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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Winter 2004 • Question 3

Question

Math 6610 • Analysis of Numerical Methods I Winter 2004 • Question 3

Gaussian Quadrature Consider the quadrature formula \[ \int _a^b w(x)f(x)\,dx=\sum _{i=1}^n \alpha _i f(\xi _i)+E. \tag {5} \] \(w\) is a given positive weight function. How do you pick the knots \(\xi _i\) and the weights \(\alpha _i\) such that the formula is exact (i.e., \(E=0\)) for all polynomials of degree up to \(2n-1\)? Show that your choice works and that you can find suitable weights and knots for all intervals \([a,b]\) and positive weight functions \(w\).

Raw LaTeX

{Gaussian Quadrature}
Consider the quadrature formula
\[
  \int_a^b w(x)f(x)\,dx=\sum_{i=1}^n \alpha_i f(\xi_i)+E. \tag{5}
\]
\(w\) is a given positive weight function. How do you pick the knots \(\xi_i\) and the
weights \(\alpha_i\) such that the formula is exact (i.e., \(E=0\)) for all polynomials
of degree up to \(2n-1\)? Show that your choice works and that you can find suitable
weights and knots for all intervals \([a,b]\) and positive weight functions \(w\).

Topic Tags

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