi _ n ] $
 
$  {\operatorname\Spec} \, \mathbf Z [ 1 / n ,\, \xi _ n ] $
−
 
+
[https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451074.png from this m06451074.png]
 
-----------------------------------------------------------------
 
-----------------------------------------------------------------
  
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\$\S  
 
\$\S  
 
\beta ( \mathcal A, \mathcal B ) =  \displaystyle\sup _ {C \in \mathcal A \otimes \mathcal B}
 
\beta ( \mathcal A, \mathcal B ) =  \displaystyle\sup _ {C \in \mathcal A \otimes \mathcal B}
−
\left \| {\mathrm P _ {\mathcal A \otimes \mathcal B} ( C ) -
+
\left | {\mathrm P _ {\mathcal A \otimes \mathcal B} ( C ) -
−
( \mathrm P _ \mathcal A - \mathrm P _ \mathcal B ) ( C )} \right \|
+
( \mathrm P _ \mathcal A - \mathrm P _ \mathcal B ) ( C )} \right |
 
=
 
=
 
\$\$
 
\$\$
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=
 
=
 
{1 \over 2}  \displaystyle\sup \sum _ {i = 1} ^ I \sum _ {j = 1} ^ J
 
{1 \over 2}  \displaystyle\sup \sum _ {i = 1} ^ I \sum _ {j = 1} ^ J
−
\left \| {\mathrm P ( A _ i \cap B _ j ) - \mathrm P ( A _ i ) \mathrm P ( B _ j )} \right \| ,
+
\left | {\mathrm P ( A _ i \cap B _ j ) - \mathrm P ( A _ i ) \mathrm P ( B _ j )} \right | ,
 
\$\$
 
\$\$
 
</pre>
 
</pre>
  
−
TeX Code displayed:
+
TeX Code displayed [https://www.encyclopediaofmath.org/legacyimages/a/a110/a110060/a11006022.png from this a11006022.png] and
 +
[https://www.encyclopediaofmath.org/legacyimages/a/a110/a110060/a11006023.png this a11006023.png]
 
$$  
 
$$  
 
\beta ( \mathcal A, \mathcal B ) =  \displaystyle\sup _ {C \in \mathcal A \otimes \mathcal B}
 
\beta ( \mathcal A, \mathcal B ) =  \displaystyle\sup _ {C \in \mathcal A \otimes \mathcal B}
−
\left \| {\mathrm P _ {\mathcal A \otimes \mathcal B} ( C ) -
+
\left | {\mathrm P _ {\mathcal A \otimes \mathcal B} ( C ) -
−
( \mathrm P _ \mathcal A - \mathrm P _ \mathcal B ) ( C )} \right \| =
+
( \mathrm P _ \mathcal A - \mathrm P _ \mathcal B ) ( C )} \right | =
 
$$
 
$$
  
 
$$
 
$$
−
= {1 \over 2}  \displaystyle\sup \sum _ {i = 1} ^ I \sum _ {j = 1} ^ J \left \| {\mathrm P ( A _ i \cap B _ j ) - \mathrm P ( A _ i ) \mathrm P ( B _ j )} \right \| ,
+
= {1 \over 2}  \displaystyle\sup \sum _ {i = 1} ^ I \sum _ {j = 1} ^ J \left | {\mathrm P ( A _ i \cap B _ j ) - \mathrm P ( A _ i ) \mathrm P ( B _ j )} \right | ,
 
$$
 
$$

Latest revision as of 22:16, 21 November 2019

Markup:
.BDIS {A15}
p \eql {fnnme max} from {1 \leq i \leq n} 
{{left \Lmi {b sub i \mns \sum from {j \eql 1} to n a sub {ij} {x hat} sub j} right \Rmi} 
over {BN \pls AN \cdt \sum from {j \eql 1} to n left \Lmi {{x hat} sub j} right \Rmi}} ,
.EDIS

TeX Code:
\$\$ \tag{A15}
p =  \displaystyle\max _ {1 <= i <= n} 
{{\left | {b _ i - \sum _ {j = 1} ^ n a _ {ij} {\hat x} _ j} \right |} 
\over {BN + AN \cdot \sum _ {j = 1} ^ n \left | {{\hat x} _ j} \right |}} ,
\$\$

TeX Code displayed from this a110010108.png $$ \tag{A15} p = \displaystyle\max _ {1 <= i <= n} {{\left | {b _ i - \sum _ {j = 1} ^ n a _ {ij} {\hat x} _ j} \right |} \over {BN + AN \cdot \sum _ {j = 1} ^ n \left | {{\hat x} _ j} \right |}} , $$ 

Markup:
\BMI  H sub {\FgA / \FgA sub 1} \LpaA,\sph B\Rpa \EMI

TeX Code:
\$ H _ {\mathfrak A / \mathfrak A _ 1} (A,\, B) \$

TeX rendered (inline) $ H _ {\mathfrak A / \mathfrak A _ 1} (A,\, B) $ from this a01020061.png 

Markup:
\BMI fnnme Spec \sph \FbZ \Lbk 1 / n ,\sph \Gxi sub n \Rbk\EMI

TeX Code:
\$  {\operatorname\Spec} \, \mathbf Z [ 1 / n ,\, \xi _ n ] \$

TeX rendered (inline) $ {\operatorname\Spec} \, \mathbf Z [ 1 / n ,\, \xi _ n ] $ from this m06451074.png 

Markup:
.BDIS
\Gba \Lpa \FfA, \FfB \Rpa \eql {fnnme sup} from {C \seo \FfA \otm \FfB}
left \Lmi {\FsP sub {\FfA \otm \FfB} \Lpa
 C \Rpa \mns \Lpa \FsP sub \FfA \tms \FsP sub \FfB \Rpa \Lpa C \Rpa} right \Rmi
.CDIS {@\eql@}
{1 over 2} {fnnme sup} \sum from {i \eql 1} to I \sum from {j \eql 1} to J
left \Lmi {\FsP \Lpa A sub i \cap B sub j \Rpa \mns
\FsP \Lpa A sub i \Rpa \FsP \Lpa B sub j \Rpa} right \Rmi ,
.EDIS

TeX Code:
\$\S 
\beta ( \mathcal A, \mathcal B ) =  \displaystyle\sup _ {C \in \mathcal A \otimes \mathcal B}
\left | {\mathrm P _ {\mathcal A \otimes \mathcal B} ( C ) -
( \mathrm P _ \mathcal A - \mathrm P _ \mathcal B ) ( C )} \right |
=
\$\$

\$\$ 
=
{1 \over 2}  \displaystyle\sup \sum _ {i = 1} ^ I \sum _ {j = 1} ^ J
\left | {\mathrm P ( A _ i \cap B _ j ) - \mathrm P ( A _ i ) \mathrm P ( B _ j )} \right | ,
\$\$

TeX Code displayed from this a11006022.png and this a11006023.png $$ \beta ( \mathcal A, \mathcal B ) = \displaystyle\sup _ {C \in \mathcal A \otimes \mathcal B} \left | {\mathrm P _ {\mathcal A \otimes \mathcal B} ( C ) - ( \mathrm P _ \mathcal A - \mathrm P _ \mathcal B ) ( C )} \right | = $$

$$ = {1 \over 2} \displaystyle\sup \sum _ {i = 1} ^ I \sum _ {j = 1} ^ J \left | {\mathrm P ( A _ i \cap B _ j ) - \mathrm P ( A _ i ) \mathrm P ( B _ j )} \right | , $$

How to Cite This Entry:
Ulf Rehmann/TEST7. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ulf_Rehmann/TEST7&oldid=44222
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