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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2017 • Question 3

Question

Math 6610 • Analysis of Numerical Methods I Fall 2017 • Question 3
**Numerical Integration** **a)** Establish a numerical integration formula of the form \[ \int_a^b f(x)\,dx \approx Af(a)+Bf'(b) \] that is accurate for polynomials of as high a degree as possible. **b)** Determine the quadrature formula of the form \[ \int_{-2}^{2} f(x)\,dx \approx A_0f(-1)+A_1f(0)+A_2f(1) \] that is accurate for polynomials of degree \(\le2\).

Raw LaTeX

{Numerical Integration}

\partitem{a} Establish a numerical integration formula of the form
\[
  \int_a^b f(x)\,dx \approx Af(a)+Bf'(b)
\]
that is accurate for polynomials of as high a degree as possible.

\partitem{b} Determine the quadrature formula of the form
\[
  \int_{-2}^{2} f(x)\,dx \approx A_0f(-1)+A_1f(0)+A_2f(1)
\]
that is accurate for polynomials of degree \(\le2\).

Topic Tags

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