View source for Two-sided estimate

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" />.

===Examples.===


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" />:

<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

<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]] <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:

<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 <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 <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

<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" />.

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:

<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>

<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>

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:

<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>

Two-sided methods for solving ordinary differential equations are now in a most advanced stage of development [[#References|[5]]]–[[#References|[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====
<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====


====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>