Numerical Differentiation and Integration
          Numerical Differentiation
                ◦ Finite Differences
                ◦ Interpolating Polynomials
                ◦ Taylor Series Expansion
                ◦ Richardson Extrapolation
          Numerical Integration
                ◦ Basic Numerical Integration
                ◦ Improved Numerical Integration
                    ⇒ Trapezoidal, Simpson’s Rules
                ◦ RhombergIntegration
        ITCS4133/5133: Numerical Comp. Methods               1                 Numerical Differentiation and Integration
              Numerical Differentiation and Integration
          Many engineering applications require numerical estimates of
              derivatives of functions
          Especially true, when analytical solutions are not possible
          Differentiation: Use finite differences
          Integration (definite integrals): Weighted sum of function values
              at specified points (area under the curve).
        ITCS4133/5133: Numerical Comp. Methods               2                 Numerical Differentiation and Integration
          Application:Integral of a Normal Distribution
                                                                                                         2
          ◦ Represented as a Gaussian, a scaled form of f(x) = e−x , very
              important function in statistics
          ◦ Noteasytodetermineindefinite integral - use numerical techniques
                                                         Z b        2
                                                  A=           e−x
                                                           a
        ITCS4133/5133: Numerical Comp. Methods               3                 Numerical Differentiation and Integration
                            Application:Integral of a Sinc
                                                 f(x) = sin(x)
                                                                x
        ITCS4133/5133: Numerical Comp. Methods               4                 Numerical Differentiation and Integration