11. Computational Numerical Methods

Reading Assignment

Please read chapter 6 of Physical Modeling in MATLAB, by Allen B. Downey [DOWNEY11].

This chapter covers methods for calculating numerical solutions to problems. The term numerical is used as the antithesis to analytical. Analytical solutions are sought after in math classes. Analytic solutions seem assuring because they give us an equation, but they are not always available. The analytic solution may be very difficult to determine, and may not be as helpful as we had hoped. In the final analysis of most applications we need a number. Thus, computational numerical methods are often required. As we will see, numerical methods can yield very accurate results.

Contents:

• 11.1. Optimization
  • 11.1.1. Numerical Roots of a Function
  • 11.1.2. Finding a Minimum
• 11.2. Data Interpolation
• 11.3. Numerical Differentiation
  • 11.3.1. Euler Derivative
  • 11.3.2. Spectral Derivative
• 11.4. Numerical Integration
  • 11.4.1. MATLAB’s Integral Function
  • 11.4.2. Fixed Width Subinterval Algorithms
  • 11.4.3. Recursive Adaptive Integral
• 11.5. Numerical Differential Equations
  • 11.5.1. Euler’s Method
  • 11.5.2. Heun’s Method
  • 11.5.3. Runge-Kutta Method
  • 11.5.4. Algorithm Comparison
  • 11.5.5. Stability Constraints
  • 11.5.6. Implicit Backward Euler Method
  • 11.5.7. Adaptive Algorithms
  • 11.5.8. The Logistic Equation
  • 11.5.9. Stiff ODEs
  • 11.5.10. MATLAB’s Suite of ODE Solvers