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.

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:

\(A x = b\)

where

\(A\) is the coefficient matrix of dimension \(m \times n\)

\(x\) is the variable / unknown vector of dimension (length) \(n\)

\(b\) is the constant / right-hand side vector of dimension \(m\).

In most cases in this course we will be dealing with \(n\) equations and \(n\) unkowns. In that case, \(m=n\) and \(A\) is called \(square\).