New Data Storage Formats for Dense Matrices Lead to Variety of High-Performance Algorithms

LAPACK has two different kinds of algorithms for dense symmetric matrices. The full storage data format and the packed storage data format. The full storage data format requires $n^2$ memory locations, and performs quickly. The LAPACK packed storage data format algorithms require minimal storage of size $n(n+1)/2$ but their speed is several times slower than of the LAPACK full storage data format algorithms. Thus, the user's program should perform quickly and require minimum memory. The Algorithms using the new packed storage data formats are very successful. They run with almost the same speed as the LAPACK full storage data format algorithms and only use $n(n+1)/2$ memory locations. Several factorization algorithms for symmetric positive definite and indefinite matrices will be explained and illustrated by performance results. The history of our a few years research work will be presented.