Double inequalities for quadrature formula of Gauss Type

Marius Heljiu\(^\S \)

September 13, 2011

\(^\S \)Department of Mathematics, University of Petroşani, Romania,
e-mail: mheljiu@upet.ro.

Double inequalities for the remainder term of the Gauss quadrature formula are given. These inequalities are sharp. It also will consider particular cases for n = 1, 2.

MSC. 26D10, 65D30

Keywords. Quadrature rule, double integral inequalities, numerical integration.

1 Introduction

In this work we will consider Gauss’s quadrature rule(see [ 2 ] )

\begin{equation} \label{1} \int _a^b f(x){\rm d}x= \sum _{i=1}^{n} C_if(x_i)+\ R[f] \end{equation}
1.1

where the remainder term for \(f:[a,b]\to \mathbb {R}\), \(f\in C^{2n}[a,b]\), has the representation

\begin{equation} \label{2} R[f]=\int _a^b \varphi (x)f^{(2n)}(x){\rm d}x. \end{equation}
1.2

The nodes \(x_i\), \(i=\overline{1,n}\) from quadrature formula (1.1) are given by the relation

\begin{equation} \label{3} x_i=\displaystyle \tfrac {a+b}{2}+\displaystyle \tfrac {b-a}{2}\xi _i \end{equation}
1.3

where \(\xi _i \) is replaced by the roots \(\xi _1, \xi _2,..., \xi _n\) of Legendre polynomial

\begin{equation} \label{4} X_n(\xi )=\displaystyle \tfrac {1}{2^n n!}\displaystyle \tfrac {d^n(1-\xi ^2)^n}{d\xi ^n}. \end{equation}
1.4

and the coefficients \(C_i\), \(i=\overline{1,n}\), from Gauss’s formula shall be determined by putting the conditions that the quadrature formula must be accurate for any polynomial of degree \(2n-1\).

The function \(\varphi (x)\), from remaining term expression, being symmetric to the line \(x=\tfrac {a+b}{2}\), is sufficient to provide the equation \(y=\varphi (x)\) in the intervals \([a,x_1], [x_1,x_2],[x_2,x_3]...\) to middle of interval (a,b). So the function \(\varphi (x)\) coincide on all the the intervals \([a,x_1], [x_1,x_2],[x_2,x_3]...,\) with the relations:

\begin{equation} \begin{array}{l} \varphi _1(x)=\displaystyle \int _a^x \tfrac {(x-s)^{2n-1}}{(2n-1)!}{\rm d}s\\ \varphi _2(x)=\displaystyle \int _a^x \tfrac {(x-s)^{2n-1}}{(2n-1)!}{\rm d}s-C_1\tfrac {(x-x_1)^{2n-1}}{(2n-1)!}\\ \varphi _3(x)=\displaystyle \int _a^x \tfrac {(x-s)^{2n-1}}{(2n-1)!}{\rm d}s-C_1\tfrac {(x-x_1)^{2n-1}}{(2n-1)!}-C_2\tfrac {(x-x_2)^{2n-1}}{(2n-1)!}\\ \cdots \end{array} \end{equation}
1.5

In the next paragraph we will establish double integral inequalities for the remainder term of the Gauss quadrature formula. We will also establish conditions under which the results are sharp. At the end of the paper we will analyze the particular cases of Gauss formula with one respectively two nodes.

2 Main results

In this section we prove the following theorems:

Theorem 2.1

If \(f\in C^{2n}[a,b]\) then

\begin{align} \label{5} & M(\gamma -S_{2n-1})(b-a)-\displaystyle \tfrac {\gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{2n+1}}{(2n+1)}\leq \\ & \le \sum _{i=1}^{n}C_if(x_i)-\int _a^b f(x){\rm d}x\leq \nonumber \\ & \le M(\Gamma -S_{2n-1})(b-a)-\displaystyle \tfrac {\Gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{2n+1}}{(2n+1)}\nonumber \end{align}

where \(\gamma ,\Gamma \in \mathbb {R}\), \(\gamma \le f^{(2n)}(x)\le \Gamma \), for all \(x\in [a,b]\) and \(S_{2n-1}=\displaystyle \tfrac {f^{(2n-1)}(b)-f^{(2n-1)}(a)}{b-a}\). Moreover,

\begin{equation} \label{6} \gamma =\min _{x\in [a,b]}f^{(2n)}(x),\quad \Gamma =\max _{x\in [a,b]}f^{(2n)}(x) \end{equation}
2.2

the inequalities (5) are sharp.

Proof â–¼
Using (1.1) and (1.2), we obtain

\begin{equation} \label{7} \int _a^b \varphi (x)f^{(2n)}(x){\rm d}x=\int _a^b f(x){\rm d}x-\sum _{i=1}^{n}C_if(x_i). \end{equation}
2.3

But it is well-known from [3] pp 283 that

\begin{equation} \label{8} \int _a^b \varphi (x){\rm d}x=\tfrac {(n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{(2n+1)}}{(2n+1)}. \end{equation}
2.4

From (2.3) and (2.4) it follows

\begin{align} \label{9} \int _a^b [f^{(2n)}(x)-\gamma ]\varphi (x){\rm d}x & =\int _a^b f(x){\rm d}x-\sum _{i=1}^{n}C_if(x_i)\\ & \quad -\displaystyle \tfrac {\gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{(2n+1)}}{(2n+1)}\nonumber \end{align}

and

\begin{align} \label{10} \int _a^b [\Gamma -f^{(2n)}(x)]\varphi (x){\rm d}x & =-\int _a^b f(x){\rm d}x+\sum _{i=1}^{n}C_if(x_i)\\ & \quad +\displaystyle \tfrac {\Gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{(2n+1)}}{(2n+1)}\nonumber \end{align}

On the other hand, we have

\begin{equation} \label{11} \int _a^b [f^{(2n)}(x)-\gamma ]\varphi (x){\rm d}x\le \max _{x\in [a,b]}|\varphi (x)|\int _a^b |f^{(2n)}(x)-\gamma |{\rm d}x. \end{equation}
2.7

By calculating the right member of the inequality(2.7), we get

\begin{align} \label{12} \int _a^b |f^{(2n)}(x)-\gamma |{\rm d}x & =\int _a^b (f^{(2n)}(x)-\gamma ){\rm d}x\\ & =f^{(2n-1)}(b)-f^{(2n-1)}(a)-\gamma (b-a)\nonumber \\ & =(S_{2n-1}-\gamma )(b-a).\nonumber \end{align}

From (2.7), (2.8) and noting

\begin{equation} \label{13} \max _{x\in [a,b]}|\varphi (x)|=M, \end{equation}
2.9

we obtain

\begin{align} \label{14} \int _a^b [f^{(2n)}(x)-\gamma ]\varphi (x){\rm d}x & \leq M(S_{2n-1}-\gamma )(b-a). \end{align}

From (2.5), by using previous relation, result

\begin{equation} \label{144} \int _a^b f(x){\rm d}x-\displaystyle \sum _{i=1}^{n}C_if(x_i)\\ \le \displaystyle M(S_{2n-1}-\gamma )(b-a)+\displaystyle \tfrac {\gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{(2n+1)}}{(2n+1)} \end{equation}
2.11

In the same way we have

\begin{equation} \label{15} \int _a^b [\Gamma -f^{(2n)}(x)]\varphi (x){\rm d}x \le \max _{x\in [a,b]}|\varphi (x)|\int _a^b |\Gamma -f^{(2n)}(x)|{\rm d}x \end{equation}
2.12

and

\begin{align} \label{16} \int _a^b |\Gamma -f^{(2n)}(x)|{\rm d}x & =\int _a^b (\Gamma -f^{(2n)}(x)){\rm d}x\\ & =\Gamma (b-a)-f^{(2n-1)}(b)+f^{(2n-1)}(a)\nonumber \\ & =(\Gamma -S_{2n-1})(b-a)\nonumber \end{align}

Using (2.6), (2.9), (2.12) and (2.13) we obtain the inequality

\begin{align} \label{17} & -\int _a^b f(x){\rm d}x+\displaystyle \sum _{i=1}^{n}C_if(x_i)\\ & \leq M(\Gamma -S_{2n-1})(b-a)-\displaystyle \tfrac {\Gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{(2n+1)}}{(2n+1)}.\nonumber \end{align}

The inequalities (2.1) follow from the inequalities (2.11) and (2.14).

To prove the second part of the theorem we consider the function \(f(x)=(x-a)^{2n}\). I have \( f^{(2n)}(x)=(2n)!, \gamma =\Gamma =(2n)!\) and \(S_{2n-1}=(2n)!\). It is easy to show that all the three members of the double inequality (2.1) are equal. This completes the proof.

Proof â–¼
The following theorem offer us \(R[f]\) and analogous inequality with the one given by Theorem 2.1.
Theorem 2.2

Under the assumptions of Theorem 2.1, we have

\begin{align} \label{18} & M(\gamma -S_{2n-1})(b-a)+\displaystyle \tfrac {\gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{2n+1}}{(2n+1)}\leq \\ & \le \int _a^b f(x){\rm d}x-\sum _{i=1}^{n}C_if(x_i)\leq \nonumber \\ & \le M(\Gamma -S_{2n-1})(b-a)+\displaystyle \tfrac {\Gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{2n+1}}{(2n+1)}\nonumber \end{align}

Moreover,

\[ \gamma =\min _{x\in [a,b]}f^{(2n)}(x),\quad \Gamma =\max _{x\in [a,b]}f^{(2n)}(x) \]

the inequalities (18) are sharp.

Proof â–¼
By using the relations (2.3), (2.5), (2.7) and (2.8) it follows

\begin{align} \label{19} & -\int _a^b f(x){\rm d}x+\displaystyle \sum _{i=1}^{n}C_if(x_i)\\ & \le -M(\gamma -S_{2n-1})(b-a)-\displaystyle \tfrac {\gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{2n+1}}{(2n+1)}\nonumber \end{align}

Analogous by using the relations (2.3), (2.6), (2.12) and (2.13) we obtain

\begin{align} \label{20} & \int _a^b f(x){\rm d}x-\displaystyle \sum _{i=1}^{n}C_if(x_i)\\ & \le M(\Gamma -S_{2n-1})(b-a)+\displaystyle \tfrac {\Gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{(2n+1)}}{(2n+1)}\nonumber \end{align}

From the relations (2.16) and (2.17) result the inequalities (2.15). To prove that the double inequalities (2.15) are exact we follow the steps of the proof from Theorem 2.1.

Proof â–¼

Theorem 2.3 gives us the inequalities which do not depend on \(S_3\).

Theorem 2.3

Under the assumptions of Theorem 1, we have

\begin{align} \label{21} & \displaystyle \tfrac {1}{2}M(\gamma -\Gamma )(b-a)-\displaystyle \tfrac {\gamma +\Gamma }{2}\displaystyle \tfrac {(n!)^{4}}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{2n+1}}{2n+1}\\ & \le \sum _{i=1}^{n}C_if(x_i)-\int _a^b f(x){\rm d}x\nonumber \\ & \le \displaystyle \tfrac {1}{2}M(\Gamma -\gamma )(b-a)-\displaystyle \tfrac {\gamma +\Gamma }{2}\displaystyle \tfrac {(n!)^{4}}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{2n+1}}{2n+1}.\nonumber \end{align}

If

\[ \gamma =\min _{x\in [a,b]}f^{(2n)}(x),\quad \Gamma =\max _{x\in [a,b]}f^{(2n)}(x) \]

then the inequalities (21) are sharp.

Proof â–¼
Multiplying the inequality (2.15) with \((-1)\) we obtain

\begin{align} \label{22} & -M(\Gamma -S_{2n-1})(b-a)-\displaystyle \tfrac {\Gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{2n+1}}{(2n+1)}\leq \\ & \le -\int _a^b f(x){\rm d}x+\sum _{i=1}^{n}C_if(x_i)\leq \nonumber \\ & \le -M(\gamma -S_{2n-1})(b-a)-\displaystyle \tfrac {\gamma (n!)^4}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{(2n+1)}}{(2n+1)}\nonumber \end{align}

From the inequalities (2.1) and (2.19) we have

\begin{align} \label{23} & \displaystyle M(\gamma -\Gamma )(b-a)-\displaystyle (\gamma +\Gamma )\displaystyle \tfrac {(n!)^{4}}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{2n+1}}{2n+1}\\ & \le 2\left[ \sum _{i=1}^{n}C_if(x_i)-\int _a^b f(x){\rm d}x \right]\nonumber \\ & \le \displaystyle M(\Gamma -\gamma )(b-a)-\displaystyle (\gamma +\Gamma )\displaystyle \tfrac {(n!)^{4}}{[(2n)!]^3}\displaystyle \tfrac {(b-a)^{2n+1}}{2n+1}.\nonumber \end{align}

Multiplying (2.20) with \(\tfrac {1}{2}\) results the double inequality (2.18). Considering the function \(f(x)=(x-a)^{2n}\) we will show that the inequality is sharp. This completes the proof.

Proof â–¼

3 Particular cases

Using the results of the before paragraph, we will present the following double inequalities for the rest of Gauss quadrature formulas in the cases n=1, n=2.

The Gauss’s formula with a single node (n = 1), also called the mid-point, has the form

\begin{equation} \int _a^b f(x){\rm d}x=(b-a)\left[f\left(\tfrac {a+b}{2}\right)\right] +R[f], \end{equation}
3.1

where

\begin{equation} R[f]=\int _a^b \varphi (x)f''(x){\rm d}x, \end{equation}
3.2

The function \(\varphi (x)\) is given by the relation

\begin{equation} \label{24} \varphi (x) =\left\{ \begin{array}{lll} \displaystyle \tfrac {(x-a)^2}{2}, & \mbox{if} & x\in \left[a,\displaystyle \tfrac {a+b}{2}\right]\medskip \\ \displaystyle \tfrac {(b-x)^2}{2}, & \mbox{if} & x\in \left[\displaystyle \tfrac {a+b}{2},b\right]. \end{array}\right. \end{equation}
3.3

Applying Theorem 2.1 for n=1, we get a result established by Ujevic [3].

Theorem 3.1

Let \(f:[a,b]\to \mathbb {R}\), \(f\in C^2(a,b)\), then we have the inequality

\begin{align} \label{25} & \displaystyle \tfrac {2\gamma -3S_1}{24}(b-a)^3\\ \nonumber & \le (b-a)f\left(\displaystyle \tfrac {a+b}{2}\right)-\int _a^b f(x){\rm d}x\\ \nonumber & \le \displaystyle \tfrac {2\Gamma -3S_1}{24}(b-a)^3 \end{align}

where \(\gamma ,\ \Gamma \in \mathbb {R},\) \(\gamma \le f^{(2)}(x)\le \Gamma \), \(\forall \ x\in [a,b]\) and

\[ S_1=\displaystyle \tfrac {f’(b)-f’(a)}{b-a}. \]

If \(\gamma =\min \limits _{x\in [a,b]}f^{(2)}(x),\) \(\Gamma =\max \limits _{x\in [a,b]}f^{(2)}(x)\) then inequalities (25) are sharp.

Similar with Theorem 2.2 and 2.3, the Gauss’s formula with a single node are given in the next theorems :

Theorem 3.2

Under the assumptions of Theorem 3.1 we have

\begin{align} \label{26} & \displaystyle \tfrac {4\gamma -3S_1}{24}(b-a)^3\le \\ \nonumber & \le \int _a^b f(x){\rm d}x-(b-a)f\left(\displaystyle \tfrac {a+b}{2}\right)\\ \nonumber & \le \displaystyle \tfrac {4\Gamma -3S_1}{24}(b-a)^3. \end{align}

If \(\gamma =\min \limits _{x\in [a,b]}f^{(2)}(x)\) and \(\Gamma =\max \limits _{x\in [a,b]}f^{(2)}(x)\) then inequalities (26) are sharp.

Theorem 3.3

Under the assumptions of Theorem 3.1 we have

\begin{align} \label{27} & \displaystyle \tfrac {(b-a)^3}{24}(\gamma -2\Gamma )\le \\ \nonumber & \le (b-a)f\left(\displaystyle \tfrac {a+b}{2}\right)-\int _a^b f(x){\rm d}x\le \\ \nonumber & \le \displaystyle \tfrac {(b-a)^3}{24}(\Gamma -2\gamma ). \end{align}

If \(\gamma =\min \limits _{x\in [a,b]}f^{(2)}(x)\) and \(\Gamma =\max \limits _{x\in [a,b]}f^{(2)}(x)\) then inequalities (27) are sharp.

Similar results, we obtain in the case of Gauss formula with two nodes. The results was established by the author in the paper [1].

Let \(f:[a,b]\to \mathbb {R}\), \(f\in C^4[a,b]\) and \(x_1,x_2\in [a,b]\) so that \(x_1=\displaystyle \tfrac {a+b}{2}-\displaystyle \tfrac {b-a}{2}\cdot \xi \), \(x_2=\displaystyle \tfrac {a+b}{2}+\displaystyle \tfrac {b-a}{2}\cdot \xi \), where \(\xi =\displaystyle \tfrac {1}{\sqrt3}=0,57735027\dots .\)

Gauss’s quadrature formula with two nodes has the following form

\begin{equation} \label{28} \int _a^b f(x){\rm d}x=\displaystyle \tfrac {b-a}{2}[f(x_1)+f(x_2)]+R[f]. \end{equation}
3.7

The error \(R[f]\) from the formula (3.7) is given by

\begin{equation} \label{29} R[f]=\int _a^b \varphi (x)f^{(4)}(x){\rm d}x, \end{equation}
3.8

where the function \(\varphi \) has the form

\begin{equation} \label{30} \varphi (x)= \left\{ \begin{array}{lll} \displaystyle \tfrac {(x-a)^4}{4!}, & \mbox{if} & x\in [a,x_1]\\ \displaystyle \tfrac {(x-a)^4}{4!}-\displaystyle \tfrac {b-a}{2}\displaystyle \tfrac {(x-x_1)^3}{3!}, & \mbox{if} & x\in ]x_1,x_2[\\ \displaystyle \tfrac {(b-x)^4}{4!}, & \mbox{if} & x\in [x_2,b] \end{array}\right.\end{equation}
3.9

In this case we obtain the following theorems:

Theorem 3.4

If \(f\in C^4[a,b]\) then

\begin{align} \label{31} & \displaystyle \tfrac {1}{17280}(b-a)^5(41\gamma -45S_3+180\xi ^3 S_3-180\xi ^3 \gamma )\\ & \le \displaystyle \tfrac {b-a}{2}[f(x_1)+f(x_2)]-\int _a^b f(x){\rm d}x\nonumber \\ & \le \displaystyle \tfrac {1}{17280}(b-a)^5 (41\Gamma -45S_3+180\xi ^3 S_3-180\xi ^3 \Gamma )\nonumber \end{align}

where \(\gamma ,\Gamma \in \mathbb {R}\), \(\gamma \le f^{(4)}(x)\le \Gamma \), for all \(x\in [a,b]\) and \(S_3=\displaystyle \tfrac {f”’(b)-f”’(a)}{b-a}\). Moreover,

\begin{equation} \label{32} \gamma =\min _{x\in [a,b]}f^{(4)}(x),\quad \Gamma =\max _{x\in [a,b]}f^{(4)}(x) \end{equation}
3.11

the inequalities (31) are sharp.

Theorem 3.5

Under the assumptions of Theorem 3.4 we have

\begin{align} \label{33} & \displaystyle \tfrac {1}{17280}(b-a)^5(49\gamma -45S_3+180\xi ^3 S_3-180\xi ^3 \gamma )\le \\ \nonumber & \le \int _a^b f(x){\rm d}x-\displaystyle \tfrac {b-a}{2}[f(x_1)+f(x_2)]\le \\ \nonumber & \le \displaystyle \tfrac {1}{17280}(b-a)^5(49\Gamma -45S_3+180\xi ^3 S_3-180\xi ^3 \Gamma )\\ \nonumber \end{align}

If

\[ \gamma =\min \limits _{x\in [a,b]}f^{(4)}(x),\quad \Gamma =\max \limits _{x\in [a,b]}f^{(4)}(x) \]

the inequalities (33) are sharp.

Theorem 3.6

Under the assumptions of Theorem 3.4 we have

\begin{align} \label{34} & \displaystyle \tfrac {1}{34560}(b-a)^5(41\gamma -49\Gamma -180\xi ^3\gamma +180\xi ^3\Gamma )\le \\ \nonumber & \le \displaystyle \tfrac {b-a}{2}[f(x_1)+f(x_2)]-\int _a^b f(x){\rm d}x\le \\ \nonumber & \le \displaystyle \tfrac {1}{34560}(b-a)^5(41\Gamma -49\gamma +180\xi ^3\gamma -180\xi ^3\Gamma ) \nonumber \end{align}

If

\[ \gamma =\min \limits _{x\in [a,b]}f^{(4)}(x),\quad \Gamma =\max \limits _{x\in [a,b]}f^{(4)}(x) \]

the inequalities (34) are sharp.

Bibliography

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M. Heljiu, Double Inequalities for quadrature formula of Gauss type with two nodes, Rev. Anal. Numér. Théor. Approx., XXXVII, no. 1, pp. 53–57, 2008. \includegraphics[scale=0.1]{ext-link.png}

2

D. V. Ionescu, Numerical Integration, Ed. Tehnică, Bucureşti, 1957 (in Romanian).

3

N. Ujevic, Some Double Integral Inequalities and Applications, Acta Math. Univ. Comenianae, 71(2), pp. 187, 2002.

4

N. Ujevic, Double Integral Inequalities of Simpson Type and Applications, J. Appl. Math. & Computing, 14, Nos. 1–2, pp.  213–223, 2004.