4.7/

Main Idea

1. What?

   • To Measure Accuracy:

   • The Coefficients \(c_1, c_2, ... , c_n\) in the approximation formula are \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) ,
     and,
     The Nodes \(x_1, x_2, ... , x_n\) are restricted by/to \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)
     This gives,
     The number of Parameters to choose is \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\)

   • If the coefficients of a polynomial are considered parameters,

     This, then, is

2. Why?

Legendre Polynomials

1. What?
2. Why?
3. Properties:
   
   3. The roots of these polynomials are:
4. The first Legendre Polynomials:
   \(P_0(x) = \ \ \ \ \ , \ \ \ \ \ \ \ \ \ \ \ \ \ \ P_1(x) = \ \ \, \ \ \ \ \ \ \ \ \ \ \ \ \ \ P_2(x) = \ \ \ \ \ \ \ \ \ \ ,\)
   \(P_3(x) = \ \ \ \ \ \ \ \ ,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P_4(x) = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)
5. Determining the nodes:
   
   • PROOF:

   The nodes \(x_1, x_2, ... , x_n\) needed to

Gaussian Quadrature on Arbitrary Intervals

1. What?
   
2. The Change of Variables:
   
3. Gaussian quadrature [arbitrary interval]: