### Mathematics 16:642:573 Numerical Analysis I

### Schedule

The course is offered during the Fall semester.- Class meeting dates: Please visit the University's academic calendar.
- Schedule and Instructor: Please visit the University's schedule of classes for the instructor, time, and room.
- Instructor and Teaching Assistant Office Hours: Please visit the Mathematical Finance program's office hour schedule.

### Course Abstract

This is the first part of a general survey of the basic topics in numerical analysis -- the study and analysis of numerical algorithms for approximating the solution of a variety of generic problems which occur in applications. In the fall semester, we will consider the approximation of functions by polynomials and piecewise polynomials, numerical integration, and the numerical solution of initial value problems for ordinary differential equations, and see how all these problems are related.In the spring semester (642:574), we will study the numerical solution of linear systems of equations, the approximation of matrix eigenvalues and eigenvectors, the numerical solution of nonlinear systems of equations, numerical techniques for unconstrained function minimization, finite difference and finite element methods for two-point boundary value problems, and finite difference methods for some model problems in partial differential equations.

Despite the many solution techniques presented in elementary calculus and differential equations courses, mathematical models used in applications often do not have the simple forms required for using these methods. Hence, a quantitative understanding of the models requires the use of numerical approximation schemes. This course provides the mathematical background for understanding how such schemes are derived and when they are likely to work.

To illustrate the theory, in addition to the usual pencil and paper problems, some short computer programs will be assigned. To minimize the effort involved, however, the use of Matlab will be encouraged. This program has many built in features which make programming easy, even for those with very little prior programming experience.

### Pre-requisites and Co-requisites

Advanced Calculus, Linear Algebra, and familiarity with differential equations.### Required Textbooks

No required text. Lecture notes will be posted online.### Grading

Please contact the instructor.### Class Policies

Please see the MSMF common class policies.### Assignments

Assignments: Homework assignments in the course consist of both theoretical and computational work. For the computational component, the students should use a language/environment that possesses high level data types so that the students spend more time working with algorithms and not worrying about the details of writing computer code. MATLAB is a good choice. Fortran 77/90/95 and C++ with appropriate class libraries can also be used.### Previous Instructor Course Websites

2015 Richard Falk2014 Duk-soon Oh

2013 Abdolrez Tahvildar-Zadeh

2012 Abdolrez Tahvildar-Zadeh

2011 Young-Ju Lee

2010 Young-Ju Lee

2009 Richard Falk

2008 Young-Ju Lee

2007 Michael Vogelius

2006 Richard Falk

### Weekly Lecturing Agenda and Readings

The lecture schedule below is a sample; actual content may vary depending on the instructor.

This page will record the topics we cover in each lecture and reading assignments. Material on most of the topics listed below can be found in A. Quarteroni, R. Sacco, and F. Saleri: Numerical Mathematics, referred to as **[QSS]** or Kendall Atkinson, An Introduction to Numerical Analysis, referred to as **[A]**.

Lecture | Topics | Reading Assignments |
---|---|---|

1 | General course outline and Background for Programming projects | |

Approximation by Polynomials, Piecewise Polynomials, and Trigonometric Functions | ||

2 | Weierstrass approximation theorem, Lagrange and Newton forms of the interpolating polynomial | [A: 4.1, 3.1, 3.2], [QSS: 8.1, 8.2] |

3 | Polynomial interpolation error, divided differences for repeated points | [A: 3.2, 3.6], [QSS: 8.5] |

4 | Interpolation of moments, Runge example | [A: 3.5], [QSS: 8.1] |

5 | Piecewise polynomial approximation: C^{0} and C^{1} piecewise polynomial approximation and error estimates; construction of basis functions |
[A: 3.7], [QSS: 8.3] |

6 | Piecewise Polynomial Approximation: cubic spline approximation, basis functions, and error estimates | [A:3.7], [QSS: 8.7] |

7 | Trigonometric interpolation; fast Fourier transform | [A:3.8], [QSS: 10.9] |

8 | Piecewise polynomial approximation in higher dimensions | [QSS: 8.6] |

Numerical Differentiation and Integration | ||

9 | Numerical differentiation | [A: 5.7], [QSS: 10.10] |

10 | Basic and composite rules for numerical integration | [A: 5.1, 5.2], [QSS: 9.1, 9.2, 9.3, 9.4] |

11 | Extrapolation and Romberg integration | [A: 5.4], [QSS: 9.6] |

12 | Orthogonal polynomials and Gaussian quadrature | [A: 4.4, 5.3], [QSS: 10.1, 10.2, 10.3, 10.4, 10.5, 10.6] |

13 | Orthogonal polynomials and Gaussian quadrature (continued) | [A: 4.4, 5.3], [QSS: 10.1, 10.2, 10.3, 10.4, 10.5, 10.6] |

14 | Adaptive quadrature | [A: 5.5], [QSS: 9.7] |

15 | Singular integrals | [A: 5.6], [QSS: 9.8] |

Numerical Methods for Ordinary Differential Equations | ||

16 | The initial value problem for ordinary differential equations | [A: 6.1], [QSS: 11.1] |

17 | Euler and general Taylor series methods | [A: 6.2], [QSS: 11.1, 11.2, 11.3] |

18 | Runge-Kutta methods | [A: 6.10], [QSS: 11.8] |

19 | Estimation of local error and adaptive methods | [A: 6.10], [QSS: 11.8.2] |

20 | Linear multistep methods (derivation, order, consistency, local truncation error) | [A: 6.3, 6.4, 6.5], [QSS: 11.5, 11.6] |

21 | Convergence of linear multistep methods | [A: 6.8], [QSS: 11.4, 11.6] |

22 | Stability of linear multistep methods | [A: 6.8], [QSS: 11.6] |

23 | Stability of linear multistep methods (continued) | [A: 6.8], [QSS: 11.6] |

24 | Predictor-corrector methods | [A: 6.6, 6.7], [QSS: 11.7] |

25 | First order systems of odes | [QSS: 11.9] |

26 | Stiff systems of odes | [A: 6.9], [QSS: 11.10] |

27 | The discontinuous Galerkin method |