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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2014 • Question 2

Question

Math 6610 • Analysis of Numerical Methods I Fall 2014 • Question 2

Consider the quadrature rule \[ \int _a^b w(x)f(x)dx = \sum _{j=1}^n w_jf(x_j)+E_n. \] Assume that \(w\) is a positive weight function. Discuss how to pick the knots \(x_1,\ldots ,x_n\) and weights \(w_1,\ldots ,w_n\) to make the error \(E_n\) zero for all polynomials \(f\) up to a degree as high as possible.

Raw LaTeX

Consider the quadrature rule
\[
  \int_a^b w(x)f(x)dx
  =
  \sum_{j=1}^n w_jf(x_j)+E_n.
\]
Assume that \(w\) is a positive weight function. Discuss how to pick the knots
\(x_1,\ldots,x_n\) and weights \(w_1,\ldots,w_n\) to make the error \(E_n\) zero for
all polynomials \(f\) up to a degree as high as possible.

Topic Tags

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