### Lecture 6: Numerical Methods for Ordinary Differential Equations

#### Summary

This lecture is given by MIT Applied Math Instructor, Dr. Christopher Rackauckas. In this lecture we will discuss the current state of software in differential equations and see how the continued advances in computer science and numerical methods are likely to impact our software in the near future. Issues such as efficiency improvements for stiff and non-stiff differential equations will be addressed from a numerical analysis standpoint but backed with recent benchmarking results. Newer mathematical topics like random ordinary differential equations, jump diffusion equations, and adaptivity for stochastic differential equations will be introduced and the successes and limitations in current automatic software solutions will be discussed. We will close with a discussion on how recent computational advancements have been influencing the software implementations, specifically showing the effects of generic typing over abstract algorithms and implicit parallelism.

- Lecture 6 handout: Modern Differential Equations Solver Software: Where We Are and Where We’re Headed (PDF - 2.4MB) (Courtesy of Christopher Rackauckas. Used with permission.)

#### Further Reading

- A Comparison Between Differential Equation Solver Suites In MATLAB, R, Julia, Python, C, Mathematica, Maple, and Fortran by Christopher Rackauckas.
- Differential Equations package for Julia by Christopher Rackauckas.
- Video of Intro to Solving Differential Equations in Julia by Christopher Rackauckas.

### Lecture 7: The SVD, its Applications, and Condition Numbers

#### Summary

This lecture is given by guest lecturer, Prof. Alan Edelman. In this lecture we discussed SVD, relationship to L_{2} norms and condition numbers, as well as applications (e.g. principal components analysis).

#### Further Reading

- Read “Lectures 4 and 5” in the textbook
*Numerical Linear Algebra*.

### Lecture 8: Linear Regression and the Generalized SVD

#### Summary

This lecture is given by guest lecturer, Prof. Alan Edelman. In this lecture we discussed generalized SVD (GSVD), least-square problems (via QR or SVD) and different viewpoints on linear regression: linear algebra, optimization, statistics, and machine learning.

- Lecture 8 notebook: Many Viewpoints on Linear Regression

#### Further Reading

- Read “Lectures 7 and 11” in the textbook
*Numerical Linear Algebra*. - The GSVD: Where are the Ellipses?, Matrix Trigonometry, and more by Alan Edelman and Yuyang Wang.