# Exam 2012

### Final Results

The final grades have been sent to the secretary. I got the permission to display them here:

10001 | C |

10002 | C |

10003 | C |

10004 | A |

10005 | E |

10006 | D |

10007 | C |

10010 | B |

10011 | E |

10015 | F |

10020 | B |

10021 | A |

10022 | A |

10023 | C |

10025 | B |

10026 | C |

10027 | B |

10029 | A |

10030 | C |

10031 | D |

10033 | F |

10034 | A |

10035 | C |

10037 | D |

10038 | C |

10039 | F |

10040 | C |

10041 | A |

10042 | B |

10043 | C |

10044 | A |

10045 | A |

10046 | B |

10049 | C |

10050 | C |

10051 | B |

10052 | A |

10053 | A |

10055 | C |

10057 | A |

### Statistics

The statistics are as follows:

A | B | C | D | E | F |
---|---|---|---|---|---|

11 | 7 | 14 | 3 | 2 | 3 |

Congratulations to you all!!!

It has been a great pleasure for me to have you as student this year. I wish you a good continuation!

GOD SOMMER!

### Solutions

### Standard Mistakes

Some comments on the classical mistakes:

- A Lagrange polynomial is a polynomial which is one at one of the interpolation points, and zero at the other ones; there is no such thing as a "interpolating Lagrange polynomial
- many made a sign mistake for the Newton's method; Newton's method is (notice the sign in front of \(F(x_n)\): \[ F'(x_n) ∆x = - F(x_n) \]
- To my surprise many did mistakes in 5b; in particular, many found solutions that were not defined for \(ε=0\), or even when \(ε\) converges to zero; this is incorrect: see the solution
- by far the hardest question was 5c); only a few of you did it correctly; I have written a very detailed solution

### Grading

The grading of the exam is finished, but you will have to wait for the final grade, because those grades have to be consolidated with the project grades. You should get your grade during this week though.

### Statistics

Only one student has made an error-free exam.

Average points for each question (in percentage):

Question | Average (%) |
---|---|

1a | 75 |

1b | 89 |

2 | 84 |

3a | 87 |

3b | 54 |

4a | 72 |

4b | 69 |

5a | 89 |

5b | 64 |

5c | 21 |

# Exam Information

The exam will take place on **Friday 08.06.2012, 15:00–19:00**.

### Exam Rules

The only printed material allowed is the official course book by Cheney and Kincaid. The notes on the FFT algorithm will be printed out along with the exam sheet, so you do not have to take them with you.

No hand-written notes are allowed.

**you are not allowed to write anything else than you name, anywhere in the book**- you may mark pages with paper marks (
*lapper*) without annotation or by dog-earings (*eselører*) - you may mark text in the book by underlining/highlighting it (
*understreking/overstreking*)

### Things to know for the exam

- Fixed point theorem
- Newton's method to find root of equations: how to use it in various cases, what are its limits
- Fundamental theorem of interpolation: there is a unique polynomial of degree \(k\) which interpolates through \(k+1\) points
- Lagrange polynomials
- Neville formula: build an approximation polynomial from two other interpolation polynomials
- Neville algorithm to compute the value of an interpolation polynomial without know the polynomial
- Newton's divided differences
- Interpolation Error formula: what it means, how to use it, what are its limits.
- Interpret a log-log graph
- Use Taylor's formula to show the order of a given approximation formula
- Richardson's extrapolation algorithm
- Gauss Elimination
- Cost of Gauss Elimination
- Gauss Elimination is equivalent to an LU decomposition
- Iterative Methods: Jacobi and Gauss-Seidel methods
- Linear fixed point methods
- Two different formulae for integration: one
*quadrature formula*and one*integration formula* - Be able to compute quadrature formula weights
- Order of quadrature formula
- Gauss Points
- Be able to use the Fast Fourier Transform algorithm