#### CS3414 Afterclass Notes --- 18 June, 2002

**Linear Systems (Chapter 6)**
- Introduction

- The problem
*Ax=b*: find *x*, given
*A* and *b*.

- Context
- This is a
*very* important problem, arising in many
scientific computing applications.
- Typical computational science codes need to solve very
large instances of this problem, and often many
different instances of this problem.
- An important theme in algorithms for
*Ax=b* is
exploiting *structure*, e.g., sparsity, symmetry,
bandedness.
- Another important theme is algorithms that are
*resource-aware*, e.g., algorithms that use the memory
hierarchy efficiently.

- Some facts from linear algebra.

We looked at several equivalent properties that imply that
there is a unique solution. Most important for our purposes is
the idea that *A* is *nonsingular* (invertible) and
that Gaussian Elimination works (with the huge assumption of
infinite precision).