Applied Numerical Methods Using MATLAB, 2nd Edition

5Numerical Differentiation/Integration

Chapter Outline

• 5.1 Difference Approximation for the First Derivative
• 5.2 Approximation Error of the First Derivative
• 5.3 Difference Approximation for Second and Higher Derivative
• 5.4 Interpolating Polynomial and Numerical Differential
• 5.5 Numerical Integration and Quadrature
• 5.6 Trapezoidal Method and Simpson Method
• 5.7 Recursive Rule and Romberg Integration
• 5.8 Adaptive Quadrature
• 5.9 Gauss Quadrature
  • 5.9.1 Gauss‐Legendre Integration
  • 5.9.2 Gauss‐Hermite Integration
  • 5.9.3 Gauss‐Laguerre Integration
  • 5.9.4 Gauss‐Chebyshev Integration
• 5.10 Double Integral
• 5.11 Integration Involving PWL Function
• Problems

5.1 Difference Approximation for the First Derivative

For a function f(x) of a variable x, its first derivative is defined as

(5.1.1)

However, this gives our computers a headache, since they do not know how to take a limit. Any input number given to computers must be a definite number and can be neither too small nor too large to be understood by the computer. The ‘theoretically’ infinitesimal number h involved in this equation is a problem.

A simple approximation that computers might be happy with is the forward difference approximation.

(5.1.2)

How far away is this approximation from the true value of Eq. (5.1.1)? In order to do ...