**Lectures:** MWF 8:10-9:00 AM, 105 Stanley

**Prerequisites:** Math 53 and 54 or equivalent

**Required Text:** Numerical Analysis, 9th Edition, by Burden/Faires

**Matlab resources:**

- J. Dorfman,
*Introduction to MATLAB Programming*, Decagon Press, Inc. Available at Krisha Copy Center on University Avenue ($20 + tax). orders@krishnacopy.com. - Course material for Math 98: Introduction to MATLAB programming from 2012.
- Otto and Denier, An Introduction to Programming and Numerical Methods in MATLAB (online)
- Quarteroni and Saleri, Scientific Computing with MATLAB and Octave (online version)
- K. Sayood, Learning programming using MATLAB (online version)
- How to set up a UC Berkeley Library Proxy Server (off-campus access to online books)

**Ways to run matlab:**

- MATLAB will be available during discussion sections in the computer lab B3A Evans
- On computers owned by the university, UC Berkeley Software Central provides MATLAB for free
- The Mathworks provide student editions of MATLAB at a discounted rate
- The free alternative Octave has some limitations (especially for graphics), but is sufficient for all excercises in the class

**Syllabus:** Programming for numerical calculations, round-off
error, approximation and interpolation, numerical quadrature, matrix
computations, and
solution of ordinary differential equations.

- Error anlysis, roundoff error and computer arithmetic, algorithms and convergence (Chapter 1)
- Nonlinear equations: bisection, Newton and secaont methods (Chapter 2)
- Polynomial interpolation and approximation (Chapter 3)
- Numerical differentiation and integration (Chapter 4)
- Initial value problems for ordinary differential equations (Chapter 5)
- Matrix computations: linear systems, matrix factorizations, norms (Chapters 6, 7.1)
- Approximation theory, least squares approximation (8.1-8.3, if time permits)

**Course Material:** I will post handouts and assignments on
bCourses.
Please e-mail me if you do not have access to the bCourses page.

**Piazza forum: **link to 128A

**Grading:**

programming assignments: | 10% | (all scores count) |

homework: | 5% | (lowest score dropped) |

quizzes: | 15% | Sep 17, Oct 1, Oct 15, Oct 29, Nov 12, Dec 3 (in section, lowest score dropped) |

Midterm 1: | 20% | Monday, October 6 (in class) |

Midterm 2: | 20% | Friday, November 14 (in class) |

Final exam: | 30% | Monday, Dec 15, 7-10 PM (location TBA) |

**More Details:** 11 homework assignments, 4 programming
assignments, 6 quizzes. My grade cutoffs are usually around 90 A, 85
A-, 80 B+, 75 B, 70 B-, 65 C+, 60 C, 50 D. Your lowest midterm grade
will be replaced by your grade on the final if you do better on the
final. If you miss a midterm for any reason (illness, family
emergency, didn't study, etc.), the final will count for both.
Homework and programming assignments are due at the beginning of
discussion section. Quizzes will be given in section.
* Late assignments and missed quizzes cannot be made up.*
Collaboration is encouraged in discussing ideas, but you are not
allowed to share code or written solutions of homework. If you are
caught cheating, you will receive an F in the course and be reported
to the university.

**Detailed syllabus** (will be updated as the semester progresses)

Mon | Wed | Fri | |||
---|---|---|---|---|---|

8/29 1.1 |
Overview, example of unstable recurrence, MVT, Rolle's theorem, Extreme value theorem/algorithm | ||||

9/1 | holiday | 9/3 1.1 |
IVT, Taylor's theorem, tricks for bounding R, maximum error, meaning of R | 9/5 1.1,1.2 |
avoiding circular reasoning, error bounds for integrals, floating-point arithmetic |

9/8 1.2, 1.3 |
floating-point arithmetic, absolute and relative error, quadratic formula, two algorithms for evaluating polynomials | 9/10 1.3, 2.1 |
error propagation, rate of convergence,
bisection method HW 1 due |
9/12 1.3, 2.1 |
convergence order examples, ln(1+x)=O(x), convergence rate of bisection method |

9/15 2.2 |
fixed point iteration, discuss HW 2.2.11, proof of fixed point theorem, error estimates | 9/17 2.3 |
Newton's method: graphical derivation, proof of
convergence, why it's so fast HW 2 due, Quiz 1 |
9/19 2.3, 2.4 |
secant method, false position, order of convergence, quadratic convergence in fixed point iteration when g'(p)=0 |

9/22 2.4, 2.5 |
multiple roots, modified Newton, accelerating convergence, Aitken's method/proof, Steffensen's method/proof | 9/24 2.6, 3.1 |
Horner's method, Mueller's method, polynomial
interpolation
HW 3 due, PA 1 due |
9/26 3.1, 3.2 |
polynomial interpolation, Neville's method |

9/29 3.2, 3.3 |
recap of Neville's method, divided differences | 10/1 3.3, 3.4 |
forward/backward differences, Hermite interpolation HW 4 due, Quiz 2 |
10/3 | Review |

10/6 | Midterm 1, (chapters 1-3) | 10/8 4.1 |
numerical differentiation, n+1 point formula, examples | 10/10 4.1-2 |
2nd order derivatives, Richardson extrapolation |

10/13, 4.3 | quadrature: trapezoidal rule, Simpson's rule, degree of accuracy, Newton-Cotes formulas | 10/15, 4.4-5 | composite quadrature, Romberg integration, Euler-Maclaurin formula HW 5 due, Quiz 3 |
10/17, 4.5 | recap/overview of quadrature, Euler-Maclaurin, Romberg integration |

10/20, 4.7 | Gaussian quadrature, Legendre polynomials, Gram-Schmidt procedure, proof that degree of precision = 2n-1 | 10/22, 4.7, 4.6 | recap of Gaussian quadrature, proof that zeros are real and distinct, arbitrary intervals, adaptive quadrature
HW 6 due, PA 2 due |
10/24, 4.8, 4.9 | multiple integrals, improper integrals |

10/27, 5.1 | (guest lecture by Danny Hermes) theory of ODE's, Lipschitz continuity, well-posed problems, examples |
10/29, 5.2 | (guest lecture by Ming Gu) Euler's method, convergence proof HW 7 due, Quiz 4 |
10/31, 5.2, 5.3 | How to compute/bound derivatives of the unknown solution, Taylor methods, local truncation error |

11/3, 5.4 | Runge-Kutta methods, Butcher array/tableau, truncation error of general explicit 2-stage RK scheme | 11/5, 5.6 | (guest lecture by Weihua Liu) multistep methods, derivation of Adams-Bashforth and Adams-Moulton HW 8 due |
11/7, 5.10 |
(guest lecture by Weihua Liu) stability of multistep methods |

11/10, 5.6, 5.10 | truncation error of multistep methods, stability+consistency=convergence | 11/12 | Review HW 9 due, Quiz 5 |
11/14 | Midterm 2 (chapters 4 and 5, only sections in which homework has been assigned) |

11/17, 5.9, 5.11 | higher-order equations and systems, stiff equations, Prothero-Robinson example | 11/19, 5.11, 5.5 | linear stability analysis, adaptive stepsize control PA 3 due |
11/21, 6.1-3 | Gaussian elimination, pivoting, matrix inversion |

11/24, 6.5 | LU factorization, forward/back-substitution, counting flops | 11/26 | no main lecture, yes discussion sections HW 10 due |
11/28 | holiday |

12/1, 6.6 | positive definite matrices, Cholesky factorization, tridiagonal matrices | 12/3, 3.5 | cubic spline interpolation PA 4 due, Quiz 6 (5.9, 5.11, 6.1, 6.2, 6.3, 6.5) |
12/5 | Review |

12/8 | RRR Week. (HW 11 due Monday in my office hours, or arrange with your GSI) | ||||

12/15 | Final Exam, 7-10 PM, 105 Stanley |