Numerical Quadrature

1. What?

   The basic method involved in approximating \(\int_{a}^{b} f(x) dx\).

2. How?

   It uses a sum \(\sum_{i=0}^{n} a_i f(x_i)\) to approximate \(\int_{a}^{b} f(x) dx\).

3. Based on:

   The methods of quadrature in this section are based on the interpolation polynomials given in Chapter 3.

4. Method:
   • Select a set of distinct nodes \({x_0, ... , x_n}\) from the interval \([a, b]\).
   • Then integrate the Lagrange interpolating polynomial
     \(P_n(x) = \sum_{i=0}^{n} f(x_i)L_i(x)\)
     and its truncation error term over \([a, b]\) to obtain
     Show Derivation

5. The Quadrature Formula:
   

6. The Error:
   

The Trapezoidal Rule

1. What?

   Approximation method for integrals produced by using first (linear) Lagrange polynomials with equally-spaced nodes.

2. The Trapezoidal Rule:
   Show Derivation

3. Error:
   The error term for the Trapezoidal rule involves \(f\) , so the rule gives the exact result when applied to any function whose second derivative is identically zero, that is, any polynomial of degree one or less.

Simpson’s Rule

1. What?
   A method to approximate an integral that results from integrating over \([a, b]\) the second Lagrange polynomial with equally-spaced nodes \(x_0 = a, x_2 = b,\) and \(x_1 = a + h\), where \(h = \dfrac{(b − a)}{2}\).
2. Simpson’s Rule:
   Show Derivation
3. Error:
   The error term in Simpson’s rule involves the fourth derivative of \(f\) , so it gives exact results when applied to any polynomial of degree three or less.

Measuring Precision

1. What?
   The standard derivation of quadrature error formulas is based on determining the class of polynomials for which these formulas produce exact results.
2. Precision [degree of accuracy]:
   

   Definition 4.1 implies that the Trapezoidal and Simpson’s rules have degrees of precision one and three, respectively.

3. Precision of Quadrature Formulas:
   
   • The degree of precision of a quadrature formula is n if and only if the error is zero for all polynomials of degree \(k = 0, 1, ... , n\), but is not zero for some polynomial of degree \(n + 1\).
   • The Trapezoidal and Simpson’s rules are examples of a class of methods known as Newton-Cotes formulas.
   • Types of Newton-Cotes formulas: There are 2 types, Open, and Closed.

Closed Newton-Cotes Formulas

1. What?
   
   • The \((n+1)\)-point closed Newton-Cotes formula uses nodes \(x_i = x_0 +ih\), for \(i = 0, 1, ... , n,\) where \(x_0 = a, x_n = b\) and \(h = \dfrac{(b − a)}{n}\) .
   • It is called closed because the endpoints of the closed interval [a, b] are included as nodes.
2. Form of the Formula:
   

   where,
   

3. The Error Analysis:
   

4. Degree of Preceision:
   
   • Even-n: the degree of precision is \(n + 1\) (although the interpolation polynomial is of degree at most n)
   • Odd-n: the degree of precision is only \(n\).
5. Closed Form Formulas:
   
   • \(n = 1\): Trapezoidal rule
     
   • \(n = 2\): Simpson’s rule
     
   • \(n = 3\): Simpson’s Three-Eighths rule
     
   • n = 4:
     

Open Newton-Cotes Formulas

1. What?
   
   • They do not include the endpoints of \([a, b]\) as nodes.
   • They use the nodes \(x_i = x_0 + ih\), for each \(i = 0, 1, ... , n\), where \(h = \dfrac{b − a}{n + 2}\) and \(x_0 = a + h\).
   • This implies that \(x_n = b − h\), so we label the endpoints by setting \(x_{−1} = a\) and \(x_{n+1} = b\).
   • Open formulas contain all the nodes used for the approximation within the open interval \((a, b)\)
2. Form of the Formula:
   

   where,
   

3. The Error Analysis:
   

4. Degree of Preceision:
   
   • Even-n: Higher.
   • Odd-n: Lower.
5. Open Form Formulas:
   
   • \(n = 0\): Midpoint rule
     
   • \(n = 1\):
     
   • \(n = 2\):
     
   • n = 3: