11. Computational Numerical Methods

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 mathematics courses. Analytic solutions seem assuring because they give us an equation, but are not always available. The analytic solution may be difficult to determine and may not be as helpful as we had hoped. In the final analysis of most engineering applications, we need a number. Thus, computational numerical methods are often required. As we will see, numerical methods can yield accurate results.

With the algorithms considered in this chapter, there are two sources of error: the intrinsic or truncation error due to algorithmic approximations and the round-off error from the computation. There is a degree of control over the truncation error by algorithm selection and appropriate use of tolerance values. Good algorithms can usually find solutions with an acceptable level of truncation error for most applications. However, efforts to eliminate the truncation error can have diminishing returns, requiring excessive computation and possibly encountering unacceptable round-off errors. For example, the algorithms for numerical differentiation, integration, and solving differential equations covered in Numerical Differentiation, Numerical Integration, and Numerical Differential Equations evaluate functions at a sequence of evenly spaced points, \(\{x_0, x_1, \ldots, x_n\}\). The computation and the round-off error increase if \(n\) is excessively increased. The best strategy is to establish accuracy requirements appropriate to the application and acceptable error tolerance values.

Contents:

• 11.1. Optimization
  • 11.1.1. Numerical Roots of a Function
  • 11.1.2. Finding a Minimum
  • 11.1.3. CVX for Disciplined Convex Programming
• 11.2. Data Interpolation
  • 11.2.1. One-Dimensional Interpolation
  • 11.2.2. Two-Dimensional 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