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Numerical Methods (CS 357) Spring 2017

What	Where
Time/place	Section M: TTh 12:30pm-1:45pm 1320 DCL / Catalog (Andreas)
	Section N: TTh 2:00pm-3:15pm 1320 DCL / Catalog (Edgar)
Class URL	https://bit.ly/cs357-s17
Class recordings	Watch » (Section M)
	Watch » (Section N)
Web forum	Discuss » · Suggestions · Instant message
Calendar	View »

Quizzes

Please find the quizzes under their corresponding lecture in the class calendar.

Take Quiz 29 »

Exams

Please find information on our upcoming exams in the corresponding section of the class calendar. Reserve your time slots in the testing facility as soon as possible--otherwise your preferred times may no longer be available.

Homework

Please see the class calendar for homework deadlines (generally Fridays at 5 pm, except for examlet weeks).

Course Outline

Part 1: Models, Errors, and Numbers
- Objectives
  - Use numpy for more efficient computations
  - Construct examples that show the benefits of contigous memory (numpy arrays)
  - Identify the advantages of numerical libraries
  - Create a numerical experiment that measures the cost of basic array operations
  - Represent a real number in a floating point system
  - Measure the error in rounding numbers in the IEEE-754 floating point standard
  - Compute floating point vaules in different portions of the floating point range
  - Use the IEEE-754 standard to quantify error floating point arithmetic
  - Compute the memory storage of single versus double precision
  - Describe the accuracy of several examples in terms of machine epsilon
  - Determine whether data grows algebraically versus exponentially and at which rate
  - Design a numerical experiment with reproducibility, randomness, consistent timing in mind
  - Compute a random process
  - Use randomness to confirm a numerical hypothesis
  - Approximate a function using a Taylor series approximation
  - Quantify the error in using a Taylor series approximation at several points
- 0. Introduction
- 1. Python, Numpy, and Matplotlib
- 2. Making Models with Polynomials: Taylor Series
- 3. Making Models with Polynomials: Interpolation
- 4. Making Models with Monte Carlo
- 5. Error, Accuracy and Convergence
- 6. Floating Point
Part 2: Arrays--Computing with Many Numbers
- Objectives
  - Take a linear algebra operation and apply the operation in a computational setting
  - Represent a problem description in terms of matrices and vectors
  - Load a collection of data into matrix and vector data structures
  - Compute the solution to a linear system of equations
  - Detail the pieces of an LU factorization of a linear system
  - Use an LU factorization to solve a problem with many right-hand sides
  - Measure the number of digits of accuracy for poorly conditioned problems
  - Compute eigenvalues/eigenvectors for different applications
  - Use the Power Method to find a specific eigenvector
  - Compute singular value decomposition for different applications
  - Use singular values to identify dominant subspaces
  - Represent a linear system as a sparse linear system
  - Represent a graph as a sparse system
  - Compute the cost of a sparse matrix-vector multiply
- 7. Modeling the World with Arrays
- 8. Norms and Errors
- 9. The 'Undo' Button for Linear Operations: LU
- 10. Repeating Linear Operations: Eigenvalues and Steady States
- 11. Eigenvalues: Applications
- 12. Approximate Undo: SVD
- 13. SVD: Applications
  - Solving Funny-Shaped Linear Systems
    - Notes
    - Quiz: Least Squares and Pseudoinverse (Quiz 24)
  - Data Fitting
    - Notes
  - Norms and Condition Numbers
    - Notes
  - Low Rank Approximation
  - Demo: Data Fitting with Least Squares
  - Demo: Image compression
  - Demo: Least Squares using the Reduced SVD
  - Demo: Least Squares using the SVD
  - Demo: Pseudoinverse and Least Squares
  - Demo: Rank-1 approximation
Part 3: Approximation: When the Exact Answer is Out Of Reach
- Objectives
  - Interpolate or fit a piece of data with a given model (or function)
  - Find the derivative of noisy data
  - Compute the intgral of a data set
  - Use least-squares data fit to determine a trend
  - Calculate the the cost of a least-squares solution
  - Measure the error in a least-squares solution
  - Approximate the minimum to an objective function
  - Approximate the minimum to an objective function in multiple dimensions
- Interpolation: A Second Look
- 14. Iteration and Convergence
- 15. Solving One Equation
- 16. Solving Many Equations
- 17. Finding the Best: Optimization in 1D
- 18. Optimization in n dimensions

Class scribbles (click to view)

CAUTION!

These scribbled PDFs are an unedited reflection of what we wrote during class. They need to be viewed in the context of the class discussion that led to them. See the lecture videos for that.

If you would like actual, self-contained class notes, look in the outline above.

These scribbles are provided here to provide a record of our class discussion, to be used in perhaps the following ways:

as a way to cross-check your own notes
to look up a formula that you know was shown in a certain class
to remind yourself of what exactly was covered on a given day

By continuing to read them, you acknowledge that these files are provided as supplementary material on an as-is basis.

Team

Andreas Kloeckner

(Instructor (Section M))

Email: andreask@illinois.edu

Office: 4318 Siebel

Edgar Solomonik

(Instructor (Section N))

Email: solomon2@illinois.edu

Office: 4229 Siebel

Nathaniel Bowman

(TA)

Email: nlbowma2@illinois.edu

Office: 0209 Siebel

Nicholas Christensen

(TA)

Email: njchris2@illinois.edu

Office: 0209 Siebel

David Raju

(TA)

Email: draju2@illinois.edu

Office: 0209 Siebel

Ammar Sakarwala

(TA)

Email: ammarks2@illinois.edu

Office: 0209 Siebel

Bhavesh Shrimali

(TA)

Email: bshrima2@illinois.edu

Office: 0209 Siebel

Yizhi Zhu

(TA)

Email: yzhu44@illinois.edu

Office: 0209 Siebel

Computing

We will be using Python with the libraries numpy, scipy and matplotlib for in-class work and assignments. No other languages are permitted. Python has a very gentle learning curve, so you should feel at home even if you've never done any work in Python.

Virtual Machine Image

While you are free to install Python and Numpy on your own computer to do homework, the only supported way to do so is using the supplied virtual machine image.

Download Virtual Machine »