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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2008 • Question 2

Question

Math 6610 • Analysis of Numerical Methods I Fall 2008 • Question 2

Gaussian Quadrature Let \(w\) be a positive weight function. Explain how to pick the knots \(x_i\) and the weights \(w_i\), \(i=1,\ldots ,n\) such that the integration formula \[ \int _a^b w(x)f(x)\,dx=\sum _{i=1}^n w_i f(x_i) \tag {2} \] is exact for polynomials \(f\) of degree as high as possible, and show why this works. How high a polynomial degree is possible?

Raw LaTeX

{Gaussian Quadrature} Let \(w\) be a positive weight function. Explain
how to pick the knots \(x_i\) and the weights \(w_i\), \(i=1,\ldots,n\) such that the
integration formula
\[
  \int_a^b w(x)f(x)\,dx=\sum_{i=1}^n w_i f(x_i) \tag{2}
\]
is exact for polynomials \(f\) of degree as high as possible, and show why this
works. How high a polynomial degree is possible?

Topic Tags

Gaussian Elimination and Pivoting Linear Programming and Optimization Numerical Quadrature Quadrature Rules