Table of Contents

Main Idea

  1. What?
    • A technique that is used to determine the nodes and coefficients
      for formulas that give exact results for higher-degree polynomials.
    • Gaussian quadrature chooses the points for evaluation in an optimal, rather than
      equally-spaced, way.
    • The nodes \(x_1, x_2, ... , x_n\) in the interval \([a, b]\) and coefficients \(c_1, c_2, ... , c_n\), are chosen to minimize the expected error obtained in the approximation, \(\int_{a}^{b} f(x) dx = \sum_{i=1}^{n} c_i f(x_i).\)

    • To Measure Accuracy: we assume that the best choice of these values produces the exact result for the largest class of polynomials, that is, the choice that gives the greatest degree of precision.

    • The Coefficients \(c_1, c_2, ... , c_n\) in the approximation formula are arbitrary,
      and,
      The Nodes \(x_1, x_2, ... , x_n\) are restricted only by the fact that they must lie in \([a, b]\), the interval of integration.
      This gives,
      The number of Parameters to choose is \(2n\).

    • If the coefficients of a polynomial are considered parameters, the class of polynomials of degree at most \(2n − 1\) also contains \(2n\) parameters.
      This, then, is The Largest Class of Polynomials for which it is reasonable to expect a formula to be exact.

  2. Why?
    The fact that, Newton-Cotes Formulas use values at equally-spaced nodes, can significantly decrease the accuracy of the approximation.

Legendre Polynomials

  1. What?
    A series of solutions to “Legendre’s differential equation” that form a polynomial
    sequence of orthogonal polynomials.
  2. Why?
    The roots of the nth-deg Legnedre Polynomial are the nodes needed for the approximation formula that gives exact results for any polynomial of degree less than \(2n\).
  3. Properties:
    1. For each \(n\), \(P_n(x)\) is a monic polynomial of degree \(n\).
    2. \(\int_{-1}^{1} P(x) P_n(x) dx = 0,\) whenever \(P(x)\) is a polynomial of degree less than \(n\).
    3. The roots of these polynomials are:
      • Distinct,
      • lie in the interval \((−1, 1)\),
      • have a symmetry with respect to the origin,
      • the correct choice for determining the parameters that give us the nodes and coefficients for our quadrature method.
  4. The first Legendre Polynomials:
    \(P_0(x) = 1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ P_1(x) = x, \ \ \ \ \ \ \ \ \ \ \ \ \ \ P_2(x) = x^2 − \dfrac{1}{3},\)
    \(P_3(x) = x^3 − \dfrac{3}{5}x,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P_4(x) = x^4 − \dfrac{6}{7}x^2 +\dfrac{3}{35}.\)

  5. Determining the nodes:
    formula

    The nodes \(x_1, x_2, ... , x_n\) needed to produce an integral approximation formula that gives exact results for any polynomial of degree less than \(2n\) are the roots of the nth-degree Legendre polynomial.

Gaussian Quadrature on Arbitrary Intervals

  1. What?
    • Change of Intervals: An integral \(\int_{a}^{b} f(x) dx\) over an arbitrary \([a, b]\) can be transformed into an integral over \([−1, 1]\) by using a change of variables.
  2. The Change of Variables:
    \(\ \ \ t = \dfrac{2x − a − b}{b − a} \ \ \ \iff \ \ \ x = \dfrac{1}{2}[(b − a)t + a + b].\)
  3. Gaussian quadrature [arbitrary interval]:
    formula