Date: December 15, 2014

Place: HL230

Time: 7:00 to 9:00 pm

Part 1

General information:

Open-text & open-notes event. This portion of the Final will test your theoretical knowledge and require some "manual" computations: calculators (not MATLAB, not Maple, etc.) will be needed for arithmetic operations.

Subjects:

Methods for numerical solution of initial value problems and boundary value problems.

Concepts which should be under unconditional control:

Initial value problems:

• Euler's and the Taylor's methods
• The Runge-Kutta methods.

Boundary value problems:

• The shooting method for linear BVPs with Dirichlet and general/mixed boundary conditions
• The finite-difference method for linear BVPs with Dirichlet boundary conditions

Contents:

1. Application of the Taylor's or Euler's method to a given IVP
2. Using one of the Runge-Kutta methods for solving a given IVP
3. A BVP with Dirichlet boundary conditions: analytical portion of the Shooting method
4. Numerical solution of a given BVP by the finite-difference method
5. Bonus

Similar problems:

1. 5.2.1, 5.2.5, 5.3.1, 5.3.5
2. 5.4.5, 5.4.6, 5.4.13, 5.4.15
3. 11.1.1 - 11.1.3
4. 11.3.1 - 11.3.3

Part 2

General information:

This part of the Final will test your skills in solving course-related problems with MATLAB - more specifically, with the functions in your MATLAB Library compiled during the term. Therefore, the MATLAB part will be comprehensive - all five topics of the course will be covered.

Subjects:

MATLAB's *.m files - your versions of the collected MATLAB Procedures: implementations of numerical methods, algorithms, and supplementary functions related to interpolation, approximation, numerical integration, initial value problems and boundary value problems.

Contents:

1. ...interpolation
2. ...approximation
3. ...numerical integration
4. ...numerical method for an IVP
5. ...numerical method for an BVP

Technical instructions:

• Operating systems on the machines in HL230?

  Windows

• How to get access to MATLAB during the Exam?

  MATLAB is installed in all computers in the computer lab HL230. Alternatively, one can use his/her own laptop with MATLAB.

• How to get access to your MATLAB Library in HL230?

  If you are using the lab computer, probably the easiest way to do that is to bring to the Exam all the scripts on a memory stick. Alternatively, the files could be put in your home (Linux) directory and could be accessed through the network.

• How to present/format the solutions?

  The solutions should be presented in a report made as a Word file. A template of the report, a Word file with formulations of all problems, will be provided - it will be e-mailed to the whole class during the First Part of the Final.
  
  Include in the report the MATLAB commands (or short scripts) which were used for solving the problems. Show graphical results/output required to be generated. Do not include excessive/intermediate/supplementary numerical data or big tables of numbers. Add some concise verbal comments clarifying the solution and what is shown in the report.

• How to print the report?

  Use regular Word's Print command with the default HL230 printer. Your job will be released for printing from the printer's monitor system.

• Are there any ways to check the computers prior to the Final?

  It would be a good idea to visit HL230 during the day hours to familiarize yourself with the facilities. The room will be made available for the Exam soon after 6:30 pm.

Date: Tuesday, December 2, 2014

Place: OH223

Time: 2:00 pm

General information:

Open-text & open-notes event. Theoretical knowledge (necessarily including the skills in the relevant "manual" computations) will be tested. Calculators, therefore, will be needed to do arithmetic operations.

Subjects:

Numerical integration - the Composite Trapezoid and Simpson's Rules, Romberg Integration, and Gaussian quadratures.

Concepts which should be under unconditional control:

• Idea and principles of numerical integration
• Composite Trapezoid Rule and an error generated by this method
• Composite Simpson's Rule and an error generated by this method
• Romberg integration; using Richardson extrapolation to reduce the truncation error
• Gaussian quadratures and related change of variable

Contents:

1. Finding an approximate value for the definite integral with either the Composite Trapezoid Rule, or the Composite Simpson's Rule, or Romberg integration
2. Finding an approximate value for the definite integral with either the Composite Trapezoid Rule, or the Composite Simpson's Rule, or Romberg integration
3. Finding errors in approximations of definite integrals by the Composite Trapezoid and Simpson's Rules
4. Using Gaussian quadratures for approximation of the definite integral
5. Bonus

Similar problems in the HW assignments:

1. 4.4.2b, 4.4.4d, 4.4.9
2. [16], 4.5.1a, 4.5.1b
3. [17], [18]
4. 4.7.2d, [19]

Date: Tuesday, November 18, 2013

Place: OH223

Time: 2:00 pm

General information:

Open-text & open-notes event. Theoretical knowledge (necessarily including the skills in the relevant "manual" computations) will be tested. Calculators, therefore, will be needed to do arithmetic operations.

Subjects:

Interpolation, approximations, numerical differentiation.

Concepts which should be under an unconditional control:

• Polynomial interpolation: Lagrange and Newton polynomials
• Construction of Newton interpolating polynomial with divided differences
• Interpolation difficulties and errors
• Spline interpolation
• Numerical differentiation: backward and forward difference formulas; central difference formulas of different accuracy
• Richardson extrapolation
• Linear, quadratic, and cubic least squares approximations
• Normal equations for discrete and continuous approximations
• Approximation using Legendre and Chebyshev orthogonal polynomials

Contents:

1. Construction of an interpolation polynomial (either Lagrange or Newton formats) for the given data; determination of the error bound.
2. Approximation of the derivative of a function at the given point (with different difference formulas); determination of accuracy of the approximate values.
3. Finding a formula of the best fit (in the least squares sense) for the set of data points.
4. Bonus.

Similar problems in the HW assignments:

1. [2] - [5]
2. [7] - [10]
3. [11] - [14]