MA 4424

Spring 2020

Mike O'Neil (oneil@cims.nyu.edu)

TTh 2pm - 3:15pm JABS 473

T 11am - 12pm, Th 12:30pm - 1:30pm

2 MetroTech 854

Gaston Gonzalez (gmg404@nyu.edu)

F 1:30pm - 2:45pm RGHS 503

Office hours

F 12:25pm - 1:25pm

2 MetroTech, 8th floor

The main textbook for the course is Suli and Mayers,

Additionally, a one-stop source for floating point arithmetic is Michael Overton,

Overall numerical grades will be computed from homework (10%), two preliminary exams (25% each), and one final exam (40%). The final letter grade will be determined from the overall numerical grade.

UPDATE: Since the course has moved online, the new grade weighting will be: homeworks one through three (10%), one preliminary exam (25%), homeworks four through six (65%).

- The first class will meet on Jan 28, 2020.
- Recitations will start on Feb 7, 2020.
- The course syllabus is available here.
- As of March 12, the course has moved online. Please attend lectures, office hours, etc. via the Zoom links accessible via NYU Classes.

This course will serve as an introduction to modern numerical analysis and will cover subjects such as the solution of systems of nonlinear equations, numerical linear algebra, numerical differentiation and integration, interpolation, Monte Carlo methods, and numerical methods for ordinary differential equations. The stability and accuracy of all the previous methods will also be analyzed in the context of ﬂoating-point arithmetic. The course will have a focus on the analysis of numerical methods, but also require you to use numerical software (Matlab, Python, or Julia).

Homework will be assigned roughly every other week and due by the start of class on the day on which it is due. A PDF of the homework may be sent in lieu of a hard-copy but the instructor must receive it by the start of class. No late homework is accepted without prior approval from the instructor, and in those cases, generally exemptions will only be made for medical reasons with documentation. Students are encouraged to work together on their homework, but each student must write-up and submit the assignments independently.

Item | Due date | Materials |
---|---|---|

HW 1 | Feb 18 | hw01.pdf hw01.tex |

HW 2 | Mar 3 | hw02.pdf hw02.tex |

HW 3 | Mar 13 | hw03.pdf hw03.tex |

Below is an updated list of discussion topics along with any documents that were distributed, notes, or relevant code.

Date | Topics | Materials |
---|---|---|

Jan 28 | Intro to the course Bisection method |
Suli & Mayers: 1.1, 1.6 Lecture notes |

Jan 30 | Secant and Newton's methods | Suli & Mayers: 1.4-1.5 Lecture notes |

Feb 4 | Convergence of Newton's method | Suli & Mayers: 1.4 Lecture notes |

Feb 6 | Multidimensional Newton's method | Lecture notes |

Feb 11 | Fixed point iterations, Contraction mapping theorm |
Suli & Mayers: 1.2 Lecture notes |

Feb 13 | Contractions, stability, rates of convergence | Suli & Mayers: 1.2-1.3 Lecture notes |

Feb 18 | Matlab demo Floating point calculations |
newton.m Lecture notes |

Feb 20 | Floating point calculations | macheps.m Lecture notes |

Feb 25 | Gaussian elimination, LU factorization | Suli & Mayers: 2.1-2.3 Lecture notes |

Feb 27 | Prelim Exam 1 | |

Mar 3 | Pivoted LU, norms | Suli & Mayers: 2.4-2.7 Lecture notes |

Mar 5 | Matrix norms, conditioning | Suli & Mayers: 2.7 Lecture notes |

Mar 10 | Condition number of a matrix | Suli & Mayers: 2.7 Lecture notes |

Mar 12 | Least squares | Suli & Mayers: 2.9 Lecture notes |

Mar 17 | No class: Spring break | |

Mar 19 | No class: Spring break | |

Mar 24 | Least squares, QR factorization | Suli & Mayers: 2.9 Lecture notes |

Mar 26 | Gerschgorin's Thm, Power method | Suli & Mayers: 5.1-5.2, 5.4 Lecture notes |

Mar 31 | Inverse power method, Jacobi's method | Suli & Mayers: 5.8, 5.3 Lecture notes |

Apr 2 | Convergence of Jacobi's method | Suli & Mayers: 5.3 Lecture notes |

Apr 7 | Lagrange interpolation | Suli & Mayers: 6.1-6.2 Lecture notes |

Apr 9 | Barycentric interpolation, Runge effect,
minimax approximation |
Suli & Mayers: 6.3,
8.1-8.3 Lecture notes runge.m |

Apr 14 | Chebyshev polynomials, 2-norm function approximation | Suli & Mayers: 8.4-8.5, 9.1-9.3 Lecture notes |

Apr 16 | Orthogonal polynomials | Suli & Mayers: 9.4 Lecture notes |

Apr 21 | Trapezoidal rule, Newton-Cotes, Euler-MacLaurin | Suli & Mayers: 7.1-7.3, 7.6 Lecture notes mybessel.m |

Apr 23 | Gaussian quadrature | Suli & Mayers: 10.1-10.4 Lecture notes |

Apr 28 | ODEs: Forward Euler, finite differences, Richardson extrapolation | Suli & Mayers: 12.1-12.2 Lecture notes |

Apr 30 | ODEs: Local vs. global error; Midpoint, trapezoidal methods | Suli & Mayers: 12.3-12.4 Lecture notes |

May 5 | ODEs: Consistency, stability, convergence | Suli & Mayers: 12.7-12.11 Lecture notes |

May 7 | ODEs: A-stability Fast Fourier transform |
Suli & Mayers: 12.7-12.11 Lecture notes |