# Syllabus

When you click the Amazon logo to the left of any citation and purchase the book (or other media) from Amazon.com, MIT OpenCourseWare will receive up to 10% of this purchase and any other purchases you make during that visit. This will not increase the cost of your purchase. Links provided are to the US Amazon site, but you can also support OCW through Amazon sites in other regions. Learn more. |

This section contains documents that could not be made accessible to screen reader software. A "#" symbol is used to denote such documents.

## Description

This course offers an advanced introduction to numerical linear algebra. Topics include direct and iterative methods for linear systems, eigenvalue decompositions and QR/SVD factorizations, stability and accuracy of numerical algorithms, the IEEE floating point standard, sparse and structured matrices, preconditioning, linear algebra software. The problem sets require some knowledge of MATLAB®.

### Prerequisites

Differential Equations (18.03) and Linear Algebra (18.06).

## Texts

The required textbook for this class is:

Bau III, David, and Lloyd N. Trefethen. *Numerical Linear Algebra*. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1997. ISBN: 0898713617.

Other readings include:

Bai, et al. *Templates for the Solution of Algebraic Eigenvalue Problems: a Practical Guide*. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2000. ISBN: 0898714710.

Barrett, et al. *Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods**.* Philadelphia, PA: Society for Industrial and Applied Mathematics, 1993. ISBN: 0898713285.

Shewchuk, Jonathan R. "An Introduction to the Conjugate Gradient Method Without the Agonizing Pain." Carnegie Mellon University (August 1994). (PDF)^{#}

Goldberg, David. "What Every Computer Scientist Should Know About Floating Point Arithmetic." *ACM Computing Surveys* 23, no. 1 (March 1991): pp. 5-48.

## Grading

ACTIVITIES | PERCENTAGES |
---|---|

Homework Assignments | 60% |

Midterm Exam | 40% |

## Policies

Collaboration on the homeworks is encouraged, but each student must write his/her own solutions, understand all the details of them, and be prepared to answer questions about them.

No books, notes, or calculators are allowed on the Midterm exam.

## Calendar

LEC # | TOPICS | KEY DATES |
---|---|---|

1 | Introduction, Basic Linear Algebra | |

2 | Orthogonal Vectors and Matrices, Norms | |

3 | The Singular Value Decomposition | |

4 | The QR Factorization | |

5 | Gram-Schmidt Orthogonalization | Homework 1 due |

6 | Householder Reflectors and Givens Rotations | |

7 | Least Squares Problems | |

8 | Floating Point Arithmetic, The IEEE Standard | |

9 | Conditioning and Stability I | Homework 2 due |

10 | Conditioning and Stability II | |

11 | Gaussian Elimination, The LU Factorization | |

12 | Stability of LU, Cholesky Factorization | Homework 3 due |

13 | Eigenvalue Problems | |

14 | Hessenberg / Tridiagonal Reduction | |

15 | The QR Algorithm I | |

16 | The QR Algorithm II | Homework 4 due |

17 | Other Eigenvalue Algorithms | |

Midterm Exam | ||

18 | The Classical Iterative Methods | |

19 | The Conjugate Gradients Algorithm I | |

20 | The Conjugate Gradients Algorithm II | |

21 | Sparse Matrix Algorithms | Homework 5 due |

22 | Preconditioning, Incomplete Factorizations | |

23 | Arnoldi / Lanczos Iterations | |

24 | GMRES, Other Krylov Subspace Methods | |

25 | Linear Algebra Software | Homework 6 due |