Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 1999 • Question 1

Question

Math 6610 • Analysis of Numerical Methods I Fall 1999 • Question 1

Quadrature Consider the quadrature formula \[ \int _a^b w(x)f(x)\,dx=\sum _{i=0}^n \alpha _i f(\xi _i)+E. \tag {1} \] \(w\) is a given positive weight function. The knots \(\xi _i\) and the weights \(\alpha _i\) are at your disposal. How do you choose them so that the error \(E\) is 0 for all polynomials \(f\) of degree as high as possible? How high is possible? Give reasons for your answers.

Raw LaTeX

{Quadrature}
Consider the quadrature formula
\[
  \int_a^b w(x)f(x)\,dx=\sum_{i=0}^n \alpha_i f(\xi_i)+E. \tag{1}
\]
\(w\) is a given positive weight function. The knots \(\xi_i\) and the weights
\(\alpha_i\) are at your disposal. How do you choose them so that the error \(E\) is 0
for all polynomials \(f\) of degree as high as possible? How high is possible? Give
reasons for your answers.

Topic Tags

Linear Programming and Optimization Numerical Quadrature Polynomial Interpolation and Error Formulas Quadrature Rules