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The set of estimates of a given quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t0946001.png" /> from above and from below. An estimate from above is an inequality of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t0946002.png" />; an estimate from below is an inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t0946003.png" />, which has the opposite sense. The quantities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t0946004.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t0946005.png" /> with the aid of which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t0946006.png" /> is estimated usually have a simpler form or can be much more readily calculated than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t0946007.png" />.
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The set of estimates of a given quantity  $  a $
 +
from above and from below. An estimate from above is an inequality of the form $  a \leq  A _ {1} $;  
 +
an estimate from below is an inequality $  a \geq  A _ {0} $,  
 +
which has the opposite sense. The quantities $  A _ {0} $,  
 +
$  A _ {1} $
 +
with the aid of which $  a $
 +
is estimated usually have a simpler form or can be much more readily calculated than $  a $.
  
 
===Examples.===
 
===Examples.===
  
 +
1) Let  $  m $,
 +
$  M $
 +
be, respectively, the minimum and the maximum of a function  $  f $
 +
on an interval  $  [ \alpha , \beta ] $.
 +
The following two-sided estimate will then be valid for the integral  $  \int _  \alpha  ^  \beta  f ( x)  d x $:
  
1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t0946008.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t0946009.png" /> be, respectively, the minimum and the maximum of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460010.png" /> on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460011.png" />. The following two-sided estimate will then be valid for the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460012.png" />:
+
$$
 +
m ( \beta - \alpha ) \leq  \int\limits _  \alpha  ^  \beta 
 +
f ( x)  dx  \leq  M ( \beta - \alpha ) ;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460013.png" /></td> </tr></table>
+
here
  
here
+
$$
 +
A _ {0}  =  m ( \beta - \alpha ) ,\ \
 +
a  =  \int\limits _  \alpha  ^  \beta  f ( x)  dx ,\ \
 +
A _ {1}  =  M ( \beta - \alpha ) .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460014.png" /></td> </tr></table>
+
2) A two-sided estimate for the [[Lebesgue constants|Lebesgue constants]]  $  L _ {n} $
 +
for all  $  n = 0 , 1 \dots $
 +
is:
  
2) A two-sided estimate for the [[Lebesgue constants|Lebesgue constants]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460015.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460016.png" /> is:
+
$$
 +
0 . 9897 \dots  < L _ {n} -
 +
\frac{4}{\pi  ^ {2} }
 +
  \mathop{\rm ln}
 +
( 2n + 1 )  \leq  1 .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460017.png" /></td> </tr></table>
+
3) A two-sided estimate of eigenvalues. Consider the eigenvalue problem for a linear self-adjoint operator  $  T $,
 +
$  Tu = \lambda u $,
 +
in a Hilbert space  $  H $.
 +
One constructs an iterative process  $  Tf _ {n+} 1 = f _ {n} $,
 +
where  $  f _ {0} \neq 0 $.
 +
Since the operator  $  T $
 +
is self-adjoint, the scalar products  $  ( f _ {m} , f _ {k} ) $
 +
depend only on the sum  $  m+ k $
 +
of the indices. The numbers  $  a _ {n} = ( f _ {0} , f _ {n} )= ( f _ {m} , f _ {n-} m ) $
 +
are known as Schwartz constants, while the numbers  $  \mu _ {n+} 1 = a _ {n} / a _ {n+} 1 $
 +
are known as Rayleigh–Schwartz ratios. If the operator  $  T $
 +
is positive, the  $  \mu _ {n} $
 +
form a monotone non-decreasing convergent sequence.
  
3) A two-sided estimate of eigenvalues. Consider the eigenvalue problem for a linear self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460019.png" />, in a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460020.png" />. One constructs an iterative process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460022.png" />. Since the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460023.png" /> is self-adjoint, the scalar products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460024.png" /> depend only on the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460025.png" /> of the indices. The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460026.png" /> are known as Schwartz constants, while the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460027.png" /> are known as Rayleigh–Schwartz ratios. If the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460028.png" /> is positive, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460029.png" /> form a monotone non-decreasing convergent sequence.
+
If  $  \lambda _ {0} $
 +
is an eigenvalue of $  T $,
 +
a < \lambda _ {0} < b $,  
 +
a < \mu _ {2k} < b $,  
 +
and the interval  $  ( a , b ) $
 +
does not comprise other points of the spectrum of $  T $,  
 +
then
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460030.png" /> is an eigenvalue of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460033.png" />, and the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460034.png" /> does not comprise other points of the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460035.png" />, then
+
$$
 +
\mu _ {2k} -
 +
\frac{\rho  ^ {2} }{b - \mu _ {2k} }
 +
  \leq  \lambda _ {0}  \leq  \mu _ {2k} +
 +
\frac{\rho  ^ {2} }{\mu _ {2k} - a }
 +
,\ \
 +
\rho  ^ {2}  =
 +
\frac{\mu _ {2k-} 1 - \mu _ {2k} }{\mu _ {2k} }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460036.png" /></td> </tr></table>
+
$$
  
(Temple's theorem, [[#References|[3]]]). Under certain conditions the Rayleigh–Schwartz ratios converge to an eigenvalue of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460037.png" />.
+
(Temple's theorem, [[#References|[3]]]). Under certain conditions the Rayleigh–Schwartz ratios converge to an eigenvalue of $  T $.
  
Numerical methods for obtaining two-sided estimates (two-sided approximations) are known as two-sided methods [[#References|[4]]]. The method of constructing Rayleigh–Schwartz ratios just described is an example of a two-sided method. Some two-sided methods are based on the use of a pair of approximate formulas, with residual terms of opposite signs. For instance, let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460038.png" /> be interpolated at the points (interpolation nodes) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460039.png" /> by the Lagrange polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460040.png" /> with nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460041.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460042.png" /> be the Lagrange interpolation polynomial with nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460043.png" /> (cf. [[Lagrange interpolation formula|Lagrange interpolation formula]]). The following relations will then be valid for the residual terms:
+
Numerical methods for obtaining two-sided estimates (two-sided approximations) are known as two-sided methods [[#References|[4]]]. The method of constructing Rayleigh–Schwartz ratios just described is an example of a two-sided method. Some two-sided methods are based on the use of a pair of approximate formulas, with residual terms of opposite signs. For instance, let a function $  f $
 +
be interpolated at the points (interpolation nodes) $  x _ {0} < x _ {1} < \dots < x _ {n} $
 +
by the Lagrange polynomial $  L _ {0} ( x) $
 +
with nodes $  x _ {0} , x _ {1} \dots x _ {n-} 1 $,
 +
and let $  L _ {1} ( x) $
 +
be the Lagrange interpolation polynomial with nodes $  x _ {1} , x _ {2} \dots x _ {n} $(
 +
cf. [[Lagrange interpolation formula|Lagrange interpolation formula]]). The following relations will then be valid for the residual terms:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460044.png" /></td> </tr></table>
+
$$
 +
R _ {0} ( x)  = f ( x) - L _ {0} ( x)  =
 +
\frac{f ^ { ( n) } ( \xi _ {0)} }{n!}
 +
( x - x _ {0} ) \dots ( x - x _ {n-} 1 ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460045.png" /></td> </tr></table>
+
$$
 +
R _ {1} ( x)  = f ( x) - L _ {1} ( x)  =
 +
\frac{f ^ { ( n) } ( \xi _ {1} ) }{n!}
 +
( x - x _ {1} ) \dots ( x - x _ {n} ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460046.png" />. If the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460047.png" /> does not change sign in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460048.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460050.png" /> have opposite signs. The following two-sided estimate is valid:
+
where $  \xi _ {0} , \xi _ {1} \in [ x _ {0} , x _ {n} ] $.  
 +
If the derivative $  f ^ { ( n) } $
 +
does not change sign in $  [ x _ {0} , x _ {n} ] $,  
 +
then $  R _ {0} $
 +
and $  R _ {1} $
 +
have opposite signs. The following two-sided estimate is valid:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094600/t09460051.png" /></td> </tr></table>
+
$$
 +
\min ( L _ {0} ( x) , L _ {1} ( x) )  \leq  f ( x)
 +
\leq  \max ( L _ {0} ( x) , L _ {1} ( x) ) .
 +
$$
  
 
Two-sided methods for solving ordinary differential equations are now in a most advanced stage of development [[#References|[5]]]–[[#References|[9]]].
 
Two-sided methods for solving ordinary differential equations are now in a most advanced stage of development [[#References|[5]]]–[[#References|[9]]].
Line 40: Line 124:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.V. Galkin,  "Estimates for the Lebesgue constants"  ''Proc. Steklov Inst. Math.'' , '''1–4'''  (1971)  ''Trudy Mat. Inst. Steklov.'' , '''109'''  (1971)  pp. 3–5</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Collatz,  "Eigenwertaufgaben mit technischen Anwendungen" , Akad. Verlagsgesell. Geest u. Portig K.-D.  (1949)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Collatz,  "Functional analysis and numerical mathematics" , Acad. Press  (1966)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.S. Bakhvalov,  "Numerical methods: analysis, algebra, ordinary differential equations" , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.A. Volkov,  "Effective error estimates for difference solutions of boundary value problems in ordinary differential equations"  ''Proc. Steklov Inst. Math.'' , '''112'''  (1971)  pp. 143–155  ''Trudy Mat. Inst. Steklov.'' , '''112'''  (1971)  pp. 141–151</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.Ya. Remez,  ''Zap. Prirodn.-Tekhn. Viddilu Akad. Nauk UkrSSR'' , '''1'''  (1931)  pp. 1–38</TD></TR><TR><TD valign="top">[7a]</TD> <TD valign="top">  A.D. Gorbunov,  Yu.A. Shakhov,  "On the approximate solution of Cauchy's problem for ordinary differential equations to a number of correct figures"  ''USSR Comp. Math. Math. Phys.'' , '''3''' :  2  (1963)  pp. 316–335  ''Zh. Vychisl. Mat. i Mat. Fiz.'' , '''3''' :  2  (1963)  pp. 239–253</TD></TR><TR><TD valign="top">[7b]</TD> <TD valign="top">  A.D. Gorbunov,  Yu.A. Shakhov,  "On the approximate solution of Cauchy's problem for ordinary differential equations to a number of correct figures II"  ''USSR Comp. Math. Math. Phys.'' , '''4''' :  3  (1964)  pp. 37–47  ''Zh. Vychisl. Mat. i Mat. Fiz.'' , '''4''' :  3  (1964)  pp. 426–433</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  V.I. Devyatko,  "On a two-sided approximation for the numerical integration of ordinary differential equations"  ''USSR Comp. Math. Math. Phys.'' , '''3''' :  2  (1963)  pp. 336–350  ''Zh. Vychisl. Mat. i. Mat. Fiz.'' , '''3''' :  2  (1963)  pp. 254–265</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  N.P. Salikhov,  "Polar difference methods of solving Cauchy's problem for a system of ordinary differential equations"  ''USSR Comp. Math. Math. Phys.'' , '''2''' :  4  (1962)  pp. 535–553  ''Zh. Vychisl. Mat. i Mat. Fiz.'' , '''2''' :  4  (1962)  pp. 515–528</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.V. Galkin,  "Estimates for the Lebesgue constants"  ''Proc. Steklov Inst. Math.'' , '''1–4'''  (1971)  ''Trudy Mat. Inst. Steklov.'' , '''109'''  (1971)  pp. 3–5</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Collatz,  "Eigenwertaufgaben mit technischen Anwendungen" , Akad. Verlagsgesell. Geest u. Portig K.-D.  (1949)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Collatz,  "Functional analysis and numerical mathematics" , Acad. Press  (1966)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.S. Bakhvalov,  "Numerical methods: analysis, algebra, ordinary differential equations" , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.A. Volkov,  "Effective error estimates for difference solutions of boundary value problems in ordinary differential equations"  ''Proc. Steklov Inst. Math.'' , '''112'''  (1971)  pp. 143–155  ''Trudy Mat. Inst. Steklov.'' , '''112'''  (1971)  pp. 141–151</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.Ya. Remez,  ''Zap. Prirodn.-Tekhn. Viddilu Akad. Nauk UkrSSR'' , '''1'''  (1931)  pp. 1–38</TD></TR><TR><TD valign="top">[7a]</TD> <TD valign="top">  A.D. Gorbunov,  Yu.A. Shakhov,  "On the approximate solution of Cauchy's problem for ordinary differential equations to a number of correct figures"  ''USSR Comp. Math. Math. Phys.'' , '''3''' :  2  (1963)  pp. 316–335  ''Zh. Vychisl. Mat. i Mat. Fiz.'' , '''3''' :  2  (1963)  pp. 239–253</TD></TR><TR><TD valign="top">[7b]</TD> <TD valign="top">  A.D. Gorbunov,  Yu.A. Shakhov,  "On the approximate solution of Cauchy's problem for ordinary differential equations to a number of correct figures II"  ''USSR Comp. Math. Math. Phys.'' , '''4''' :  3  (1964)  pp. 37–47  ''Zh. Vychisl. Mat. i Mat. Fiz.'' , '''4''' :  3  (1964)  pp. 426–433</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  V.I. Devyatko,  "On a two-sided approximation for the numerical integration of ordinary differential equations"  ''USSR Comp. Math. Math. Phys.'' , '''3''' :  2  (1963)  pp. 336–350  ''Zh. Vychisl. Mat. i. Mat. Fiz.'' , '''3''' :  2  (1963)  pp. 254–265</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  N.P. Salikhov,  "Polar difference methods of solving Cauchy's problem for a system of ordinary differential equations"  ''USSR Comp. Math. Math. Phys.'' , '''2''' :  4  (1962)  pp. 535–553  ''Zh. Vychisl. Mat. i Mat. Fiz.'' , '''2''' :  4  (1962)  pp. 515–528</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Collatz,  "Numerical treatment of differential equations" , Springer  (1966)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Rektorys (ed.) , ''Survey of applicable mathematics'' , Iliffe  (1969)  pp. Sect. 32A</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T.J. Rivlin,  "An introduction to the approximation of functions" , Dover, reprint  (1969)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.E. Moore,  "Interval analysis" , Prentice-Hall  (1966)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D.M. Young,  R.T. Gregory,  "A survey of numerical mathematics" , Dover, reprint  (1988)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Collatz,  "Numerical treatment of differential equations" , Springer  (1966)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Rektorys (ed.) , ''Survey of applicable mathematics'' , Iliffe  (1969)  pp. Sect. 32A</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T.J. Rivlin,  "An introduction to the approximation of functions" , Dover, reprint  (1969)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.E. Moore,  "Interval analysis" , Prentice-Hall  (1966)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D.M. Young,  R.T. Gregory,  "A survey of numerical mathematics" , Dover, reprint  (1988)</TD></TR></table>

Latest revision as of 08:27, 6 June 2020


The set of estimates of a given quantity $ a $ from above and from below. An estimate from above is an inequality of the form $ a \leq A _ {1} $; an estimate from below is an inequality $ a \geq A _ {0} $, which has the opposite sense. The quantities $ A _ {0} $, $ A _ {1} $ with the aid of which $ a $ is estimated usually have a simpler form or can be much more readily calculated than $ a $.

Examples.

1) Let $ m $, $ M $ be, respectively, the minimum and the maximum of a function $ f $ on an interval $ [ \alpha , \beta ] $. The following two-sided estimate will then be valid for the integral $ \int _ \alpha ^ \beta f ( x) d x $:

$$ m ( \beta - \alpha ) \leq \int\limits _ \alpha ^ \beta f ( x) dx \leq M ( \beta - \alpha ) ; $$

here

$$ A _ {0} = m ( \beta - \alpha ) ,\ \ a = \int\limits _ \alpha ^ \beta f ( x) dx ,\ \ A _ {1} = M ( \beta - \alpha ) . $$

2) A two-sided estimate for the Lebesgue constants $ L _ {n} $ for all $ n = 0 , 1 \dots $ is:

$$ 0 . 9897 \dots < L _ {n} - \frac{4}{\pi ^ {2} } \mathop{\rm ln} ( 2n + 1 ) \leq 1 . $$

3) A two-sided estimate of eigenvalues. Consider the eigenvalue problem for a linear self-adjoint operator $ T $, $ Tu = \lambda u $, in a Hilbert space $ H $. One constructs an iterative process $ Tf _ {n+} 1 = f _ {n} $, where $ f _ {0} \neq 0 $. Since the operator $ T $ is self-adjoint, the scalar products $ ( f _ {m} , f _ {k} ) $ depend only on the sum $ m+ k $ of the indices. The numbers $ a _ {n} = ( f _ {0} , f _ {n} )= ( f _ {m} , f _ {n-} m ) $ are known as Schwartz constants, while the numbers $ \mu _ {n+} 1 = a _ {n} / a _ {n+} 1 $ are known as Rayleigh–Schwartz ratios. If the operator $ T $ is positive, the $ \mu _ {n} $ form a monotone non-decreasing convergent sequence.

If $ \lambda _ {0} $ is an eigenvalue of $ T $, $ a < \lambda _ {0} < b $, $ a < \mu _ {2k} < b $, and the interval $ ( a , b ) $ does not comprise other points of the spectrum of $ T $, then

$$ \mu _ {2k} - \frac{\rho ^ {2} }{b - \mu _ {2k} } \leq \lambda _ {0} \leq \mu _ {2k} + \frac{\rho ^ {2} }{\mu _ {2k} - a } ,\ \ \rho ^ {2} = \frac{\mu _ {2k-} 1 - \mu _ {2k} }{\mu _ {2k} } $$

(Temple's theorem, [3]). Under certain conditions the Rayleigh–Schwartz ratios converge to an eigenvalue of $ T $.

Numerical methods for obtaining two-sided estimates (two-sided approximations) are known as two-sided methods [4]. The method of constructing Rayleigh–Schwartz ratios just described is an example of a two-sided method. Some two-sided methods are based on the use of a pair of approximate formulas, with residual terms of opposite signs. For instance, let a function $ f $ be interpolated at the points (interpolation nodes) $ x _ {0} < x _ {1} < \dots < x _ {n} $ by the Lagrange polynomial $ L _ {0} ( x) $ with nodes $ x _ {0} , x _ {1} \dots x _ {n-} 1 $, and let $ L _ {1} ( x) $ be the Lagrange interpolation polynomial with nodes $ x _ {1} , x _ {2} \dots x _ {n} $( cf. Lagrange interpolation formula). The following relations will then be valid for the residual terms:

$$ R _ {0} ( x) = f ( x) - L _ {0} ( x) = \frac{f ^ { ( n) } ( \xi _ {0)} }{n!} ( x - x _ {0} ) \dots ( x - x _ {n-} 1 ) , $$

$$ R _ {1} ( x) = f ( x) - L _ {1} ( x) = \frac{f ^ { ( n) } ( \xi _ {1} ) }{n!} ( x - x _ {1} ) \dots ( x - x _ {n} ) , $$

where $ \xi _ {0} , \xi _ {1} \in [ x _ {0} , x _ {n} ] $. If the derivative $ f ^ { ( n) } $ does not change sign in $ [ x _ {0} , x _ {n} ] $, then $ R _ {0} $ and $ R _ {1} $ have opposite signs. The following two-sided estimate is valid:

$$ \min ( L _ {0} ( x) , L _ {1} ( x) ) \leq f ( x) \leq \max ( L _ {0} ( x) , L _ {1} ( x) ) . $$

Two-sided methods for solving ordinary differential equations are now in a most advanced stage of development [5][9].

Two-sided methods make it possible to identify the boundaries of the domain in which the solution of the problem is known to be contained. This necessarily entails a more complicated algorithm, and a further complication of the algorithm must be accepted if the method is used in practical computations, in view of the rounding-off errors involved. Two-sided methods are used mainly in cases where a guaranteed estimate of the error is required.

References

[1] P.V. Galkin, "Estimates for the Lebesgue constants" Proc. Steklov Inst. Math. , 1–4 (1971) Trudy Mat. Inst. Steklov. , 109 (1971) pp. 3–5
[2] L. Collatz, "Eigenwertaufgaben mit technischen Anwendungen" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1949)
[3] L. Collatz, "Functional analysis and numerical mathematics" , Acad. Press (1966) (Translated from German)
[4] N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)
[5] E.A. Volkov, "Effective error estimates for difference solutions of boundary value problems in ordinary differential equations" Proc. Steklov Inst. Math. , 112 (1971) pp. 143–155 Trudy Mat. Inst. Steklov. , 112 (1971) pp. 141–151
[6] E.Ya. Remez, Zap. Prirodn.-Tekhn. Viddilu Akad. Nauk UkrSSR , 1 (1931) pp. 1–38
[7a] A.D. Gorbunov, Yu.A. Shakhov, "On the approximate solution of Cauchy's problem for ordinary differential equations to a number of correct figures" USSR Comp. Math. Math. Phys. , 3 : 2 (1963) pp. 316–335 Zh. Vychisl. Mat. i Mat. Fiz. , 3 : 2 (1963) pp. 239–253
[7b] A.D. Gorbunov, Yu.A. Shakhov, "On the approximate solution of Cauchy's problem for ordinary differential equations to a number of correct figures II" USSR Comp. Math. Math. Phys. , 4 : 3 (1964) pp. 37–47 Zh. Vychisl. Mat. i Mat. Fiz. , 4 : 3 (1964) pp. 426–433
[8] V.I. Devyatko, "On a two-sided approximation for the numerical integration of ordinary differential equations" USSR Comp. Math. Math. Phys. , 3 : 2 (1963) pp. 336–350 Zh. Vychisl. Mat. i. Mat. Fiz. , 3 : 2 (1963) pp. 254–265
[9] N.P. Salikhov, "Polar difference methods of solving Cauchy's problem for a system of ordinary differential equations" USSR Comp. Math. Math. Phys. , 2 : 4 (1962) pp. 535–553 Zh. Vychisl. Mat. i Mat. Fiz. , 2 : 4 (1962) pp. 515–528

Comments

References

[a1] L. Collatz, "Numerical treatment of differential equations" , Springer (1966) (Translated from German)
[a2] K. Rektorys (ed.) , Survey of applicable mathematics , Iliffe (1969) pp. Sect. 32A
[a3] T.J. Rivlin, "An introduction to the approximation of functions" , Dover, reprint (1969)
[a4] R.E. Moore, "Interval analysis" , Prentice-Hall (1966)
[a5] D.M. Young, R.T. Gregory, "A survey of numerical mathematics" , Dover, reprint (1988)
How to Cite This Entry:
Two-sided estimate. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Two-sided_estimate&oldid=49056
This article was adapted from an original article by V.V. Pospelov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article