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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Spring 2020 • Question 3

Question

Math 6610 • Analysis of Numerical Methods I Spring 2020 • Question 3

Numerical Integration

Determine a formula \[ \int _{-\pi }^{\pi } f(x)\cos (x)\,dx \approx A_0 f\left (-\frac {3}{4}\pi \right ) +A_1 f\left (-\frac {1}{4}\pi \right ) +A_2 f\left (\frac {1}{4}\pi \right ) +A_3 f\left (\frac {3}{4}\pi \right ) \] that is exact when \(f\) is a polynomial of degree 3.

Raw LaTeX

{Numerical Integration}

Determine a formula
\[
  \int_{-\pi}^{\pi} f(x)\cos(x)\,dx
  \approx
  A_0 f\left(-\frac{3}{4}\pi\right)
  +A_1 f\left(-\frac{1}{4}\pi\right)
  +A_2 f\left(\frac{1}{4}\pi\right)
  +A_3 f\left(\frac{3}{4}\pi\right)
\]
that is exact when \(f\) is a polynomial of degree 3.

Topic Tags

Linear Programming and Optimization Numerical Differentiation and Finite Difference Operators Numerical Quadrature Polynomial Interpolation and Error Formulas