Linear systems

Linear system solvers are the workhorse of scientific computing, and includes the canonical solve.

E.g. the trivial equation for $x$ with scalars $a$ and $b$:

$ax = b$

$x=b/a$

In contrast: $x^2 = 1$ is nonlinear.

A more complicated example is when you have multiple equations and unknowns:

In general, we will have $n$ unknowns that must be solved simultaneously and a set of $m$ linear equations. The group of such equations is termed a linear system and is written compactly in matrix form:

where

• $\mathbf{A}$ is the coefficient matrix of dimension $m \times n$

• $\mathbf{x}$ is the variable / unknown vector of dimension (length) $n$

• $\mathbf{b}$ is the constant / right-hand side vector of dimension $m$.

In most cases in this course we will be dealing with $n$ equations and $n$ unknowns. In that case, $m=n$ and $\mathbf{A}$ is called $square$. We will discuss the properties of matricies (and by extension, linear systems) in the context of finite precision, direct and iterative methods to solve linear systems, and conclude with a generalization to rectangular matricies.