## Metric and Probability-Based Recommendations

How many of you like to eat, play, and most importantly, drink? (Hopefully, everyone!)

I helped create an app called LocALL. It gives 50,000 recommendations for short trips to eat, play, and drink and support local businesses in Austin. We can extend our app to cover ~400 metropolitan cities in the US and create 20 million recommendations. (Big data and machine learning!)

At the heart of our solution is math. We used spherical geometry and probability to create simple, fast recommendations.

## Iterative Methods: Part 3

Let’s look at one more way to solve the equation $A\vec{x} = \vec{b}$. We assume that $A \in \mathbb{R}^{n \times n}$ is nonsingular, and define the $k$-th Krylov subspace as follows:

$\mathscr{K}_{k} \,:=\, \mbox{span}\{\vec{b},\,A\vec{b},\,\cdots,\,A^{k - 1}\vec{b}\}$.

Krylov subspace methods are efficient and popular iterative methods for solving large, sparse linear systems. When $A$ is symmetric, positive definite (SPD), i.e.

$\left\{\begin{array}{l} A^{T} = A \\[12pt] \vec{x}^{T}A\vec{x} > 0,\,\,\,\forall\,\vec{x} \neq \vec{0}, \end{array}\right.$

we can use a Krylov subspace method called Conjugate Gradient (CG).

Today, let’s find out how CG works and use it to solve 2D Poisson’s equation.