Newton-Cotes Integration

Introduction to Quadrature

    We now approach the subject of numerical integration. The goal is to approximate the definite integral of  f(x)  over the interval  [a,b]  by evaluating  f(x)  at a finite number
of sample points.

Definition (Quadrature Formula)  Suppose that  .   A formula of the form

(1)        

with the property that

(2)        

is called a numerical integration or quadrature formula.  The term  E[f]  is called the truncation error for integration.  The values are called the quadrature nodes and are called the weights.

    Depending on the application, the nodes    are chosen in various ways.  For the Trapezoidal Rule, Simpson�s Rule, and Boole�s Rule, the nodes are chosen to be equally spaced.  For Gauss-Legendre quadrature, the nodes are chosen to be zeros of certain Legendre polynomials.  When the integration formula is used to develop a predictor formula for differential equations, all the nodes are chosen less than b.  For all applications, it is necessary to know something about the accuracy of the numerical .  This leads us to the next definition.

Definition (Degree of Precision)  The degree of precision of a quadrature formula is the positive integer  n  such that    for all polynomials   of degree  ,  but for which    for some polynomial    of degree  n+1.  That is

           when degree  ,  
and
           when degree  .  

    The form of   can be anticipated by studying what happens when  f(x)  is a polynomial.  Consider the arbitrary polynomial

        
    
of degree i.  If  ,  then    for all  x,  and    for all  x.  Thus it is not surprising that the general form for the truncation error term is

(3)        ,  

where  K  is a suitably chosen constant and  n  is the degree of precision.  The proof of this general result can be found in advanced books on numerical integration.  The derivation of quadrature formulas is sometimes based on polynomial interpolation.  Recall that there exists a unique polynomial    of degree  ,  passing through the  m+1  equally spaced points  .   When this polynomial is used to approximate  f(x)  over  [a,b],  and then the integral of  f(x) is approximated by the integral of  ,  the resulting formula is called a Newton-Cotes quadrature formula.  When the sample points    and    are used, it is called a closed Newton-Cotes formula.  The next result gives the formulas when approximating polynomials of degree    are used.

Theorem (Closed Newton-Cotes Quadrature Formula)  Assume that    are equally spaced nodes and  .  The first four closed Newton-Cotes quadrature formulas:

(4) Trapezoidal Rule      

(5) Simpson�s Rule      

(6) Simpson 3/8 Rule      

(7) Boole�s Rule      

Corollary (Newton-Cotes Precision)  Assume that  f(x)  is sufficiently differentiable; then  E[f]  for Newton-Cotes quadrature involves an appropriate higher derivative.

(8) The trapezoidal rule has degree of precision  n=1.  If  , then

        .

(9) Simpson�s rule has degree of precision  n=3.  If  , then

        .

(10) Simpson�s rule has degree of precision  n=3.  If  , then

        .

(11) Boole�s rule has degree of precision  n=5.  If  , then

        .

Example  Consider the function  ,  the equally spaced quadrature nodes  , ,  ,  , and ,  and the corresponding function values  ,  ,  ,  ,  and  .  Apply the various quadrature formulas (4) through (7).

        Trapezoidal Rule                                                Simpson�s Rule 

        Simpson�s 3/8 Rule                                                Boole�s Rule