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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2011 • Question 4

Question

Math 6610 • Analysis of Numerical Methods I Fall 2011 • Question 4

Numerical Quadrature

Show that there is no set of nodes \(x_1,x_2,\ldots ,x_n\) and coefficients \(\alpha _1,\alpha _2,\ldots ,\alpha _n\) such that the quadrature rule \[ \sum _{j=1}^n \alpha _j f(x_j) \] exactly equals to the integral \(\int _a^b f(x)w(x)dx\) for all polynomials \(f(x)\) of degree less than or equal to \(2n\). \(w(x)\) is the weight function.

Raw LaTeX

{Numerical Quadrature}

Show that there is no set of nodes \(x_1,x_2,\ldots,x_n\) and coefficients
\(\alpha_1,\alpha_2,\ldots,\alpha_n\) such that the quadrature rule
\[
  \sum_{j=1}^n \alpha_j f(x_j)
\]
exactly equals to the integral \(\int_a^b f(x)w(x)dx\) for all polynomials
\(f(x)\) of degree less than or equal to \(2n\). \(w(x)\) is the weight function.

Topic Tags

Linear Programming and Optimization Numerical Differentiation and Finite Difference Operators Numerical Quadrature Quadrature Rules