Excerpts from notes:

Numerical methods for solving large-scale linear algebraic problems are necessary to solve various partial/ordinary differential equations by the finite element method, finite difference method, spectral method, and etc. Also in the solution of optimization problems, such methods are essential.

Difficulties in practical computation of large linear systems arise from the following observations:

* usually the computational costs are too expensive;
* there will be possible loss in accuracy with a fixed number of digits computation;
* the methods are not applicable to different problems.

The main questions in numerical methods for linear systems are

* How fast is the numerical method in the sense of operation counts (flops: 1 flop=1 multiplication + 1 addition)?
* What is the accuracy? Can a priori and a posteriori estimates be given?
* What is the coverage of the method?