# Math 625 Numerical Analysis

## Solutions to Problem Set 2

*Friday, May 24 2024*

Here are my solutions to three problems about pagerank and the numerical methods for finding eigenvalues and eigenvectors: problem1.ipynb, problem2.ipynb, and problem3.ipynb.

In January, Scientific American published an article Math explains why your friends are more popular than you. You should read it --- it's an interesting read with a simple explanation. I added a footnote in my solution to problem1 that empirically checked this popularity fact for the Facebook data given in the pagerank problem.

## Problem set 2

*Sunday, April 21 2024*

Here are three problems for you to work on about pagerank and the numerical methods for finding eigenvalues and eigenvectors: ps2.pdf. Our next class is on Thursday, May 2. These problems are due on Tuesday, May 14. I may add or modify these problems depending on what we cover after Spring Break.

## Class notes on the powermethod

*Tuesday, April 8 2024*

Here are the notes I covered in class on Tuesday about the power method for self-adjoint operators on an inner product space: pm2.ipynb. These notes include an exercise at the end.

## Class notes on pagerank

*Monday, April 8 2024*

Here are the notes I covered in class on Thursday: pagerank.ipynb.

## Pagerank

*Thursday, April 4 2024*

Check out Google's pagerank:

## The first problem set

*Thursday, Feb 29 2024*

Here's the first problem set in pdf: ps1.pdf and LaTeX ps1.tex. It's due on Thursday, March 14.

## Lagrange polynomials and Chebyshev points

*Wednesday, Feb 28 2024*

Here's the notebook that I went over in class yesterday. It defines the Lagrange polynomials, states and proves the error formula, and introduces the Chebyshev points: lagrange.ipynb. I added an exercise at the end in which I invite you to analyze the error formula for one of the examples discussed.

## Notebook and homework for the weekend

*Thursday, Feb 15 2024*

Here's the notebook that contains the additional examples of orthogonal projection that I illustrated in class on Tuesday: inner_products2.ipynb.

Also, here are a few problems from an old Numerical Analysis final exam that you can work on over the weekend: poly_approx_quiz.pdf. I haven't covered Lagrange's error formula yet, so you might not be able to do problem 5 yet, but treat the other prolems like a take-home quiz and bring your solutions to the next class, which will be on **Tuesday, February 20.**

## No class on Thursday, Feb 15

*Tuesday, Feb 13 2024*

Just a reminder that the next class will be Tuesday, Feb 20. There is no class on Thursday, Feb 15.

## Orthogonal projection and the minimization problem

*Sunday, Feb 11 2024*

I revised and expanded the jupyter notebook on inner products inner_products.ipynb.

Be aware that you *do not* need to have Python and Jupyter installed on your own computer to run or create Jupyter notebooks. You can use an online Jupyter notebook environment like Google's Colab and do everything from within a web browser. In order to get started with my notebook on inner products, you'll want to

- Create a Colab account. This is free to do if you have a Google account.
- Download my notebook inner_products.ipynb on your computer.
- Log into Colab and select File -> Upload Notebook, select the notebook you just downloaded and get started.

There is more information about Python, Jupyter, Colab, and other resources on the Math 625 syllabus.

## Orthogonal projection and the minimization problem

*Tuesday, Feb 6 2024*

Here are lecture notes on Orthogonal projection and the minimization problem.

If you'd like to read more about the context that I provided in class, check out the Wikipedia pages on Metric Space, Normed Vector Space, and Inner Product Space. You may also want to read the entry on Semi-norm which is what you get when you weaken a norm to allow some nonzero vectors to have norm zero.

## Homework for the weekend

*Thursday, Feb 1 2024*

Do problems 8a and 13 in Section 6b and study example 6.63 in Section 6c of Axler's book Linear Algebra Done Right.

You may want to take a look at this jupyter notebook inner_products.ipynb which is a slightly expanded version of the demonstration I gave in class today.

## Linear algebra references

*January 25, 2024*

I'm going to start with a review of linear algebra. I recommend Sheldon Axler's Linear Algebra Done Right. It's an open access book, so you can obtain an unabridged pdf. There's also an abridged version without the proofs, examples, and exercises that might be a handy reference.

I suggest:

- Chapters 2 and 6 for vector spaces and inner products
- Chapters 3, 5, 7 for linear maps, eigenvectors, and operators on inner product spaces

## Welcome to Math 625

*January 25, 2024*

This page will contain announcements for Math 625. For now, take a look at the syllabus, which contains an overview for the course, has information on prerequisites, references, important dates, and information about computing.