# Two-sided estimate

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