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1. Fundamentals

• Computer arithmetic and round-off errors
• Accuracy, consistency, stability, convergence
• Advantages and limitations of numerical methods

2. Solution of non-linear equations

Chapter 2 deals with finding the zeros of a nonlinear function of one or several variables or of system of nonlinear equations.

• Bisection method
• Fixed-point iteration
• Newton’s method
• Derivative-free methods
• Convergence

3. Optimization

Chapter 3 deals with methods for the minimization of nonlinear functions.

• Unconstraint optimization without derivatives
o Golden section search
o Successive parabolic interpolation
• Unconstrained optimization with derivatives
o Newton’s method
o Steepest descent
o Conjugate gradient search

4. Numerical interpolation & approximation

Chapter 4 deals with methods for numerical interpolation and least-squares approximation. It includes both 1D and 2D/3D problems.

• 1D interpolation
o Polynomial interpolation
o Spline interpolation
• 2D interpolation
o Polynomial interpolation
o Spline interpolation
o Patch interpolation
• Least-squares approximation
o Least squares and normal equations
o QR factorization
• Perspective on alternative concepts and more advanced methods


5. Numerical differentiation and integration

Chapter 5 deals with methods for numerical differentiation and integration.

• Numerical differentiation
o Finite difference formulas
o Rounding error
o Extrapolation
• Numerical integration
o Interpolatory quadrature (Newton Cotes)
o Composed integration formulas
o Gauss-Legendre quadrature
o Error estimates
o 2D integration (Cartesian products and product rules)

6. Numerical methods for solving ordinary differential equations

Chapter 7 deals with numerical methods for solving ordinary differential equations, beginning with basic techniques such as Euler-Cauchy, Heun, Runge-Kutta, and multistep methods. The methods are then extended to solving systems of first-order ordinary differential equations, and the concept of stiffness and stability are introduced.

• Basic concepts and classification
• Single-step methods (Euler-Cauchy, Heun, Runge-Kutta)
• Multistep methods (Adams Bashforth, Adams-Moulton, predictor-corrector)
• Stability and convergence
• Choice of a method

At the end of this course, the student will be able to:

By attending the lectures, completing the assignments, and participating in class, the student should accomplish the following general goals:

1. Gain experience in basic numerical methods
2. Explain how and when the numerical methods can be expected to work
3. Gain a firm basis for future study in numerical analysis
4. Solve a numerical problem with confidence using basic methods and algorithms.
5. Be aware of limitations and potential extensions.
6. Be familiar with the implementation and application of basic numerical methods to problems in Aerospace Engineering.
7. Has developed a critical attitude towards methods and results.

Upon completion of this course, a student should be able to perform the following materials:

1. computer arithmetic and round-off errors
2. solution of non-linear equations
3. finding the minimum of a non-linear function
4. interpolation using polynomials and splines in 1D and 2D
5. discrete least-squares approximation of experimental data
6. numerical differentiation and integration
7. numerical solution of ordinary differential equations

Lectures, assignments, supervised labs

C.F. Gerald & P.O. Wheatley, Applied Numerical Analysis, 7th edition, Pearson, Addison Wesley, 2004

Weekly quiz built upon the weekly assignment. The final mark is computed based on the best five of the six in-lecture quizzes. No written exam.

Students attend a lecture, where they gain a basic knowledge about the numerical algorithm under consideration.

The lecturer, in particular, emphasizes potential problems and occasionally, poses questions to activate students.

At home, students download from Blackboard the assignments (supplied with necessary instructions) and complete them.