Two-sided estimate

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

Examples.

1) Let , be, respectively, the minimum and the maximum of a function on an interval . The following two-sided estimate will then be valid for the integral :

here

2) A two-sided estimate for the Lebesgue constants for all is:

3) A two-sided estimate of eigenvalues. Consider the eigenvalue problem for a linear self-adjoint operator , , in a Hilbert space . One constructs an iterative process , where . Since the operator is self-adjoint, the scalar products depend only on the sum of the indices. The numbers are known as Schwartz constants, while the numbers are known as Rayleigh–Schwartz ratios. If the operator is positive, the form a monotone non-decreasing convergent sequence.

If is an eigenvalue of , , , and the interval does not comprise other points of the spectrum of , then

(Temple's theorem, [3]). Under certain conditions the Rayleigh–Schwartz ratios converge to an eigenvalue of .

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 be interpolated at the points (interpolation nodes) by the Lagrange polynomial with nodes , and let be the Lagrange interpolation polynomial with nodes (cf. Lagrange interpolation formula). The following relations will then be valid for the residual terms:

where . If the derivative does not change sign in , then and have opposite signs. The following two-sided estimate is valid:

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

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)