Numerical integration using local Taylor expansions in nodes

Abstract

In this paper we introduce a new formula for approximating the integration based on locally Taylor expansions in each one of the nodes and reduce the results in respect of difference equation. Finally some examples are given to illustrate the application of the formulae for comparison the results with other related formulae.

Introduction

It is known that in the general form of Gauss quadrature rules the nodes x_i and the corresponding weights w_i are usually known [3], or are predetermined [1], [2]. In the next section we are going to introduce the integration formula which in each node the Taylor expansion is used in the related nodes. In this formula we try to derive explicit formulae in numerical integration for nodes and coefficient of integration. These new integration formulae can be introduced by the following:

$${\int }_a^{b}w(x)f(x)\mathrm{d}x\simeq \sum _{i=1}^n{a}_i\left(\sum _{j=0}^{n+1}\frac{f^{(j)}(x_i)}{j!}\right)\text{.}$$

Defining the following operator changes the above formula to

$$D^n[f(x_i)]=\sum _{j=0}^{n+1}\frac{f^{(j)}(x_i)}{j!}\text{.}$$

Eq. (1) can be written as

$${\int }_a^{b}w(x)f(x)\mathrm{d}x\simeq \sum _{i=1}^n{a}_i{D}^n[f(x_i)]\text{.}$$

For example when n = 2, Eq. (1) will be changed to

$${\int }_a^{b}w(x)f(x)\mathrm{d}x\simeq a_1\left(f(x_1)+\frac{f^{\prime }(x_1)}{1!}+\frac{f^{″}(x_1)}{2!}+\frac{f^{‴}(x_1)}{3!}\right)+a_2\left(f(x_2)+\frac{f^{\prime }(x_2)}{1!}+\frac{f^{″}(x_2)}{2!}+\frac{f^{‴}(x_2)}{3!}\right)\text{.}$$

According to the concept of precision degree in integration given in (1) we will have

$${\int }_a^{b}w(x)x^k\mathrm{d}x=\sum _{i=1}^n{a}_i\left(\sum _{j=0}^k\frac{(k-1)(k-2)\cdots (k-j)}{j!}{x}_i^k-j\right)\text{,}k=0\text{,}1\text{,}\dots \text{,}2n-1$$

by defining

$${\mu }_k={\int }_a^{b}w(x)x^k\mathrm{d}x\text{,}k=0\text{,}1\text{,}\dots \text{,}2n-1\text{.}$$

We will get the non-linear system in (7):

$$\left\{\begin{array}{c}\sum _{i=1}^n{a}_i={\mu }_0\text{,}\hfill \\ \sum _{i=1}^n{a}_i(x_i+1)={\mu }_1\text{,}\hfill \\ \sum _{i=1}^n{a}_i(x_i^2+2x_i+1)={\mu }_2\text{,}\hfill \\ ⋮\hfill \\ \sum _{i=1}^n{a}_i\left(x_i^k+\left(\begin{array}{c}k\hfill \\ k-1\hfill \end{array}\right)x_i^{k-1}+\cdots +\left(\begin{array}{c}k\hfill \\ 2\hfill \end{array}\right)x_i+\left(\begin{array}{c}k\hfill \\ 0\hfill \end{array}\right)\right)={\mu }_k\text{,}\hfill \\ ⋮\hfill \\ \sum _{i=1}^n{a}_i\left(x_i^{2n-1}+\left(\begin{array}{c}2n-1\hfill \\ 2n-2\hfill \end{array}\right)x_i^{2n-2}+\cdots +\left(\begin{array}{c}2n-1\hfill \\ 2\hfill \end{array}\right)x_i^2+\left(\begin{array}{c}2n-1\hfill \\ 1\hfill \end{array}\right)x_i+\left(\begin{array}{c}2n-1\hfill \\ 0\hfill \end{array}\right)\right)={\mu }_{2n-1}\hfill \end{array}\right.$$

but according to the above relations this non-linear system gives the system in (8):

$$\left\{\begin{array}{c}\sum _{i=1}^n{a}_i={\mu }_0\text{,}\hfill \\ \sum _{i=1}^n{a}_i{x}_i={\mu }_1-{\mu }_0\text{,}\hfill \\ \sum _{i=1}^n{a}_i{x}_i^2={\mu }_2-2{\mu }_1+{\mu }_0\text{,}\hfill \\ \sum _{i=1}^n{a}_i{x}_i^3={\mu }_3-3{\mu }_2+3{\mu }_1-{\mu }_0\text{,}\hfill \\ ⋮\hfill \\ \sum _{i=1}^n{a}_i{x}_i^k={\mu }_k+\sum _{i=1}^k(-1{)}^i\left(\begin{array}{c}k\hfill \\ i\hfill \end{array}\right){\mu }_{k-i}\text{,}\hfill \\ ⋮\hfill \\ \sum _{i=1}^n{a}_i{x}_i^{2n-1}={\mu }_n+\sum _{i=1}^{2n-1}(-1{)}^{2n-1}\left(\begin{array}{c}2n-1\hfill \\ i\hfill \end{array}\right){\mu }_{2n-1-i}\text{,}\hfill \end{array}\right.$$

defining

$$\left\{\begin{array}{c}{\lambda }_0={\mu }_0\text{,}\hfill \\ {\lambda }_k={\mu }_k+\sum _{i=1}^k(-1{)}^k\left(\begin{array}{c}k\hfill \\ i\hfill \end{array}\right){\mu }_{k-i}\text{,}k=1\text{,}2\text{,}\dots \text{,}2n-1.\hfill \end{array}\right.$$

The linear system in (7) gives

$$\left\{\begin{array}{c}{a}_1+a_2+\cdots +a_n={\lambda }_0\text{,}\hfill \\ a_1{x}_1+a_2{x}_2+\cdots +a_n{x}_n={\lambda }_1\text{,}\hfill \\ a_1{x}_1^2+a_2{x}_2^2+\cdots +a_n{x}_n^2={\lambda }_2\text{,}\hfill \\ ⋮\hfill \\ a_1{x}_1^k+a_2{x}_2^k+\cdots +a_n{x}_n^k={\lambda }_k\text{,}\hfill \\ ⋮\hfill \\ a_1{x}_1^{2n-1}+a_2{x}_2^{2n-1}+\cdots +a_n{x}_n^{2n-1}={\lambda }_{2n-1}.\hfill \end{array}\right.$$

In other word instead of solving the system in (7) to obtain x_is and w_is it is sufficient to solve the system in (10) with respect to x_i and a_i, which the system cannot be easily solved even if the value x_i are known, we reach a linear system whose coefficients matrix is ill conditioned [1], [4], [5], to follow the procedure x_is will be the roots of polynomials (where c_i must be calculated as indicated in (12)) by replacing the c_is which are obtained from (12) in (11) will give x_is and replacing in (10) will give the desired results for a_is. In the next section we give some examples to illuminate the procedures.

$$c_0{x}^n+c_1{x}^{n-1}+\cdots +c_{n-1}x+1=0\text{,}$$

$$\left[\begin{array}{cccc}\hfill {\lambda }_n\hfill & \hfill {\lambda }_{n-1}\hfill & \hfill \dots \hfill & \hfill {\lambda }_1\hfill \\ \hfill {\lambda }_{n+1}\hfill & \hfill {\lambda }_n\hfill & \hfill \dots \hfill & \hfill {\lambda }_2\hfill \\ \hfill {\lambda }_{n+2}\hfill & \hfill {\lambda }_{n+1}\hfill & \hfill \dots \hfill & \hfill {\lambda }_3\hfill \\ \hfill ⋮\hfill & \hfill \hfill & \hfill \hfill & \hfill ⋮\hfill \\ \hfill {\lambda }_{2n-1}\hfill & \hfill {\lambda }_{2n-2}\hfill & \hfill \dots \hfill & \hfill {\lambda }_n\hfill \end{array}\right]\left[\begin{array}{c}\hfill c_0\hfill \\ \hfill c_1\hfill \\ \hfill c_2\hfill \\ \hfill ⋮\hfill \\ \hfill c_{n-1}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill -{\lambda }_0\hfill \\ \hfill -{\lambda }_1\hfill \\ \hfill -{\lambda }_2\hfill \\ \hfill ⋮\hfill \\ \hfill -{\lambda }_{n-1}\hfill \end{array}\right]\text{.}$$