1. Todays software libraries are reaching the core peak performance (i.e., 90% of peak performance) and thus reaching the limitats of current systems. Fast algorithms deploy new algorithmic strategies, new opportunities, thus reaching new performance and, in practice, stretching the system capabilities. 2. Fast algorithms reduce the number of operations and thus data movements. This means that for very large problems they (may) reduce off-chip communication (dominant factor in multicore systems).