Precision

QR Decompositions with Reorthogonalisation

Problem Formulation We already discussed QR decompositions and showed that using the modified formulation of Gram Schmidt significantly improves the accuracy of the results. However, there is still an error of about $10^3 M_\varepsilon$ (where $M_\varepsilon$ is the machine epsilon) when using the modified Gram Schmidt as base algorithm for the orthogonalisation.

Floating Point Accuracy and Precision

Floating point computations on computers may behave differently than one might expect. Every software developer should be aware of these since computed results may be off by orders of magnitude in the worst case.